Unit 1 Notes / Secondary 2 Honors

Day 1: Review Linear Equations … Graphing and Solving

LINEAR EQUATIONS:

Slopes of Lines:

1. slope (m) = 2. Horizontal slope = Vertical slope=

Slope Formula: Point-Slope Form of a Line: Slope-Intercept Form of a line:

Write the equation of the line that passes through the given points. Leave answers in Point-Slope form.

3. (4, 2) and (2, -1) 4. (3, -1) and (2, 5)

Horizontal and Vertical Lines:

Write the equation of the line that passes through the given points.

5. (-2, 3) and (4, 3) 6. (-2, 3) and (-2, 8)

x-intercept: y-intercept:

7. Find the x-intercept and y-intercept for: 4x – 3y = 6

x-int: y-int:

x

y

x

y

Solving Linear Equations:

Solve the following equations:

8. 5𝑝 − 14 = 8𝑝 + 4

9. 5𝑛 + 34 = −2(1 − 7𝑛)

10. −3(4𝑥 + 3) + 4(6𝑥 + 1) = 43

Day 2: Review GCF / Simplifying radicals

Greatest Common Factoring:

1. 312 6x x 2. 2 2 420 15ab c a bc 3. 4 9 8 3 8 3 2 8 39 15 3x y z x y z x y z

Simplify the radicals – assume all variables are positive. Pay close attention to the type of root!

4. 72 5. 318x 6. √75𝑥3𝑦5 7. 3 27

8. 7 53 16x y 9. √32𝑥2𝑦94 10. 2 6 3 24 11. √5𝑥𝑦 ∙ √10𝑥3𝑦3

12. 12 27 18 13. 3√8 − 5√50

Are like terms needed to multiply roots? Are like terms needed to add/subtract roots?

Change from root form to fractional exponent form.

14. y 15. 3 5 16. 5 x 17. 74 p

Change from fractional exponent form to root form.

18. 1

312 19. 2

57 20. 2

3c

x

y

Day 3: Review Exponential Functions / Systems of equations

Exponentials

Definition: An exponential function has the form xy ab where 0a and the base b is a positive

number other than 1.

If a > 0 and b > 1 then the function is an exponential growth function

If a > 0 and 0 < b < 1 then the function is an exponential decay function

Are the following functions exponential?

1. 1

23

x

y

2. 2 34y x y 3. 3.5x

y 4. 3.5xy

Graph the following exponential equations by making a table of values using x values of -2, -1, 0, 1 and 2. Plot at

least 3 points, draw the asymptote and give the equation of the asymptote, and give the domain and range.

Remember arrows where appropriate!

5. 3xy 6.

1

3

x

y

Domain: Range: Domain: Range:

Asymptote: Asymptote:

7. Using the graph from #5, investigate the graph of 4

3 1x

y

.

Asymptote equation: Domain: Range:

Describe how this graph is different from the graph in #5.

x

y

x

y

x

y

8. Describe how the graph of 𝑦 = 3𝑥−2 − 5 would be different from the graph of 𝑦 = 3𝑥.

9. Graph 𝑦 = 4𝑥+1 − 2

SYSTEMS OF EQUATIONS:

A system is two or more equations that are solved together. The solution to a system is the point(s) of intersection between

the graphs.

10. Solve by substitution. x + 3y = 6

2𝑥 − 3𝑦 = 3

11. Solve by elimination: 2𝑥 + 𝑦 = 4

−3𝑥 − 2𝑦 = −7

12. Solve using either method: 𝑥 − 2𝑦 = 9

𝑦 = 𝑥+92

“Quick” graphs of exponentials:

Graphs start with:

• (0, 1)

• (1, b)

• x-axis asymptote

Then shift graph according to “extra” numbers.

Day 4: - Midpoint / Distance / Translations

Ms. Lopez is planning a treasure hunt for her kindergarten students. She drew a model of the playground on a

coordinate plane as shown. She used this model to decide where to place items for the treasure hunt, and to

determine how to write the treasure hunt instructions. Each grid square represents one square yard on the

playground.

1. Ms. Lopez wants to place some beads in the grass halfway between the merry-go-round and the slide. a. Determine the distance between the merry-go-round and

the slide. Explain your method used.

b. How far should the beads be placed from the merry-go-round and the slide?

c. Write the coordinates for the location exactly half-way between the merry-go-round and the slide. Plot this point.

d. How do the x- and y-coordinates of the point representing the beads compare to the x and y coordinates of the points representing the slide and the merry-go-round? Explain in a complete sentence.

2. Ms. Lopez wants to place some buttons in the grass halfway between the swings and the merry-go-round.

Determine the midpoint between the swings and the merry-go-round. Show your work!

The MIDPOINT FORMULA:

(𝑥1 + 𝑥2

2,𝑦1 + 𝑦2

2)

The DISTANCE FORMULA :

2 2

2 1 2 1( ) ( )d x x y y

3. Find the midpoint and the distance for the following points. Show complete work!

( -3, 1) and (9, -7) Midpoint: Distance:

TRANSLATION: a rigid motion that “slides” each point of a figure the same distance and direction. The original

figure is called the pre-image and the new figure is called the image.

Translate each given line segment or angle on the coordinate plane as described.

