Unit 1
Equations and Functions
Math 1A
Day:_______ Name:________________________________ Date:___________
Math 1A 2
UNIT 1A: EQUATIONS, INEQUALITIES, & FORMULAS STANDARDS MATH 1
# DESCRIPTION OF STANDARD SCORE
1 Create an algebraic expression
5
2 Simplify expressions
5
3 Solving equations
5
4 Determine when you have one solution, no solution, or infinitely many
solutions when solving equations
5
5 Solving equations with fraction, decimals, proportions
5
6 Solving Word problems
5
7 Rewrite literal equations
5
8 Solve & graph inequalities
5
9 Creating Inequalities
5
10
5
8/25/16 Quiz 1 Standards 1-4
Day:_______ Name:________________________________ Date:___________
Math 1A 3
Contents Writing and Simplifying Algebraic Expressions ........................................................................................ 5
Writing Expressions .................................................................................................................................... 6
Simplifying Expressions ............................................................................................................................. 7
Writing and Simplifying Expressions ......................................................................................................... 8
Solving Multi-Step Equations ..................................................................................................................... 9
Examples of One-Step Equations ............................................................................................................. 10
Solving Two-Step Equations .................................................................................................................... 11
Solving Multi-Step Equations ................................................................................................................... 12
Solving Equations with Variables on Both sides ...................................................................................... 13
Solving Multistep Equations ..................................................................................................................... 15
Solving Equations Practice ....................................................................................................................... 16
Multi-step Equations – Special Cases ....................................................................................................... 17
Practice – Multi-step Equations – Special Cases ...................................................................................... 18
Practice - Multi-Step Equations ................................................................................................................ 19
Multi-Step Equations with Fractions, Decimals and Proportions ............................................................. 20
Solving Word Problems with Equations ................................................................................................... 21
Guided Notes - Multi-Step Equations as Word Problems ........................................................................ 23
Practice: Literal Equations ........................................................................................................................ 25
Applications of Volume Formulas ............................................................................................................ 26
Literal Equations - Volume Formulas....................................................................................................... 27
Variable Expressions & Equations REVIEW ........................................................................................... 30
More Practice – Solving Equations........................................................................................................... 32
Solving Inequalities .................................................................................................................................. 33
Solving Word Problems with Inequalities ................................................................................................ 35
Multistep Inequalities................................................................................................................................ 36
Solving Word Problems with Inequalities ................................................................................................ 38
Inequality Application Word Problems .................................................................................................... 40
Inequality Review ..................................................................................................................................... 41
Day:_______ Name:________________________________ Date:___________
Math 1A 4
Recursive Patterns ..................................................................................................................................... 42
Input-Output Machines ............................................................................................................................. 44
Input-Output Warm-Up ............................................................................................................................ 45
Activity: Patterns Lab .............................................................................................................................. 46
Recursive Sequence Warm-Up ................................................................................................................. 51
Recursive Patterns ..................................................................................................................................... 52
Sequencer Exploration Questions ............................................................................................................. 53
Explicit Rules – Input-Output Formulas ................................................................................................... 58
Function Investigation .............................................................................................................................. 59
Testing For Functions Review .................................................................................................................. 62
Functions Classwork ................................................................................................................................. 65
Notes: Function Notation ......................................................................................................................... 68
Function Notation Extra Practice .............................................................................................................. 70
Functions Warm Up .................................................................................................................................. 71
Graphs & Real World Situations .............................................................................................................. 72
Rate of Change Practice ............................................................................................................................ 76
Classwork: Rate of Change ....................................................................................................................... 77
Graphical Stories ....................................................................................................................................... 78
Walking a Fine Line.................................................................................................................................. 79
Day:_______ Name:________________________________ Date:___________
Math 1A 5
Writing and Simplifying Algebraic Expressions
Write each phrase as an algebraic expression. 1. $18 less than the sale price _________________ 2. the quotient of n and 12 _________________ 3. 8 less than 25 multiplied by a number q _________________ 4. 3 more than the difference of 20 and a number m _________________ 5. 5 less than the quotient of a number z and 16 _________________ 6. 8 times the product of 28 and a number g _________________ 7. 10 plus a number s times 5 _________________ 8. 10 less than the quantity j multiplied by 44 _________________
9. 6y + (-13y) 10. -12z + (-9z) 11. -8x + 9x - 13x
12. 15x + 2x – 12x - 13x2 – 15 13. 2p4 + 3p + 12 – 18p4 – p – 7
14. 12m + (-9) – 45m
15. -8 + 8k + 14 – 19k
16. 5(3e + 5) – 25e 17. -12n – 18n + 9(4n + 3)
18. 8(z2 + 3) – 19z2 + 14 19. -6(3m + 2) – 6m + (-13)
Day:_______ Name:________________________________ Date:___________
Math 1A 6
Writing Expressions
Write the following expressions in algebraic form. 1) 9 more than c
2) b minus 4
3) the quotient of z and 9
4) the total of n and 40
5) the sum of 8 and m
6) x divided by 5
7) the difference of h and 7
8) 23 less than p
9) the product of g and 2
10) 77 plus twice v
11) two times r increased by 12
12) 3 times j decreased by 12
Day:_______ Name:________________________________ Date:___________
Math 1A 7
Simplifying Expressions Identify the coefficient and constant(s) in expressions listed below: 1. 8x + 9 – 3x 2. 17 – 2a + 5a - 1 Coefficient(s): Coefficient(s):
Constant(s):
Constant(s):
Steps to Simplifying an Expression:
1. Distribute to get rid of any parenthesis
2. Combine like terms
3. Put terms with variables in abc order and constants at the end.
Simplify the following expressions: 1) 3(4x – 5)
2) -4(x – 2)
3) 7(b – 10)
4) 2(b-3) + 4(2b + 2)
5) 5(-3y + 5)
6) –(7y – 4)
7) -5(-8g – 3) – (5g + 3)
8) 4(2a + b) – 3(3a – 4b)
Day:_______ Name:________________________________ Date:___________
Math 1A 8
Writing and Simplifying Expressions
Key Vocabulary Algebraic Expression Coefficient Constant Distributive Property
Equivalent Expression Integers Like Terms Order of Operations
Simplify Substitute Term Variable
Writing Expressions What are some key words that represent the following operations?
Addition Subtraction
Multiplication Division
Tips to remember
Day:_______ Name:________________________________ Date:___________
Math 1A 9
Solving Multi-Step Equations
An equation states that each side of the equal sign is equivalent.
Ex: 3 = 3 2.5 =5
2 6 + 2 = 8
It is important to always keep each side equivalent.
3 = 3
3 + 5 = 3 + 5
If you add 5 to one side, you must add
5 to the other side.
This is the addition property of equality.
3 = 3
3 − 5 = 3 − 5
If you subtract 5 from one side, you must subtract 5 from the
other side.
This is the subtraction property of equality.
3 = 3
3 ∙ 5 = 3 ∙ 5
If you multiply by 5 on one side, you must multiply by 5 on the
other side.
This is the multiplication property of equality.
3 = 3
3
5=
3
5
If you divide by 5 on one side, you must divide by 5 on the other
side.
This is the division property of equality.
To solve an equation for an unknown variable, we use inverse operations. Inverse operations
are operations that “undo” one another.
● Addition and Subtraction are inverse operations.
● Multiplication and Division are inverse operations.
Your goal is to isolate the variable. This means you want to get the variable alone on one side of the equation.
Day:_______ Name:________________________________ Date:___________
Math 1A 10
Examples of One-Step Equations Solve each equation. State the property used.
1) 𝑥 + 15 = 18
2) −7 + 𝑤 = −2 3) 𝑁 − 6 = −9
Property: Property: Property:
4) 2𝑦 = −10 5) 2
5𝑎 =
8
15 6)
3
10=
𝑐
5
Property:
Property: Property:
Your turn:
a) 𝑚 − 10 = 2 b) −9 = 𝑏 − 5 c) 2
3𝑦 =
1
4
Property: Property: Property:
Day:_______ Name:________________________________ Date:___________
Math 1A 11
Solving Two-Step Equations
1. Use the Addition or Subtraction Property of Equality to get the term with a
variable alone on one side of the equation. 2. Use the Multiplication or Division Property of Equality to write an equivalent
equation in which the variable has a coefficient of 1.
Examples: Solve each equation and state each property used.
1. 10 =𝑚
4+ 2
2. −𝑏 + 6 = −11
You try:
3. 7 = 2𝑦 − 3
4. −𝑥 + 7 = 12
5. 𝑥
9− 15 = 12
6. 6 −𝑦
3= −2
Day:_______ Name:________________________________ Date:___________
Math 1A 12
Solving Multi-Step Equations
1. Use the Distributive Property to eliminate parentheses
2. Clear the equation of fractions by multiplying by the common denominator (optional step)
3. Combine like terms
4. “Undo” addition or subtraction
5. “Undo” multiplication or division
Examples: Solve each equation.
1. 2𝑥 + 𝑥 + 12 = 78 2. 2 – 2 (𝑥 – 4) = 10
You try:
3. −2𝑥 + 5 + 5𝑥 = 14 4. 15 = 9 – 3 (𝑥 – 1)
Day:_______ Name:________________________________ Date:___________
Math 1A 13
Solving Equations with Variables on Both sides 1. Clean up both sides of the equation individually (combine like terms and simplify).
