name: chapter 1: expressions, equations, and functions...

Name: Chapter 1: Expressions, Equations, and Functions

Lesson 1-1: Variables and Expressions Date:

An algebraic consists of sums and/or products of numbers and variables.

Symbols used to represent unknown/unspecified numbers or values are called . (Any letter may be used)

A of an expression may be a number, a variable, or a product or quotient of numbers and variables. (It is one piece of an expression, separated by addition or subtraction)

The numerical factor of a term is called a . (It is the number in front of the variable)

Example:

In an expression involving multiplication, the quantities being multiplied are the factors and the result is the product.

Example:

Ways to represent multiplication:

An expression like 𝑥𝑛 is called a , where 𝑛 is called the and 𝑥 is called the . The exponent indicates the number of times the base is used as a factor (how many times to repeatedly multiply). *If no exponent is shown, it is understood to be 1

Written as: Read as:

Example:


Example 1: Write Verbal Expressions

A. Write a verbal expression for

B. Write a verbal expression for

C. Write a verbal expression for

a. the sum of 3 and 𝑥

b. the product of 𝑥 squared and 3

c. the sum of 𝑥 and 3 squared

d. the sum of 𝑥 squared and 3

Example 2: Write Algebraic Expressions

A. Write an algebraic expression for

B. Write an algebraic expression for

C. Write an algebraic expression for

D. Which is an algebraic expression for

a. 6 − 4𝑥

b. (6 − 4) + 𝑥

c. 6 + 4𝑥

d. 6𝑥 − 4

Real World Example 3: Write an Expression

ENTERTAINMENT Mr. Nehru bought two adult tickets and three student tickets for the planetarium show. Write an algebraic expression that represents the cost of the tickets.

Step 1: Write a “let” statement

Step 2: Use your variables to write the algebraic expression


Lesson 1-2: Order of Operations Date:

Easy way to remember order of operations:

Example 1: Use Order of Operations

A. Evaluate B. Evaluate

Example 2: Expression with Grouping Symbols

A. Evaluate B. Evaluate

C. Evaluate


D. Evaluate

Example 3: Evaluate an Algebraic Expression

Evaluate 2(𝑥2 − 𝑦) + 𝑧2 if 𝑥 = , 𝑦 = , and 𝑧 =

Example 4: Guadalupe Peak in Texas has an altitude that is 671 feet more than double the altitude of Mount Sunflower in Kansas.

a) Write an expression that represents that altitude of Guadalupe Peak. (hint: start with a let statement)

b) Find the altitude of Guadalupe Peak if Mount Sunflower has an altitude of 4039 feet.


Lesson 1-3: Properties of Numbers Date:

Key Concept: Properties of Equality

Property Words Symbols Examples

Reflexive

Any quantity is equal to itself.

Symmetric

If one quantity equals a second quantity, then the second quantity equals the first quantity.

Transitive

If one quantity equals a second quantity and the second quantity equals a third quantity, then the first quantity equals the third quantity.

Substitution

A quantity may be substituted for its equal in any expression.

Key Concept: Addition Properties


Additive Identity

The sum of any number and zero is equal to the number.

Additive Inverse

A number and its opposite are additive inverses of each other.


Key Concept: Multiplication Properties


Multiplicative Identity

The product of any number and 1 is equal to the number.

Multiplicative Property of

Zero

The product of any number and 0 is equal to 0.

Multiplicative Inverse

Two numbers whose product is 1.

Two numbers whose product is 1 are called multiplicative inverses or (flip the fraction)

Example 1: Evaluate Using Properties

A. Evaluate . Name the property used in each step.


Key Concept: Commutative Property


Commutative

The order in which you add or multiply numbers does not change the sum or product.

Key Concept: Associative Property


Associative

The way you group three or more numbers when adding or multiplying does not change the sum or product.

Real World Example 2: Apply Properties of Numbers

HORSEBACK RIDING Virginia made a list of trail lengths to find the total miles she rode. Find the total miles Virginia rode her horse.

Trails

Name Miles

Bent Tree

Knob Hill

Meadowrun

Pinehurst

Example 3: Use Multiplication Properties

A. Evaluate using properties of numbers. Name the property used in each step.


B. Evaluate

a. 45

b. 36

c. 15

d. 180


Lesson 1-4: The Distributive Property Date:

Real-World Example 1: Distribute Over Addition

FITNESS Julio walks 5 days a week. He walks at a fast rate for 7 minutes and cools down for 2 minutes. Use the Distributive Property to write and evaluate an expression that determines the total number of minutes Julio walks.

Example 2: Mental Math

Use the Distributive Property to rewrite . Then evaluate.