4. Translate 𝑀𝑁̅̅ ̅̅ ̅ 3 units up 5. Translate ∠𝐷𝐸𝐹 12 units down. 6. Translate ∠𝑁𝑃𝑄 8 units to the

and 10 units to the right. left and 6 units down.

Day 5: Inductive & Deductive Reasoning / Logic Statements

Vocabulary:

Deductive Reasoning: ___________________________________________________________

Inductive Reasoning: ____________________________________________________________

Example:

1. Emma considered the following statements.

• 42 = 4 𝑥 4

• Nine cubed is equal to nine times nine times nine.

• 10 to the fourth power is equal to four factors of 10 multiplied together.

She concluded that raising a number to a power is the same as multiplying the number as many times as indicated by the

exponent.

Rickey read that raising a number to a power is the same as multiplying that number many times as indicated by the

exponent. He had to determine seven to the fourth power using a calculator. So, he entered 7 × 7 × 7 × 7.

Compare Emma’s reasoning to Rickey’s reasoning.

Decide whether inductive or deductive reasoning is used to reach the conclusion. Explain your reasoning. 2. Liz knows that Jason is older than Seth. She also knows that Seth is older than Makaela. Liz reasons that Jason is older than Makaela.

3. Jose is told that all garter snakes are not venomous. He sees a garter snake in his backyard and concludes that it is not venomous. 4. The rule at your school is that you must attend all of your classes in order to participate in sports after school. You played in a soccer game after school on Monday. Therefore, you went to all of your classes on Monday.

5. For the past 5 years, you have fed your neighbor’s dog on July 4th while your neighbors were out of town. You conclude that you will be asked to feed their dog on July 4th next year. 6. Caitlyn has been told that every taxi in New York City is yellow. When she sees a red car in New York City, she concludes that it cannot be a taxi.

Vocabulary:

Counterexample: a specific example that shows that a general statement is not true.

**To show a statement is false you can provide a counterexample.

Show the conjecture is FALSE by providing a counterexample.

7. The sum of two numbers is always greater than the larger number.

8. All prime numbers are odd.

9. The value of 2x is always greater than the value of x .

Vocabulary:

Conditional If Hypothesis (p), then conclusion(q) 𝑝 → 𝑞

Inverse

Converse switch the hypothesis and the conclusion

Contrapositive ~𝑞 → ~𝑝

Examples: Write the converse, inverse, and contrapositive of each statement. Then determine if the statement is true or false.

10. If a Dog is a Great Dane, then it is large.

Converse ____________________________________________________________________________ T F

Inverse ____________________________________________________________________________ T F

Contrapositive _________________________________________________________________________ T F

11. If an integer is even, then the integer is divisible by two.

Converse ____________________________________________________________________________ T F

Inverse ____________________________________________________________________________ T F

Contrapositive __________________________________________________________________________ T F

12. If an angle measure is 122o , then the angle is an obtuse angle.

Converse ____________________________________________________________________________ T F

Inverse ____________________________________________________________________________ T F

Contrapositive __________________________________________________________________________ T F

Day 6: Angle Pairs and Relationships

Vocabulary: ______________________ Angles: two angles whose sum is 90° ______________________ Angles: two angles whose sum is 180° ______________________ Angles: two angles that share a common side and vertex. ______________________ Angles: congruent, nonadjacent angles formed by two intersecting lines. ______________ _____________: adjacent (next to; share a common side), supplementary angles *Be careful, because there is a difference between supplementary angles and a linear pair. Using the diagram, identify the following: 1. Complementary angles: 2. Vertical Angles: 3. Linear Pair: 4. Supplementary angles: Identify the type of angles given and then find the value of x. 5. 6. 7. Use the diagram at the right to answer the following questions.

8. If 114m A , find , ,m B m C and m D .

9. If 57m D , find , ,m C m A m B .

Solve for x and y. 10. 11. Parallel Lines and Transversals Vocabulary: Transversal: a line that crosses at least two other lines. Parallel lines: two lines that are in the same plane and never intersect. When parallel lines are intersected by a transversal, several angle pairs are created.

Corresponding Angles: Two angles are corresponding angles if they have corresponding positions. These angles are CONGRUENT.

Alternate interior angles: Two angles are alternate interior angles if they lie between the two lines and on opposite sides of the transversal. These angles are _________________

Same-side interior angles: Two angles are same-side interior angles if they lie between the two lines and on the same side of the transversal. These angles are _________________

Alternate exterior angles: Two angles are alternate exterior angles if they lie outside the two lines and on opposite sides of the transversal. These angles are _________________

Same-side exterior angles: Two angles are same-side exterior angles if they lie outside the two lines and on the same side of the transversal. These angles are _________________

Example: Classify all possible numbered angles. 12. Corresponding: 13. Alternate interior: 14. Alternate exterior: 15. Same-side interior: 16. Same-side exterior: Example: Solve for x given m is parallel to n or a is parallel to b. Write which angle property you use to justify your math. 17. 18.

19. Given 43m DOC find the measure of:

________

________

_________

_________

COB

DOE

BOA

EOA

20. Find the missing angles given that 𝑚 ∥ 𝑛.

1 ______

2 ______

3 ______

4 ______

5 ______

m

m

m

m

m