2. Move all variables to one side of the equation.
3. Move everything else to the other side of the equation.
4. Solve for the variable!
Examples: Solve each equation.
1. 𝑛 + 4𝑛 – 5 = 2𝑛 + 12 − 2𝑛 2. 38 − 𝑦 = −3(4𝑦 + 2)
You try:
3. 2(𝑐 – 6) = 9𝑐 + 2 4. 3(2𝑥 + 1) − 𝑥 = 8 − 4𝑥 + 6
State each property used to justify each step.
5. 4(𝑥 − 1) = 5𝑥 + 3 − 2𝑥
4𝑥 − 4 = 5𝑥 + 3 − 2𝑥 ___________________________________
4𝑥 − 4 = 3𝑥 + 3 ___________________________________
-3x -3x
𝑥 − 4 = 3 ___________________________________
+4 +4
𝑥 = 7 ___________________________________
6. Justify each step.
8 − 3(𝑥 − 4) = 2𝑥
8 − 3𝑥 + 12 = 2𝑥 ___________________________________
20 − 3𝑥 = 12𝑥 ___________________________________
+3x +3x
20 = 15𝑥 ___________________________________
15 15
4
3= 𝑥 ___________________________________
Day:_______ Name:________________________________ Date:___________
Math 1A 14
Guided Practice:
Day:_______ Name:________________________________ Date:___________
Math 1A 15
Solving Multistep Equations Directions: Identify the solution for each equation. Show your work on a separate sheet of paper.
1) xx 5146 2) rr 61116 3) vv 141410
4)
152692 xx
5) xxx 39237 6)
xxx 547653
Directions: Each problem has been incorrectly solved for the variable x. Identify which step the mistake was made and complete the problem to correctly solve for x. Use a separate sheet of paper for your work. 7)
5
3)5(
35)4(
475)3(
2473)2(
42873)1(
xstep
xstep
xstep
xxstep
xxstep
8)
0)5(
00)4(
312955)3(
129535)2(
1249935)1(
xstep
xstep
xxstep
xxstep
xxxstep
9)
10)6(
10)5(
330)4(
16314)3(
1612149)2(
16121436)1(
xstep
xstep
xstep
xstep
xxstep
xxxstep
The mistake is in step _________ The correct answer is x = _______
The mistake is in step _________ The correct answer is x = _______
The mistake is in step _________ The correct answer is x = _______
Directions: Using a separate sheet of paper set up an equation to help you solve each word problem below. Check your work by substituting your answer back into your equation.
10) Four more than twice Jason’s age is the same as his age ten years from now. How old is Jason now?
11) Seven increased by the product of three and a value x is the same as the product of 3 and a value x decreased by seven.
12) The sum of two consecutive even numbers is the same as three times the smallest number. What are the two numbers?
Day:_______ Name:________________________________ Date:___________
Math 1A 16
Solving Equations Practice Solve each equation AND CHECK your solution. Show ALL work. You may need to use a separate sheet of paper.
1. 𝑥
2+ 13 = 20
3. −10 = 4 −7𝑥
4
5. −4
9(2𝑥 − 4) = 48
7. −2𝑚 = 16𝑚 − 9
9. 10(−4 + 𝑦) = 2𝑦
11. 𝑥+3
10=
5
2
2. 22 = 18 −1
4𝑥
4. 3
5𝑤 + 9 =
3
4𝑤
6. −4𝑎 − 3 = 6𝑎 + 2
8. 3(4 + 4𝑥) = 12𝑥 + 12 10. 5𝑥 − (6 − 𝑥) = 2(𝑥 − 7)
12. 𝑛+7
6=
𝑛+3
4
Explain where the mistake was made in 13 &14:
13. 1
4𝑥 − 2 = 7
𝑥 − 2 = 28 𝑥 = 30
14. 7(𝑐 − 6) = (4 − 𝑐)(3)
7𝑐 − 6 = 4 − 3𝑐
10𝑐 = 10 𝑐 = 1
Day:_______ Name:________________________________ Date:___________
Math 1A 17
Multi-step Equations – Special Cases So far we have looked at equations where there is exactly one solution. It is possible to have no solutions or infinite solutions to an equation.
No solution would mean that there is no answer to the equation. It is impossible for the equation to be true no matter what value we assign to the variable.
Example: 2x + 3 = 2x + 7 -2x -2x___ 3 = 7
You try: 9x + 3x – 10 = 3(3x + x)
Infinite solutions would mean that any value for the variable would make the equation true. Example: 2x + 3 = 2x + 3 -2x -2x___ 3 = 3 You try: -3 – 8x + 17 = -2(4x – 7)
When the solution is ZERO: Zero can be an answer! Don’t get it confused with no
solution!
Example: 2x + 3 = 3 You try: a + 5 = -5a + 5
__ -3 -3___
2x = 0, x = 0
Can you… 1. Create an equation with an answer of all solutions?
2. Create an equation with an answer of no solution?
That can’t be right! We know that three doesn’t equal seven. It is a false statement to say 3 = 7,
so we say that there can be NO SOLUTION!
When does three equal three? All the time! This means it doesn’t matter what value we substitute for x, the equation will always be true. Try two numbers to verify this is true.
The answer would be ALL SOLUTIONS!
Day:_______ Name:________________________________ Date:___________
Math 1A 18
Practice – Multi-step Equations – Special Cases 1. a + 5 = -5a + 5 2. 6 = 1 – 2n + 5
3. p - 4 = -9 + p 4. 12 = -4(-6x – 3) 5. 4m – 4 = 4m 6. 24a – 22 = -4(1 – 6a) 7. 11 + 3x – 7 = 6x + 5 – 3x 8. 6x + 5 – 2x = 4 + 4x + 1 9. 13 – (2x + 2) = 2(x + 2) + 3x 10. 7x – 4y + 12z + 4 = 5 – 3y + 7x –y + 12z 11. Create an equation that has no solutions. 12. Create an equation with an answer of all solutions. 13. Pick two problems from above and plug-in your answer to check that they were solved correctly. You can do this with any problem to check your work!
Day:_______ Name:________________________________ Date:___________
Math 1A 19
Practice - Multi-Step Equations
Solve each equation.
1. −20 = −4x − 6x
2. 6 = 1 − 2n + 5
3. 8x − 2 = −9 + 7x
4. a + 5 = −5a + 5
5. 4m − 4 = 4m
6. p − 1 = 5p + 3p − 8
7. 5p − 14 = 8p + 4
8. p − 4 = −9 + p
9. −8 = −(x + 4)
10. 12 = −4(−6x − 3)
11. 14 = −(p − 8)
12. −(7 − 4x) = 9
13. −18 − 6k = 6(1 + 3k)
14. 5n + 34 = −2(1 − 7n)
15. 2(4x − 3) − 8 = 4 + 2x
16. 3n − 5 = −8(6 + 5n)
17. −(1 + 7x) − 6(−7 − x) = 36
18. −3(4x + 3) + 4(6x + 1) = 43
19. 24a − 22 = −4(1 − 6a)
20. −5(1 − 5x) + 5(−8x − 2) = −4x − 8x
Day:_______ Name:________________________________ Date:___________
Math 1A 20
Multi-Step Equations with Fractions, Decimals and Proportions Equations with Fractions/Decimals: Steps:
1. Clear parentheses by using the distributive property. 2. If there are fractions or decimals, clear them by multiplying by the lowest
common denominator (lowest decimal place value for decimal numbers). 3. Combine like terms on each side of the equal sign. 4. Add or subtract to isolate the variable 5. Multiply or Divide to isolate the variable
Example 1: 𝟐
𝟕𝒙 +
𝟒
𝟕𝒙 = −
𝟑𝟎
𝟕 Multiply by 7 to clear the fraction. .
2x + 4x = -30 Solve like other equations. Example 2: 28 – 2.2y = 11.6y + 262.6 Multiply by 10 (since lowest decimal is tenths
280 – 22y = 116y + 2626 place). Solve.
Try on your own and check with a partner
1. −𝟏𝟕
𝟐𝟒= −
𝟒
𝟑𝒙 −
𝟕
𝟒+
𝟏
𝟐𝒙 2. 13.7b – 6.5 = -2.3b + 8.3
3. 𝟓
𝟔= −𝒙 −
𝟒
𝟑− 𝟏 4. 27.67x – 8 = 22.56x + 40
Algebraic Proportions: Steps:
1. Cross multiply
2. Distribute
3. Combine like terms
4. Add or subtract to isolate the
variable
5. Multiply or divide to isolate the
variable
Example 1: 𝒙+𝟒
𝟓=
𝒙−𝟐
𝟕
7(x + 4) = 5(x – 2)
Solve like other equations.
Example 2: 𝟏𝟐𝒙−𝟑𝟐
𝟒𝒙= 𝟓
Rewrite as 𝟏𝟐𝒙−𝟑𝟐
𝟒𝒙=
𝟓
𝟏 and Solve.
Try on your own and check with a partner.
1. 𝟐𝒙−𝟐
𝟑𝒙+𝟔=
𝟐
𝟓 2.