Example 3: Algebraic Expressions

A. Rewrite using the Distributive Property. Then simplify.

B. Rewrite using the Distributive Property. Then simplify.


C. Simplify

a. 3𝑥3 + 2𝑥2 − 5𝑥 + 7

b. 4𝑥3 + 5𝑥2 − 2𝑥 + 10

c. 3𝑥3 + 6𝑥2 − 15𝑥 + 21

d. 𝑥3 + 2𝑥2 − 5𝑥 + 21

Example 4: Combine like terms

A. Simplify B. Simplify

C. Simplify D. Simplify

Example 5: Write and Simplify Expressions

Use the expression six times the sum of x and y increased by four times the difference of 5x and y.

A. Write an algebraic expression for the verbal expression

B. Simplify the expression and indicate the properties used.


Lesson 1-5: Equations Date:

An is two expressions combined with an equals sign between them.

Example:

Example 1: Use a Replacement Set

Find the solution set for if the replacement set if {2, 3, 4, 5, 6}.

𝑎 True or False?

2

3

4

5

6

The solution set is

Standardized Test Example 2:

A. Use Order of Operations to solve

a. 19

b. 27

c. 33

d. 42

Types of solutions:

One unique solution

No solution – the equation is always false

Infinite solutions (this is called an identity) – the equation is always true

Example 3: Solutions of Equations

A. Solve B. Solve


Example 4: Identities

A. Solve

B. Solve

Example 5: Equations Involving Two Variables

GYM MEMBERSHIP Dalila pays $16 per month for a gym membership. In addition, she pays $2 per Pilates class. Write and solve an equation to find the total amount Dalila spent this month if she took 12 Pilates classes.


Lesson 1-6 Relations Date:

A is a set of ordered pairs that can be represent in many ways including: words, an equation, ordered pairs, a table, a mapping diagram, or a graph.

A is formed by the intersection of two number lines, the horizontal/ and the vertical/ .

The intersection of the two axes occurs at the or the point (0, 0).

A set of numbers or coordinates written in the form (𝑥, 𝑦) is called an .

The refers to the horizontal placement of the point.

The refers to the vertical placement of the point.

Example:

A mapping diagram uses ordered pairs in two bubbles connected with arrows.

The refers to the x-values or the variable.

The refers to the y-values or the variable.

Example 1: Representations of a Relation

A. Express the relation {( , ), ( , ), ( , ), ( , )} as a table, a graph, and a mapping.

x

y

x

y


B. Determine the domain and range for the relation {( , ), ( , ), ( , ), ( , )}.

Real-World Example 2: Independent and Dependent Variables

A. The number of calories you burn increases as the number of minutes that you walk increases. Identify the independent and the dependent variables for this function.

B. The more hours Mary works at her job, the larger her paycheck becomes. Identify the independent and the dependent variables for this function.

C. The area of a square increases as the length of a side increases. Identify the independent and dependent variable in this function.

Example 3: Analyze Graphs

A. The graph represents the temperature in Mrs. Simon’s classroom on a winter school day. Describe what is happening in the graph.

B. The graph represents Macy’s speed as she swims laps in a pool. Describe what is happening in the graph.


Lesson 1-7 Functions Date:

A is a relationship between input and output. There is exactly one output for each input (each element of the domain is paired with exactly one element of the range). That means the values do not repeat.

Example 1: Identify Functions

Determine whether each relation is a function. Explain.

A. B.

C. D.

Another way to determine if a relation is a function is to look at the graph and use the . If a vertical line intersects the graph more than once, the relation is NOT a function.

Example 2: Identify Functions

Determine whether each relation is a function. Explain.

A. B.


Example 3: Equations as Functions

A. Determine whether is a function.

𝑥 𝑦

B. Determine whether is a function.

𝑥 𝑦

Equations that are functions can be written in . This notation typically

replaces 𝑦 with 𝑓(𝑥).

Example of equation

Example in function notation

x

y

x

y


Example 4: Function Values

A. If , find 𝑓(4). B. If , find 𝑓(−5).

C. If , find 𝑓(2𝑘).


Lesson 1-8 Interpreting Graphs of Functions Date:

A is the y-coordinate of the point that intersects the y-axis (when x=0).

An is the x-coordinate of the point that intersects the x-axis (when y=0).

Example of y-intercept Example of x-intercept

Example 1: Interpret Intercepts

COLLEGE The graph shows the cost at a community college y as a function of the number of credit hours taken x. Identify the function as linear or nonlinear. Then estimate and interpret the intercepts of the graph of the function.

A graph possesses line symmetry in the y-axis or some other vertical line if each half of the graph on either side of the line matches exactly (it can be folded in half and match up).

Example 2: Interpret Symmetry

MANUFACTURING The graph shows the cost y to manufacture x units of a product. Describe and interpret any symmetry.


Example 3: Interpret Extrema and End Behavior

DEER The graph shows the population y of deer x years after the animals are introduced on an island. Estimate and interpret where the function is positive, negative, increasing, and decreasing, the x-coordinates of any relative extrema, and the end behavior of the graph.

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