𝟓
𝒓−𝟗=
𝟖
𝒓+𝟓
3. 0.07x + 9.95 = 12.47 - .05x 4. 4x + 2.5 = -28.4 – 2.2x
Day:_______ Name:________________________________ Date:___________
Math 1A 21
Solving Word Problems with Equations Consecutive and Non-Consecutive Integers 1) Find two consecutive even integers such that the sum of the larger and twice the smaller is 62.
2) Find three consecutive even integers such that the sum of the smallest and the largest is 36.
3) Find three consecutive odd integers such that the sum of the smallest and 4 times the largest
is 61.
4) Find three consecutive integers such that the sum of twice the smallest and 3 times the largest
is 126.
5) Find four consecutive odd integers who sum is 56.
6) The larger of two numbers is 1 less than 3 times the smaller. Their sum is 63. Find the
numbers.
7) The sum of two numbers is 172. The first is 8 less than 5 times the second. Find the first
number.
8) Find two numbers whose sum is 92, if the first is 4 more than 7 times the second.
9) The sum of three numbers is 61. The second number is 5 times the first, while the third is 2
less than the first. Find the numbers.
10) The sum of three numbers is 84. The second number is twice the first, and the third is 4 more
than the second. Find the numbers.
11) The sum of two numbers is 35. Three times the larger number is the same as 4 times the
smaller number. Find the numbers.
(HINT: Let x = larger number 35 – x = smaller number)
Day:_______ Name:________________________________ Date:___________
Math 1A 22
Perimeter 12) An 84-meter length cable is cut so that one piece is 18 meters longer than the other. Find the
length of each piece.
13) The length of a rectangle is 2 cm less than 7 times the width. The perimeter is 60 cm. Find the
width and length.
14) The first side of a triangle is 7 cm shorter than twice the second side. The third side is 4 cm
longer than the first side. The perimeter is 80 cm. Find the length of each side.
15) The length of a rectangle is 6 cm longer than the width. If the length is increased by 9 cm and
the width by 5 cm, the perimeter will be 160cm. Find the dimensions of the original rectangle.
16) The first side of a triangle is 8 m shorter than the second side. The third side is 4 times as long
as the first side. The perimeter is 26 m. Find the length of each side.
17) A triangular sail has a perimeter of 25 m. Side “a” is 2 m shorter than twice side “b”, and side
“c” is 3 m longer than side “b”. Find the length of each side.
18) The length of a rectangular field is 18 m longer than the width. The field is enclosed with
fencing and divided into two parts with a fence parallel to the shorter sides. If 216 m of fencing
are required, what are the dimensions of the outside rectangle?
Age and Points 19) Matthew is 3 times as old as Jenny. In 7 years, he will be twice as old as she will be then. How
old is each now?
20) Melissa is 24 years younger than Joyce. In 2 years, Joyce will be 3 times as old as Melissa will
be then. How old are they now?
21) In the Championship game, Julius scored 5 points less than Kareem, and Wilt scored 1 point
more than twice as many as Kareem. If Wilt scored 20 points more than Julius, how many
points were scored by each player?
Day:_______ Name:________________________________ Date:___________
Math 1A 23
Guided Notes - Multi-Step Equations as Word Problems Write each sentence as an algebraic equation. 1. Juan’s salary plus $125 is $600. ____s + 125 = 600___ 2. Six times as many visitors is 120 visitors. _________________ 3. Twenty-seven is seven fewer students than last year. _________________ 4. Two and one-half times the amount of interest is $2500. _________________ 5. The number of cats decreased by 17 is 19. _________________ 6. Four times the number of feet is 12 feet. _________________ 7. The price decreased by $4 is $29. _________________ 8. After dividing the money 5 ways, each person got $67. _________________ 9. Three more than 8 times as many trees is 75 trees. _________________ 10. Twice as many points as Bob would be 18 points. _________________ Define the variable, write the equation, and solve. 1. Yesterday Josh sold some boxes of greeting cards. Today he sold seven boxes. If he sold 25
boxes in all, how many did he sell yesterday? Variable: Equation: Solve: 2. After Hoshi spent $27.98 for a sweater, she had $18.76 left. How much money did she have
to begin with? Variable: Equation: Solve: 3. After Simon donated four books to the school library, he had 28 books left. How many books
did Simon have to start with? Variable: Equation: Solve:
Day:_______ Name:________________________________ Date:___________
Math 1A 24
4. One day Reeva baked several dozen muffins. The next day she made 8 dozen more muffins.
If she made 20 dozen muffins in all, how many dozen did she make the first day? Variable: Equation: Solve: 5. Twelve notebooks cost $15.48 in all. What is the price of one notebook? Variable: Equation: Solve: 6. Skylar bought seven books at $12.95 each. How much did Skylar spend? Variable: Equation: Solve: 7. Eugene has five payments left to make on his computer. If each payment is $157.90, how
much does he still owe? Variable: Equation: Solve: 8. Hugo received $100 for his birthday. He then saved $20 per week until he had a total of $460
to buy a printer. Write an equation to show how many weeks it took him to save the money. Variable: Equation: Solve: 9. A health club charges a $50 initial fee plus $2 for each visit. Mary has spent a total of $144 at
the health club this year. Use an equation to find how many visits she has made. Variable: Equation: Solve:
Day:_______ Name:________________________________ Date:___________
Math 1A 25
Practice: Literal Equations
Solve the following equations for the variable listed.
1. -3x + b = 6x; for x
2. y = mx + b; for m
3. 5𝑥+𝑦
𝑎= 2 ; for a
4. P = IRT; for T
5. 12x – 4y = 20; for y
6. A = 𝑏ℎ
2 ; for h
7. s = 𝑢+𝑣
𝑡 ; for v
8. x2 + y2 = z2; for x
1._____________ 2._____________ 3._____________ 4._____________ 5._____________ 6._____________ 7._____________ 8. _____________
Day:_______ Name:________________________________ Date:___________
Math 1A 26
Applications of Volume Formulas
1. Annie has a cylindrical container, but she does not know its radius or height. She does know that
the radius and the height are the same and that the volume of the container is 512π cubic inches.
Find the radius of Annie’s container.
2. A cone with a radius of 6 centimeters and a height of 12 centimeters is filled to capacity with liquid.
Find the minimum height of a cylinder with a 4 centimeter radius that will hold the same amount of
liquid.
3. The volume of a cylinder is 980 in. . The height of the cylinder is 20 in. What is the radius of the cylinder?
4. Picture sand filled in one upside down cone shaped container and it is pouring into another right side up cone shaped container as a timer counts down. The radius of the sand in the upside down cone container is 10mm. The height of the sand in the upside down cone container is 24 mm. You must answer a trivia question before the sand in the timer falls to the bottom. The sand falls at a rate of 50 cubic millimeters per second. How much time do you have to answer the question? Use the formula for the volume of a cone to find the volume of the sand in the timer.
5. The inside of each glass is shaped like a cone. Which glass can hold more liquid? How much more?
Day:_______ Name:________________________________ Date:___________
Math 1A 27
Literal Equations - Volume Formulas Volume of a Cone
Find the volume of each cone. Round all answers to the nearest hundredth.
What would be the volume of a cone with a diameter of 10 in and a height of 50 in?
Formula for volume of a cone:
𝑉 =1
3𝐵ℎ
Where B = area of the base and h is the height of the cone
Day:_______ Name:________________________________ Date:___________
Math 1A 28
Volume of a Pyramid
Example 1: Find the volume of the pyramid.
Formula for volume of a pyramid:
𝑉 =1
3𝐵ℎ
Where B = area of the base and h is the height of the pyramid
Day:_______ Name:________________________________ Date:___________
Math 1A 29
Volume of a Sphere
Find the volume of each sphere. Round all answers to the nearest hundredth.
What would be the volume of a sphere with a diameter of 13 cm?
Formula for volume of a sphere:
𝑉 =4
3𝜋𝑟3
Where r is the radius of the sphere.
5 cm 14
in
2.5 cm
Day:_______ Name:________________________________ Date:___________
Math 1A 30
Variable Expressions & Equations REVIEW Solve:
1. 2
6+
3𝐾
2=
7
12
3. 6 − 17𝑎 + 3 = 4𝑎 + 9
5. 3
484
w
7. 6 2 7 13y y
9. 1
43
p p
11. 4 = −2𝑥 + (−8)
10
2. 3
𝑎+1= −
5
𝑎−3
4. 7 − 6𝑥 − 28 = −3(2𝑥 + 7)
6. 3 5
13 4 6
nn
8. 7 146
p
10. 0 = 0.98b + 0.02b – b
12. 2 1 1
13 2 3
a a a
Answers 1._______________ 2._______________ 3._______________ 4._______________ 5._______________ 6._______________ 7._______________ 8._______________ 9._______________ 10.______________ 11.______________ 12.______________
24 2
4
16
Day:_______ Name:________________________________ Date:___________
Math 1A 31
Simplify.
13. -12n – 18n + 9(4n + 3)
14. -6(3m + 2) – 6m + (-13)
15. 8(z2 + 3) – 19z2 + 14
16. 2p4 + 3p + 12 – 18p4 – p – 7
17. -(4x - 2) - 3(2x + 2)
Write an expression or equation from the given
scenario.
18. 4 less than 2 multiplied by a number x
19. 2 more than 7 times a value m is 67.
20. 7 less than twice a number is the same as the
number increased by 6.
21. A value x raised to the third power is 4 less than
the product of 3 and x.
13. ________________ 14. ________________ 15. ________________ 16. ________________ 17.________________ 18.________________ 19.________________ 20.________________ 21.________________
Day:_______ Name:________________________________ Date:___________
Math 1A 32
More Practice – Solving Equations
Unit 1 – Review Solve each equation below.
1. 2
3+
3𝐾
4=
7
12
2. 18𝑥 − 5 = 3(6𝑥 − 2)
3. 9 + 5𝑎 = 2𝑎 + 9
4. −8𝑥 + 14 = −2(4𝑥 − 7)
5. 2 = −3𝑥−(−4)
8
6. 2 = −3𝑥−(−4)
8
7. 𝑎−2
𝑎+5=
3
8
8. 3y + 2 = 9x - 4
Solve each equation for x.
9. 𝑎
𝑥=
𝑏
𝑐
10. 2xy + z = 5x
Define the variable (if needed) and write an expression. 11. 4 more than twice a number 12. 12 times the quantity of x minus 8 13. The quotient of the sum of y and 3 and 5. Define a variable, write an equation or inequality and find the solution. 14. There are 4 more boys than girls in Spanish class. The class has 38 members. How many boys and girls are there separately? 15. Find three consecutive odd integers whose sum is 45. 16. Ten less than two times a number is the same as the number increased by 6.
Day:_______ Name:________________________________ Date:___________
Math 1A 33
Solving Inequalities
An equation contains an equal sign.
An inequality contains a less than (<, ≤) or greater than (>, ≥) sign.
Inequalities have more than one answer. Graphing is a way of showing all the answers. Examples: Graph the following inequalities on a number line.
1. 𝑥 > 3 2. 𝑥 ≥ 4 3. 𝑥 < −2
4. 𝑥 ≤ 5 5. 1 < 𝑥
A closed circle includes the point (=). An open circle does not include the point. Examples: Solve and graph your solution.
1. 4𝑦 + 3 > 7 2. 3𝑥 ≤ 11𝑥 + 4 3. 5 − 𝑥 < 4
Classwork Solve each inequality. Graph the solution on a number line.
1. 2𝑧 + 7 > 𝑧 + 10 2. 4(𝑘 − 1) > 4
3. 1.5 + 2.1𝑦 < 1.1𝑦 + 4.5 4. ℎ + 2(3ℎ + 4) ≥ 1
5. 𝑟 + 4 > 13 − 2𝑟 6. 6𝑢 − 18 − 4𝑢 < 22
Solving inequalities is similar to
solving equations. It is still important
to maintain a true statement.
Day:_______ Name:________________________________ Date:___________
Math 1A 34
Investigation: Starting with the given true inequality, perform the instructed operation to both sides and check to make sure the inequality is still true.
−4 < 6
Add 5 to both sides. _______________ Is the statement still true? ______
Subtract 3 from both sides _____________ Is the statement still true? ______
Multiply both sides by 2. _______________ Is the statement still true? ______
Divide both sides by 4. _________________ Is the statement still true? ______
Multiply both sides by -3. _____________ Is the statement still true? ______
What can we do to make the statement true? ________________________________
Now, divide both sides by -3. ____________ Is the statement still true? ______
What can we do to make the statement true? _______________________________
Rule: When solving inequalities, if you multiply or divide each side by a ___________ number, you must _________ the inequality symbol.
Day:_______ Name:________________________________ Date:___________
Math 1A 35
Solving Word Problems with Inequalities
1) The low temperatures for the previous two days were 62º and 58º. What would the temperature need to be for the third day such that the average daily temperature is at least 64º.
2) Gabriella is a waitress at the Hampton Grille. In one night she earned at least $75 while working a six-hour shift. If Gabriella earned $31.50 in tips, find all possibilities for the amount she earned in wages per hour.
3) There are 40 children and 12 adults going on a trip to New York City by car. Each car can hold a maximum of 5 people. What is the least number of cars needed for the trip?
4) On her last two math tests, Larisa had scores of 83 and 92. Assuming that Larisa cannot score above a 100 on any given test, determine all possible scores Larisa can score to average at least a 90 on all three tests.
5) Manuel takes a job translating English instruction manuals to Spanish. He will receive $15 per page plus $100 per month. Manuel would like to work for 3 months during the summer and make at least $1,500. Write an inequality to find the minimum number of pages Manuel must translate in order to reach his goal.
Day:_______ Name:________________________________ Date:___________
Math 1A 36
Multistep Inequalities Solve each of the following and graph the solution.
Day:_______ Name:________________________________ Date:___________
Math 1A 37
Day:_______ Name:________________________________ Date:___________
Math 1A 38
Solving Word Problems with Inequalities In this lesson we will gain an idea of how inequalities have meaning in our lives.
1) Carlos goes to the fair where it costs $5 to get in and $0.80 per ride.
a. Write an expression representing the cost of the day after buying x rides.
b. Carlos is allowed to spend at most $25. Write the inequality using the expression you
found in part a.
c. Solve the inequality for x. How many rides can he go on?
2) Kristina bought suckers for her children. She gave Linus half of the ones she bought but
ended up taking 5 away from him by the end of the day due to poor behavior.
a. Write an expression for the amount of x suckers Linus received.
She gave Nicholas one third of the ones she bought and 3 extra by the end of the day due to
good behavior.
b. Write an expression for the amount of x suckers Nicholas received.
Linus had at least as many suckers as Nicholas had at the end of the day.
c. Write an inequality comparing the two expressions you found in part a and b.
d. Solve the inequality for x. What is the minimum number of suckers Kristina bought?
3) Mr. Gleason is opening a new restaurant. He has enough booths to seat up to 40 people. He is
ordering tables to fill up the rest of the seating space. Each table can seat up to 6 people.
a. If t represents the number of tables Mr. Gleason orders, write an expression to show the
total amount of people that can be seated at booths and tables.
b. Write an inequality that could be used to determine, t, the number of tables Mr. Gleason
needs to order so that he has enough seating at booths and tables for at least 125 people.
Explain how you found this inequality.
c. Solve the inequality you found in part b.
4) Annah has an assortment of post-it notes in pink, blue, yellow and green. She has 4 times as
many blue as pink, three times as many yellow as pink, and twice as many green as pink.
Day:_______ Name:________________________________ Date:___________
Math 1A 39
a. Which does she have the least of?
b. Write an expression showing how many blue post-its she has compared to pink.
c. Write an expression showing how many yellow post-its she has compared to pink.
d. Write an expression showing how many green post-its she has compared to pink.
Annah has fewer than 120 post-its
e. Write an inequality using the expressions you found in parts b – d.
f. Solve the inequality. What is the largest number of pink post-it notes she could have?
5) The track team at Hale High School plans to sell t-shirts to raise money for new equipment. At
Ted’s Tees, printing costs are $0.80 per shirt and the cost for each T-shirt is $3.75. The shop
also charges a $125 fee per order for the silk-screen design.
a. Write an expression showing the cost to make n T-shirts.
b. Write an expression showing the income from selling n T-shirts.
In order for the track team to make a profit, the income from selling the T-shirts must be
greater than the cost of making the T-shirts.
c. Write an inequality comparing the two expressions from part a and b.
d. Solve the inequality. How many shirts must they sell to make a profit?
e. The coach has decided that $8.95 is too expensive per shirt. How will lowering the price
of the T-shirt affect the number of shirts you have to sell to make a profit?
Day:_______ Name:________________________________ Date:___________
Math 1A 40
Inequality Application Word Problems Examples: Define a variable and write an inequality for each situation.
1. If the speed limit on the highway is 55 mph, what speed should you go to avoid a ticket?
2. If a help wanted sign advertises a job paying at least $6.15 per hour, how much will you make
if you get the job?
You try: a. A bus can seat at most 48 students.
b. You must be at least 16 years old to get a driver’s license.
c. It is not safe to use a light bulb of more than 60 watts in this light fixture.
d. At least 350 students attended the band concert last night.
e. The Navy’s flying squad, the Blue Angels, makes more than 75 appearances each year.
Examples: Define a variable, write an inequality and solve each problem. 1. Your baseball team has a goal to collect at least 160 blankets for a shelter in a week. Team
members brought 42 blankets on Monday and 65 blankets on Tuesday. How many blankets
must the team donate the rest of the week in order to reach or exceed their goal?
2. The student council votes to buy food for a local food bank. A case of spaghetti sauce cost
$13.75. What is the greatest number of cases the student council can buy if they have $216 to
spend?
3. Ella is starting a tutoring business. She plans to charge $15 per hour. She will have to buy
$25 worth of supplies to start the business. How many hours must she tutor to earn at least
$600?
Day:_______ Name:________________________________ Date:___________
Math 1A 41
Inequality Review Represent each of the following as an algebraic inequality. 1) x is at most 30 ___________________ 2) the sum of 5x and 2x is at least 14 ___________________ 3) the product of x and y is less than or equal to 4 ___________________ 4) 5 less than a number y is under 20 ___________________ 5) Which statement is modeled by 2p + 5 < 11?
a) The sum of 5 and 2 times p is at least 11.
b) Five added to the product of 2 and p is less than 11.
c) Two times p plus 5 is at most 11.
d) The product of 2 and p added to 5 is 11.
6) Which statement can be modeled by x + 3 < 12?
a) Sam has 3 bottles of water. Together, Sam and Dave have at most 12 bottles of water.
b) Jennie sold 3 cookbooks. To earn a prize, Jennie must sell at least 12 cookbooks.
c) Peter has 2 baseball hats. Peter and his brothers have fewer than 12 baseball hats.
d) Kathy swam 3 laps in the pool this week. She must swim more than 12 laps.
7) Graph the following: x 3 or x 2 8) Graph the following: -2 < x < 1 9) Translate the verbal expression into an algebraic inequality, solve and graph your answer:
The sum of twice a number and 5 is at most 3 less than a number. 10) Solve and graph: 3x + 6 > 4x
Day:_______ Name:________________________________ Date:___________
Math 1A 42
Recursive Patterns
Vocabulary:
Recursive Equation:
Sequence:
Domain:
Range:
Now – Next Form of Recursive Equations Examples:
1) 5, 10, 15, 20, . . . Start = _____________
What is the pattern? __________________ each time
Now – Next form: _____________________________
2) Start = _____________
What is the pattern? __________________ each time Now – Next form: _____________________________
You Try! 1) 2, 4, 6, 8,… 2) 3, 6, 12, 24,… 3) 2, -4, -10, -16,… Start: Start: Start: NOW-NEXT form: NOW-NEXT form: NOW-NEXT form: 4) 4, -12, 36, -108,… 5) Start: Start: NOW-NEXT form: NOW-NEXT form: Example: Start = _____________ Now – Next form: _____________________________
n = 1 Perimeter = 3
n = 2 Perimeter = 4
n = 3 Perimeter = 5
n = 4 Perimeter = 6
Term Value
1 -3
2 6
3 -12
4 24
5 -48
Day:_______ Name:________________________________ Date:___________
Math 1A 43
You Try: Start = _____________ Now – Next form: ____________________________
Iterating Functions on the Graphing Calculator
Calculators can quickly iterate functions. Start by entering your initial value, then use the ANS key (found above the negative sign) to write an expression and pressing ENTER repeatedly.
Example: Find the first 5 terms of the sequence for NEXT = 5NOW + 10 when the initial value is 1.
Steps:
Start by entering your initial value 1.
Press ENTER
Replace NOW with ANS
5 2nd ( – ) + 10
Press ENTER repeatedly
Example: List the first six values generated by the recursive routine below. Then write the routine as a NOW-NEXT equation.
-32.1 Enter First six values: ____________________________
Ans + 11.8 Enter, Enter, … Now – Next form: _____________________
You Try! 1) Find the first 5 terms of the sequence for NEXT = –6 • NOW + 12 when the initial value is 2.
First five values: _________________________________________ 2) List the first six values generated by the recursive routine below. Then write the routine as a NOW-
NEXT equation.
54 Enter First six values: _____________________________ Ans - 9 Enter, Enter, … Now – Next form: _____________________
Day:_______ Name:________________________________ Date:___________
Math 1A 44
Input-Output Machines
Our input-output machine takes a number [input], operates on the number, and changes it to a new number [output]. Below are some input-output machines.
Figure out the rule for what the machine did to the input.
Day:_______ Name:________________________________ Date:___________
Math 1A 45
Input-Output Warm-Up For each of the problems below, analyze the inputs and outputs given. Then use the pattern to write the rule. Problem 1
Input Output
10 20
30 60
50 100
125 150
OUT= __________ Problem 2
Input Output
2 7
12 17
25 30
32 37
OUT= ___________ Problem 3
Input Output
100 4
75 3
50 2
25 1
OUT= ____________
Day:_______ Name:________________________________ Date:___________
Math 1A 46
Activity: Patterns Lab A recursive formula tells you how to get the next term in the sequence. But what if you wanted the 100th term? Do you want to do the formula 100 times? In this activity you are looking for a relationship between the term and the term number. Using the given diagrams, list the first three terms, the 10th term and the 100th term. Then write a formula to find the nth term based on the term number n. NOTE: The formula you are writing is not a recursive formula that will be used repeatedly, but instead a formula that
will produce the results in one action. 1. The building with one cube for its tower takes 6 cubes to build. The building with two cubes for its tower takes 7 cubes to build. The building with 3 cubes for its tower takes 8 cubes to build.
2. A kite with 1 cube for its tail takes 10 cubes to build.
A kite with 2 cubes for its tail
takes 11 cubes to build.
Cubes in Tower Cubes in All
1 6
2 7
3
10
100
n
Cubes in Tail Cubes in All
1 10
2 11
3
10
100
n
Day:_______ Name:________________________________ Date:___________
Math 1A 47
3. A tower with one floor takes 3 cubes to build. A tower with 2 floors takes 6 cubes to build.
A tower with 3 floors takes 9 cubes to build.
4. For one car, it takes 5 cubes for a carport.
For two cars, it takes 10 cubes for a carport.
For three cars, it takes 15 cars for a carport.
Floors Cubes in All
1
2
3
10
100
n
Cars Cubes in Carport
1 5
2 10
3 15
10
100
n
Day:_______ Name:________________________________ Date:___________
Math 1A 48
5. A well with one layer takes 8 bricks to build. A well with 2 layers takes 16 bricks to build A well with 3 layers takes 24 bricks to build 6. For 1 cow, it takes 8 cubes to build a fence.
For 2 cows, it takes 10 cubes to build a fence.
For 3 cows, it takes 12 cubes to build a fence.
7. One arch takes 7 cubes to build. Two arches take 11 cubes to build. Three arches take 15 cubes to build
Layers Bricks
1
2
3
10
100
n
Cars Cubes in Carport
1
2
3
10
100
n
Arches Cubes in All
1
2
3
10
100
n
Day:_______ Name:________________________________ Date:___________
Math 1A 49
8. A building with one window takes 8 cubes to build.
A building with 2 windows takes 13 cubes to build. A building with 3 windows takes 18 cubes to build.
9. A chair 2 cubes tall takes 3 cubes to build. A chair 3 cubes tall takes 5 cubes to build. A chair 4 cubes tall takes 7 cubes to build. 10. The building 2 stories tall takes 7 cubes to build. The building 3 stories tall takes 11 cubes to build. The building 4 stories tall takes 15 cubes to build.
Windows Shaded Cubes
1
2
3
10
100
n
Height of Chair Cubes in All
2
3
4
10
100
n
Stories Cubes in All
2
3
4
10
100
n
Day:_______ Name:________________________________ Date:___________
Math 1A 50
11.
The building with towers 3 stories tall takes 11 cubes to build.
The building with towers
4 stories tall takes 13 cubes to build. The building with towers
5 stories tall takes 15 cubes to build.
12.
The first staircase takes 1 cube to build
The second staircase takes 4 cubes to build.
The third staircase takes 9 cubes to build.
Stories in Tower Cubes in All
3
4
5
10
100
n
Staircase Cubes in All
1
2
3
10
100
n
Day:_______ Name:________________________________ Date:___________
Math 1A 51
Recursive Sequence Warm-Up
For each of the following recursive number sequences, find a pattern. Write a NOW-NEXT equation for the pattern. (Don’t forget to include the starting value!) Then write the next term in the sequence. 3, 7, 11, 15, 19, ... 1, 2, 4, 8, 16, ... 5, 8, 11, 14, 17, ... 1, 1, 2, 3, 5, 8, ... 1, 10, 100, 1000, ...
Day:_______ Name:________________________________ Date:___________
Math 1A 52
Recursive Patterns 1) Solve each equation. Show all work!
2) For each of the graphs below, fill in the table of values and write the NOW-NEXT equation for each relationship.
3) One hundred meter sticks are used to outline a rectangle. Write a recursive routine that generates a sequence
of ordered pairs (l, w) that lists all possible rectangles.
4) Match the iterative routine in the first column with the equation in the second column.
1) START = 0.75
NEXT = NOW +2
2) START = -0.75
NEXT = NOW +2
3) START = 0.75
NEXT = NOW – 2
4) START = -0.75
NEXT = NOW – 2
5) START = 2
NEXT = NOW – .75
6) START = -2
NEXT = NOW – .75
7) START = 2
NEXT = NOW + .75
8) START = -2
NEXT = NOW + .75
Day:_______ Name:________________________________ Date:___________
Math 1A 53
Sequencer Exploration Questions
Go to: http://www.shodor.org/interactivate/activities/Sequencer/ Choose numbers for each of the following combinations and enter them in the Sequencer applet. Some of the numbers have been chosen for you. In other cases, the type of number has been indicated (such as “positive” or “negative integer” or “decimal < 1”) but you must choose a particular value to fit that type. Then hit “Calculate Sequence!” For each sequence, record the values you chose, fill in the table, and make a rough sketch of the graph below.
Chosen Values Table Sketch of Graph
Sequence A
Term Value
1
2
3
4
5
Starting Num: (positive)
Multiplier: 1
Add-on: (positive)
Steps: 5
Chosen Values Table Sketch of Graph
Sequence B
Term Value
1
2
3
4
5
Starting Num: (negative)
Multiplier: 1
Add-on: (positive)
Steps: 5
Chosen Values Table Sketch of Graph
Sequence C
Term Value
1
2
3
4
5
Starting Num: (positive)
Multiplier: 1
Add-on: (negative)
Steps: 5
Day:_______ Name:________________________________ Date:___________
Math 1A 54
Chosen Values Table Sketch of Graph
Sequence E
Term Value
1
2
3
4
5
Starting Num: (positive)
Multiplier: (positive Integer)
Add-on:
0
Steps: 5
Chosen Values Table Sketch of Graph
Sequence F
Term Value
1
2
3
4
5
Starting Num: (negative)
Multiplier: (positive Integer)
Add-on:
0
Steps: 5
Chosen Values Table Sketch of Graph
Sequence G
Term Value
1
2
3
4
5
Starting Num: (positive)
Multiplier: (negative Integer)
Add-on:
0
Steps: 5
Chosen Values Table Sketch of Graph
Sequence H
Term Value
1
2
3
4
5
Starting Num: (negative)
Multiplier: (negative Integer)
Add-on:
0
Steps: 5
Day:_______ Name:________________________________ Date:___________
Math 1A 55
Chosen Values Table Sketch of Graph
Sequence I
Term Value
1
2
3
4
5
Starting Num: (positive)
Multiplier: (decimal <1)
Add-on:
0
Steps: 5
Chosen Values Table Sketch of Graph
Sequence J
Term Value
1
2
3
4
5
Starting Num: (negative)
Multiplier: (decimal <1)
Add-on:
0
Steps: 5
Chosen Values Table Sketch of Graph
Sequence K
Term Value
1
2
3
4
5
Starting Num: (positive)
Multiplier: (decimal >1)
Add-on:
0
Steps: 5
Chosen Values Table Sketch of Graph
Sequence L
Term Value
1
2
3
4
5
Starting Num: (negative)
Multiplier: (decimal >1)
Add-on:
0
Steps: 5
Day:_______ Name:________________________________ Date:___________
Math 1A 56
Chosen Values Table Sketch of Graph
Sequence M
Term Value
1
2
3
4
5
Starting Num: (positive)
Multiplier: (negative decimal)
Add-on:
0
Steps: 5
Chosen Values Table Sketch of Graph
Sequence N
Term Value
1
2
3
4
5
Starting Num: (negative)
Multiplier: (negative decimal)
Add-on:
0
Steps: 5
Chosen Values Table Sketch of Graph
Sequence O
Term Value
1
2
3
4
5
Starting Num:
Multiplier:
Add-on:
Steps: 5
Chosen Values Table Sketch of Graph
Sequence P
Term Value
1
2
3
4
5
Starting Num:
Multiplier:
Add-on:
Steps: 5
Day:_______ Name:________________________________ Date:___________
Math 1A 57
Write a summary of what you noticed about using different types of starting numbers, add-ons, and multipliers. The questions below may help you organize your thoughts. Starting Numbers: What effect does a negative starting number have on the sequence? What effect does a positive starting number have on the sequence? Add-ons: What effect does a negative add-on have on the sequence? What effect does a positive add-on have on the sequence? Multipliers: What effect does a positive multiplier have on the sequence? What effect does a negative multiplier have on the sequence? What effect does a decimal multiplier have on the sequence? What effect does a large multiplier have on the sequence? What combinations of starting numbers, add-ons, and/or multipliers made the sequence increasing? What combinations of starting numbers, add-ons, and/or multipliers made the sequence decreasing? What combinations produced a linear graph? What combinations produced a non-linear graph?
Day:_______ Name:________________________________ Date:___________
Math 1A 58
Explicit Rules – Input-Output Formulas Function Machine Model 1. If you input 4 into the function machine, the output is 5. If you input 5 into the function machine, then the output is 7 What is the input-output rule for the function machine?
___________________________________________
Therefore, if the input is 8 the output would be ___________
2. If the input is 1, the output is 4.
If the input is 2, the output is 7.
If the input is 3, the output is 10.
What is the input-output rule for the function machine? _____________
When looking at a sequence, an INPUT-OUTPUT formula connects the term number to the term, rather than connecting the term to the previous term like NOW-NEXT formulas. Examples:
Now-Next: Now-Next: Now-Next:
Input-Output: Input-Output: Input-Output:
Now-Next: Now-Next: Now-Next:
Input-Output: Input-Output: Input-Output:
x y
1 5
2 6
3 7
4 8
x y
1 -2
2 -1
3 0
4 1
x y
1 3
2 6
3 9
4 12
x y
1 3
2 5
3 7
4 9
x y
2 1
3 1.5
4 2
5 2.5
x y
1 1
2 8
3 27
4 64
Day:_______ Name:________________________________ Date:___________
Math 1A 59
Function Investigation
Given the sequence 3, 6, 9, 12, ….
Investigation
Make a table showing the relationship between the buttons on the
“Purple Box” machine and the movies available for rent.
Buttons Movies
A1 Gone with the breeze
Draw a mapping diagram for your table.
List each movie only once in your mapping.
Describe your relation as a set of ordered pairs.
The relationship between the
term number and the value can
be expressed by a table.
The relationship can also be expressed as
a set of ordered pairs.
Or the relationship can be
expressed by a mapping
diagram
{(1, 3), (2, 6), (3, 9), (4,12)}
Day:_______ Name:________________________________ Date:___________
Math 1A 60
Example 1: Tell whether each table represents a function. Explain why or why not.
a.
Example 2: Tell whether each mapping diagram represents a function. Explain why or why not.
a. b. c.
Domain: ______________ Domain: ______________ Domain: ______________
Range: _______________ Range: _______________ Range: _______________
Function? (Yes or No): _____ Function? (Yes or No): _____ Function? (Yes or No): _____
Example 3: Tell whether each set of ordered pairs represents a function. Explain why or why not.
a. {(–1, 3), (–3, –2), (5, 7), (0, –2)} b. {(0, 3), (3, -1), (4, 8), (0, –2)}
Domain: ______________ Domain: ______________
Range: _______________ Range: _______________
Function? (Yes or No): _____ Function? (Yes or No): _____
x y
1 3
2 6
2 9
4 12
5 15
2
-1
0
1
3
4
5
2
-6
-1
8
9
1
0
4
7
1
3
4
b.
Day:_______ Name:________________________________ Date:___________
Math 1A 61
Example 4: List the ordered pairs for the points of the each graph. Is each graph a function?
Ordered Pairs: __________________________ Ordered Pairs: ____________________________
Function? (Yes or No): ______ Function? (Yes or No): ______
Some graphs contain too many points to list. To decide if a graph is a function, use the vertical line test.
If a vertical line touches more than one point on the graph, it is not a function.
Example 5: Use the vertical line test to determine which relationships are functions.
y
x
y
x
y
x
y
x
y
x
Day:_______ Name:________________________________ Date:___________
Math 1A 62
Testing For Functions Review
Tell whether each mapping diagram represents a function. (write “function” or “not a function” on the line below each diagram) 1. 2.
___________________________ ____________________________
Use the equations to find the missing domain and range values.
2. 𝒚 =𝟏
𝟒(𝟐𝒙 + 𝟒) 4. 𝟔𝒙 − 𝟒𝒚 = 𝟏𝟐
Input
x
Output
y
4
-2
2
0
-6
6
-4
Input
x
Output
y
-2
0
-4
3
-6
0
6
1
4
9
-1
1
-2
2
-3
3
1
0
4
7
1
3
4
Day:_______ Name:________________________________ Date:___________
Math 1A 63
Determine if the relation is a function. Explain your answer. 1.{(3, 4), (4, -6), (5, -7), (3, 2), (-2, 5)}
2. {(-4, 6), (-3, 2), (1, 0), (7, 6), (8, 2)}
3. {(-3, 4), (-2, 5), (0, 0), (-2, 5), (4, 8)}
Domain: Domain: Domain: Range: Range: Range:
4. 5. 6.
7. 8. 9.
Day:_______ Name:________________________________ Date:___________
Math 1A 64
10. Fill in the tables below for each rule.
a)
Domain: Range: What is the 8th term?
11. Write the INPUT-OUTPUT rule and fill in the empty boxes.
a)
Rule: Rule:
12. Write the NOW-NEXT RULE and the INPUT-OUTPUT rule for each table.
a) b)
NOW- NEXT RULE: NOW-NEXT RULE:
INPUT-OUTPUT rule: INPUT-OUTPUT rule:
x y
–2
–1
0
1
2
x y
–2 –7
–1
0 –1
1
2 5
x y
–1
2 6
5
8 12
9
x y
1 5
2 10
3 15
4 20
5 25
x y
1 7
2 8
3 9
4 10
5 11
OUTPUT = INPUT * 3 +
1
NEXT = NOW *3 – 5
b)
Start = 5
Day:_______ Name:________________________________ Date:___________
Math 1A 65
Functions Classwork
1. What is the domain of the function represented by the list of ordered pairs?
{(0, 2), (–3, 6), (4, 8), (2, 5), (–2, 4)}
A. {2, 4, 5, 6, 8} B. {–3, –2, 0, 2, 4} C. {0, 4, 6, 8} D. {all real numbers} 2. Determine the domain and range of the function y = f(x) represented in the graph.
3. Find the range of f(x) = –3x – 3 for the domain {–2, 1, 2, 5}. A. {0, 3, –6, –18} B. {–3, –6, –9, –18} C. {3, –6, –9} D. {3, –6, –9, –18} 4. Give the domain and range of the relation.
A. Domain: 1 x 3 Range: 4 y 5
[–1 , 3] [–4 , 5]
B. Domain: 0 x 3 Range: 3 y 5
[0 , 3] [–3 , 5]
C. Domain:4 y 5 Range: 1 x 3
[–4 , 5] [–1 , 3]
D. Domain: 3 y 5 Range: 0 x 3
[–3 , 5] [0 , 3]
A. D: 1 x 6 R: 1 y 7
[1 , 6) (1 , 7]
B. D: 2 x 6 R: 4 y 6
[2 , 6] [4 , 6]
C. D: 1 x 6 R: 1 y 7
(1 , 6] [1 , 7]
D. D: 1 x 7 R: 1 y 6
(1 , 7] [1 , 6]
Day:_______ Name:________________________________ Date:___________
Math 1A 66
5. Give the domain and range of the relation.
X Y
3 7
10 21
0 0
–2 –3
6. Determine if it is a Function or Not. If a function, state domain and range.
{(–3, 5), (–2, 0), (0, –4), (2, 0), (4, 12)}
A. Function; Domain: {–3, –2, 0, 5, 12} ; Range: {5, 0, –4, –3} B. Function; Domain: {5, 0, –4, 12} ; Range: {–3, –2, 0, 2, 4} C. Not a function D. Function; Domain: {–3, –2, 0, 2, 4} ; Range: {5, 0, –4, 12} 7. Give the domain and range of the relation.
A. Domain: {–3, 0, 7, 21} Range: {–2, 0, 3, 10}
B. Domain: {–2, 0, 3, 10} Range: {–3, 0, 7, 21}
C. Domain: {–2, 3, 10} Range: {–3, 7, 21}
D. Domain: {3, 10, –2, 7, 21, –3} Range: {0}
A. Domain: 2 x 4 Range: 3 y 2
[–2 , 4) [–3 , 2)
B. Domain:3 x 2 Range: 2 y 4
[–3 , 2) [–2 , 4)
C. Domain: 3 x 2 Range: 3 y 6
[–3 , 2) [–3 , 6]
D. Domain:3 x 2 Range: 0 y 4
[–3 , 2) [0 , 4)
Day:_______ Name:________________________________ Date:___________
Math 1A 67
8. Give the domain and range of the relation.
9. This graph shows the height and distance traveled by a tennis ball.
10. Determine if it is a Function or Not. If a function, state domain and range. X Y
A. D: 3 x 4 R: 0 y 5
[–3 , 4] [0 , –5]
B. D: 3 x 4 R: y = –5
[–3 , 4] [–5 , –5]
C. D: 4 x 5 R: 6 y 0
[–4 , 5] [–6 , 0]
D. D: x = –5; R: 3 y 4
[–5 , –5] [–3 , 4]
How far does the ball travel before it is a height of 2 ft. above the ground?
A. 3 feet
B. 6 feet
C. 7 feet
D. 9 feet
Alice
Brad
Carl
snake
cat
dog
A. Function ; domain: {Alice, Brad, Carl} ; range: {snake, cat, dog}
B. Function ; domain: {snake, cat, dog} ; range: {Alice, Brad, Carl}
C. Not a function
D. Function ; domain: {Alice, dog, Brad} ; range: {Carl, cat, snake}
Day:_______ Name:________________________________ Date:___________
Math 1A 68
Notes: Function Notation
The equation y = 9 – 4x represents a function.
You can use the letter f to name this function and then use function notation to express it. Just replace y with f(x). (Note: In function notation, the parentheses do not mean multiplication.)
You read f(x) as “f of x,” which means “the output value of the function f for the input value x.”
Example: Find f(2). What output do you get when you input 2? f(2) = 9 – 4(2)
f(2) = 1
Evaluating Functions Using Function Notation
Ex. 1: Given 𝒇(𝒙) = 𝟕𝒙 − 𝟏, find 𝒇(−𝟐). Ex. 2: Given 𝒈(𝒙) = 𝒙𝟐 − 𝟒, 𝒇𝒊𝒏𝒅 𝒈(−𝟓).
Ex. 3: Given 𝒉(𝒙) = 𝟓𝒙 − 𝟏, find x if 𝒉(𝒙) = 𝟗 Ex. 4: Given 𝒇(𝒙) = −𝒙 + 𝟐, find x if 𝒇(𝒙) = 𝟔
When you input 2 into function f the output is 1.
Day:_______ Name:________________________________ Date:___________
Math 1A 69
Let’s Practice!
Evaluate the following expressions given the functions below:
a. g(10) = b. f(3) = c. h(–2) =
d. j(7/4) = e. h(a)
f. Find x if g(x) = 16 g. Find x if h(x) = –2 h. Find x if f(x) = 23
Not all functions are expressed as equations. Here is a graph of a function g. The equation is not given, but you can still use function notation to express the outputs for various inputs.
Examples:
1. g(0) =____ 2. g(4) =____ 3. g(6) =____
4. Can you find x-values for which g(x) = 3? ______
5. f(x)=6, what is x? _______
6. f(x)=0, what is x? _______
7. What is the domain of the function? _________ 8. Range? ________
Day:_______ Name:________________________________ Date:___________
Math 1A 70
You Try: Use the graph of y = f(x) at the right to answer each question.
a) f(4) = b) f(6) =
c) For what x value(s) does f(x) = 2?
d) For what x value(s) does f(x) = 1?
e) How many x-values make the statement f(x) = 0.5 true?
f) For what x-values is f(x) greater than 2?
g) What are the domain and range shown on the graph?
Function Notation Extra Practice Look at the graph below. Use it to answer questions 7 - 9
7) What is the domain?
8) What is the range?
9) Use the graph to find each function value in the table. Then do the indicated operations. Show all work to the right.
10) The height of a plant can be modeled by the function h(x) = 2x + 3, where x is the number of weeks. Find h(x) for each given value of x.
a. h(10) b. h(2) c. x when h(x) = 15 d. f(0)
e. Does your answer to part d make sense? Explain.
Notation Value
𝑓(3)
𝑓(5) + 𝑓(3)
𝑓(5) ∙ 𝑓(1)
𝑓(3) ÷ 𝑓(1)
𝑓(5) − 𝑓(0)
y = f(x)
Day:_______ Name:________________________________ Date:___________
Math 1A 71
Functions Warm Up
Express each of the following relations as a table and a graph. Then identify the domain and range.
1. {(–2,2), (–1,5), (0,1), (3, –6), (5, 8)} 2. {(–5,4), (–3,2), (0,1), (5, –2), (–3, –2)}
Domain:
Domain: Domain:
Range: Range:
Is this relation a function? Yes or No Is this relation a function? Yes or No
x y
x y
Day:_______ Name:________________________________ Date:___________
Math 1A 72
Graphs & Real World Situations
In this lesson you will
● describe graphs using the words increasing, decreasing, linear, and nonlinear
● match graphs with descriptions of real-world situations
● learn about continuous and discrete functions
● use intervals of the domain to help you describe a function’s behavior
1. This graph shows the relationship between time and the depth of water in a leaky swimming pool.
a. What is the initial depth of the water?
b.For what time interval(s) is the water level decreasing?
c.What accounts for the decrease(s)?
d.For what time interval(s) is the water level increasing?
e.What accounts for the increase(s)?
f.Is the pool ever empty? How can you tell?
g.What is the dependent variable?
h.What is the independent variable?
2 4 6 8 10 12 14
16
Time (hrs)
Dep
th (
ft)
1 2
3
4
Day:_______ Name:________________________________ Date:___________
Math 1A 73
2. This graph shows the volume of air in a balloon as it changes over time.
a.What is the independent variable? How is it
measured?
b.What is the dependent variable? How is it
measured?
c.For what time intervals is the volume increasing?
What accounts for the increases?
d.For what time intervals is the volume decreasing?
What accounts for the decreases?
e. Domain:
f. Range:
Day:_______ Name:________________________________ Date:___________
Math 1A 74
3. Given the graph of the distance from a Ranger’s station (in miles) while hiking in the Smokies.
What is the dependent variable in this graph? ________________
Using inequality notation, list the domain, range and when the function is increasing, decreasing, and constant:
Domain: _________________ Range:_________________ Increasing: _______________
Decreasing: ________________
Constant: _______________
What is the distance at 11 AM?
_____________________
Approximately how long will it take a
ranger m to be 8 miles away?
__________________
Functions that have smooth graphs, with no breaks in the domain or range, are called continuous functions.
Functions that are not continuous often involve quantities—such as people, cars, or stories of a building—that
are counted or measured in whole numbers. Such functions are called discrete functions. Below are some
examples of discrete functions.
3. Describe each graph as increasing or decreasing at a constant or changing rate
Graph 4 Graph 5
Graph 1 Graph 2 Graph 3 y
x
y
x
y
x
y
x
y
x
REMEMBER:
DOTS= DISCRETE
CONNECTED=CONTINUOUS
Day:_______ Name:________________________________ Date:___________
Math 1A 75
Matching-Up Investigation Read the description of each situation below. Identify the independent and dependent variables. Then decide which of the graphs above match the situation.
A. White Tiger Population
A small group of endangered white tigers are brought to a special reserve. The group of tigers reproduces slowly at first, and then as more and more tigers mature, the population grows more quickly.
Independent Variable:
Dependent Variable:
Matching Graph:
B. Temperature of Hot Tea Grandma pours a cup of hot tea into a tea cup. The temperature at first is very hot, but cools off quickly as the cup sits on the table. As the temperature of the tea approaches room temperature, it cools off more slowly. Independent Variable:
Dependent Variable:
Matching Graph:
C. Number of Daylight Hours over a Year’s Time
In January, the beginning of the year, we are in the middle of winter and the number of daylight hours is at its lowest point. Then the number of daylight hours increases slowly at first through the rest of winter and early spring. As summer approaches, the number of daylight hours increases more quickly, then levels off and reaches a maximum value, then decreases quickly, and then decreases more slowly into fall and early winter. Independent Variable:
Dependent Variable:
Matching Graph:
D. Height of a Person Above Ground Who is Riding a Ferris Wheel When a girl gets on a Ferris wheel, she is 10 feet above ground. As the Ferris wheel turns, she gets higher and higher until she reaches the top. Then she starts to descend until she reaches the bottom and starts going up again. Independent Variable:
Dependent Variable:
Matching Graph:
Day:_______ Name:________________________________ Date:___________
Math 1A 76
Rate of Change Practice
The diagram at the right shows the side view of a ski lift.
1. What is the rate of change from A to B?
2. What is the rate of change from B to C?
3. What is the rate of change from C to D?
4. Which section is the steepest? Explain.
The diagram at the right shows the altitude of an airplane upon landing.
What is the independent variable?
What is the dependent variable?
Find the rate of change that occurred between 60 sec. and 180 sec.
Day:_______ Name:________________________________ Date:___________
Math 1A 77
Classwork: Rate of Change 1. Find the rate of change for each leg of
the journey. Include units.
Section1:
Section2:
Section3:
Section4:
2. Every year on her birthday, Rosalita’s mom measured her height to see how much she had grown
in the past year. The table below shows Rosalita’s height in inches between the ages of 6 and 15.
Age (yrs) 6 7 8 9 10 11 12 13 14 15
Height
(in) 42.5 46.25 49 51.75 53.5 55.5 58.25 60.75 62 62.5
a) What is the average rate of change in her height between age 6 and age 9?
b) What is the average rate of change in her height between age 9 and age 12?
c) What is the average rate of change in her height between age 12 and age 15?
d) During which of these age intervals was Rosalita growing the fastest? Explain.
3. In the Mojave Desert in California, temperatures can drop quickly from daytime to nighttime.
Suppose the temperature drops from 100°F at 2:00 pm to 68°F at 5 am. Find the average rate of
change in temperature for this time period.
Day:_______ Name:________________________________ Date:___________
Math 1A 78
Graphical Stories
Below the following graphs are three stories about walking from your locker to your class. Two of the stories below correspond to two of the graphs. Match the stories with a graph. Draw your own graph to matches the third story.
1. I started to walk to class, but I realized I had forgotten my notebook, so I went back to my locker and then I went quickly at a constant rate to class.
2. I was rushing to get to class when I realized I wasn’t really late, so I slowed down a
bit. 3. I started walking at a steady, slow, constant rate to my class, and then, realizing I
was late, I ran the rest of the way at a steady, faster rate.
Day:_______ Name:________________________________ Date:___________
Math 1A 79
Walking a Fine Line
The modeling of time-distance relationships is a very useful application of algebra. In this investigation, you
will explore time-distance relationships by considering various walking scenarios. You will learn how the
starting position, speed, direction, and final position of a walker influence the graph and the equation of a time-
distance relationship. We will be using motion detectors to measure distance walked over time. Let’s consider
the two graphs below:
Steve’s Walk Jamal’s Walk
Who started further away from the motion detector? How can you tell?
Who is walking faster? How can you tell?
Fill in the table below.
Steve Jamal
Starting Position
Total Time
Speed
Direction
Final Position
0 1 2 3 4 5 6 7 8
9
Time (sec)
0 1 2 3 4 5 6 7 8
9
Time (sec)
Dis
tance
(m
)
9
8
7
6
5
4
3
2
1
0
Dis
tance
(m
)
9
8
7
6
5
4
3
2
1
0
Day:_______ Name:________________________________ Date:___________
Math 1A 80
Connect your CBR2 to your TI-84 calculator.
Below you will find a set of walking instructions. Sketch the graph produced by each one. Take turns being the
walker!
A: Stand 1 meter from the motion detector. Walk
away at a steady pace.
B: Stand 4 meters from the motion detector. Walk
towards the motion detector at a steady pace.
C: Stand 2 meters from the motion detector. Stand
still for 2 seconds, then walk away at a steady pace.
D: Stand 1.5 meters from the motion detector.
Walk away slowly at first, then speed up.
Try to replicate the shape of each of the graphs below. Take turns being the walker! For each graph,
write a set of walking instructions. Tell where the walk begins, how fast the person walks (in m/s), and
whether the person walks toward or away from the motion detector.
Elaine Tasha Hector
Day:_______ Name:________________________________ Date:___________
Math 1A 81
Without using the motion detector, draw the graph of a 6-second walk based on each set of walking
instructions or data. 1) Start at the 2.5-meter mark and stand still.
2) Start at the 3-meter mark and walk toward the sensor at a constant rate of 0.4 meter per second.
3)
Apply the Math: Let’s see if you’ve got it!
The time-distance graph shows Carol walking at a steady rate. Her friend Bobby used a motion detector
to measure her distance over time. Answer the questions about the graph. 1) According to the graph, how much time did Carol spend walking?
2) Was Carol walking toward or away from the motion detector?
Explain your thinking.
3) Approximately how far away from the motion detector
was she when she started walking?
4) If you know Carol is 2.9 m away from the motion detector after
4 s, how fast was she walking?
5) If the equipment will measure distances only up to 6 m, how many more seconds of data can be
collected if Carol continues walking at the same rate?
6) Looking only at the graph, how do you know that Carol was neither speeding up nor slowing down
during her walk?
Time (s) 0 1 2 3 4 5 6
Distance
(m)
0.8 1.0 1.2 1.4 1.6 1.8 2.0
0 1 2 3 4 5 6 7 8
9
Time (sec)
Dis
tance
(m
) 9
8
7
6
5
4
3
2
1
0
Day:_______ Name:________________________________ Date:___________
Math 1A 82
Follow up problems:
Graph a walk for the following set of instructions: “Start at the 0.5-meter mark and walk away from the
motion detector at a steady pace of 0.25 meter per second for 5 seconds.”
Write a set of walking instructions based on the table data and sketch a graph of the walk.
Write a set of walking instructions for this graph:
At what rate in ft/s would you walk so that you were moving at a constant speed of 1 mph?
Describe how the rate of walking affects the graph of each situation: a) The graph of a person walking toward a motion detector at a constant rate.
b) The graph of a person standing still.
c) The graph of a person walking slowly away from the motion detector.
Describe how you would tell someone to walk the line y = x, where x is measured in seconds and y is
measured in feet. Describe how to walk the same line where x is measured in seconds and y is measured
in meters. Which represents a faster rate? Explain.
For each situation, determine if it is possible to collect such walking data and either describe how to
collect it or explain why it is not possible.
Time (s) 0 1 2 3 4 5 6
Distance
(m)
4.0 3.6 3.2 2.8 2.4 2.0 1.6
Day:_______ Name:________________________________ Date:___________
Math 1A 83
Samantha’s walk was recorded by a motion detector. A graph of her walk and a few data points are
shown below. a) Write an equation in the form 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑓𝑟𝑜𝑚 𝑚𝑜𝑡𝑖𝑜𝑛 𝑑𝑒𝑡𝑒𝑐𝑡𝑜𝑟 = 𝑠𝑡𝑎𝑟𝑡 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 + 𝑐ℎ𝑎𝑛𝑔𝑒 to
model this walk.
b) If she continues to walk at a constant rate, at what time would she pass the motion detector?
Time
(s)
Distance
(m)
0 3.5
2 3
6 2
Match the walking instructions with their graph sketches.
i. ii. iii.
a) The walker stands still.
b) The walker takes a few steps towards the motion detector, then walks away.
c) The walker steps away from the motion detector, stops, then continues more slowly in the same
direction.
Andrei and his younger brother are having a race. Because the younger brother can’t run as fast, Andrei
lets him start out 5 m ahead. Andrei runs at a speed of 7.7 m/s. His younger brother runs at 6.5 m/s.
The total length of the race is 50 m. a) Write an equation to find how long it will take Andrei to finish the race. Solve the equation to find the
time.
b) Write an equation to find how long it will take Andrei’s younger brother to finish the race. Solve the
equation to find the time.
c) Who wins the race? How far ahead was the winner when he crossed the finish line?
d
t
d
t
d
t
