Expressions, Equations and Functions Unit 1 Expressions, Equations and Functions

1-2: Order of Operations Students will . . . evaluate numerical expressions by using the order of operations. evaluate algebraic expressions by using the order of operations.

Please Excuse My Dear Aunt Sally Order of Operations: Parentheses Exponents Multiplication/Division (going left to right) Addition/Subtraction (going left to right) Please Excuse My Dear Aunt Sally

Example: Evaluate each of the following: 20 – 7 + 82 – 7 ∙ 11 48 ÷ 23 ∙ 3 + 5 4[12 ÷ (6 – 2)]2

Example: Evaluate the following algebraic expressions for the given values: a2(3b + 5) ÷ c if a = 2, b = 6, c = 4 2(x2 – y) + z2 if x = 4, y = 3, z = 2

Example: Ann is planning a business trip for which she needs to rent a car. The car rental company charges $38 per day plus $0.50 per mile over 100 miles. Write an expression for how much it will cost Ann to rent the car. Suppose Ann rents the car for 5 days and drives 180 miles. How much will she pay the rental company? HW: Pg. 13 #16 – 38 even, 59

Review Evaluate each expression: 3 ∙ 5 + 1 – 2 14 ÷ 2 ∙ 6 – 52

1-4: The Distributive Property Students will . . . use the distributive property to evaluate expressions. use the distributive property to simplify expressions.

Like terms are terms that have EXACTLY the same variables. Example: Identify the like terms below. The coefficient of a term is the number in front of the variable. -4x3 2 -4x2 2y2 5x -3 12x -½ 5x2

Example: Simplify each of the following: An expression is in simplest form when there are no parentheses and the like terms are combined. Example: Simplify each of the following: 6n – 4n b2 + 13b + 13 4y3 + 2y – 8y + 5 7a + 4 – 6a2 – 2a

Distributive Property: a(b + c) = ab + ac (b + c)a = ba + ca a(b - c) = ab - ac (b - c)a = ba - ca

Example: Simplify each of the following: -6(r + 3g – t) (y + 3)12 -4(-8 – 3m) 4(y2 + 8y + 2)

Example: Write and simplify an algebraic expression for each of the following verbal expressions: 5 times the difference of q squared and r plus 8 times the sum of 3q and 2r 6 times the sum of x and y increased by four times the difference of 5x and y HW: Pg. 29 – 30 #12, 26 – 38 even, 56

Review Evaluate each expression: Simplify each expression: 23 – 2(17 + 33) Simplify each expression: (⅓ – 2b)27 4(8p + 4q – 7r) 4y3 + 3y3 + y4 4(x + 3) – 2x + 8

1-6: Relations Students will . . . represent relations interpret graphs of relations

A set or ordered pairs is called a relation. {(-1, 2) (-2, 3) (4, 5) (6, 7) (0, 1)} The first number in the ordered pairs (x value) is called the domain. D = {-2, -1, 0, 4, 6} The second number in the ordered pairs (y value) is called the range. R = {1, 2, 3, 5, 7}

Example: For each of the following relations identify the domain and range. { (0, 1) (2, 1) (3, 2) (3, 4) } { (2, 1) } { (4, 10) (1, 2) (3, 2) (2, 9) (5, 10) } { (1, 1) (-1, 1) }

Relation Representations Ordered Pairs { (1, 1) (2, 3) (1, 3) (3, 4)} Table x coordinates in the first column and the corresponding y coordinates in the second column (keep the pairs together) x y 1 3 2 4

Graph Graph each point

Mapping Make a list of x-values (in order with no repeats) and a list of y-values (in order with no repeats); draw arrows to corresponding values Domain Range 1 2 3 1 3 4

Example: Express each of the following relations as a table, graph and mapping. Then determine the domain and range. {(4, -3), (3, 2), (-4, 1), (0, -3)} {(4, 3), (-2, -1), (2, -4), (0, -4)}

In a relation, the independent variable is the value that is subject to choice. represented by the x values The dependent variable is the value that depends on the independent variable. represented by the y values Example: Identify the independent and dependent variables for each relation. The number of calories you burn increases as the number of minutes that you walk increases In warm climates, the average amount of electricity used rises as the daily average temperature increases and falls as the daily average temperature decreases.

Some relations can be interpreted by analyzing their shape. Example: For each of the following, describe what happens in the graph. HW: Pg. 43 – 45 #10 – 14 even, 15, 16, 19, 20, 40

Review Evaluate each expression: Simplify each expression: 7 + 2(9 – 3) [(25 – 5) ÷ 9]11 Simplify each expression: 3(x – 6) -5(2x + 9) Express the relation as a table, graph and mapping. Determine the domain and range. {(-2, 4), (-1, 3), (0, 2), (-1, 2)}

1-7: Functions Students will . . . determine whether a relation is a function find function values

Example: Determine whether each relation is a function. A function is a relationship between input and output where there is exactly one output for each input. The x-values DO NOT repeat. Example: Determine whether each relation is a function. Domain Range -2 4 1 7 {(2, 1), (3, -2), (3, 1), (2, -2)}

Example: Determine whether each graph represents a function. You can use the vertical line test to tell if a graph is a function or not. If a vertical line intersects the graph more than once, then it is NOT a function. Otherwise, it IS a function. Example: Determine whether each graph represents a function.

Equations that are functions can be written in function notation. Replace y with f(x) Example: Write each of the following in function notation. y = 3x – 8 y = 3x2 – 4x + 10 A function with a graph that is not a straight line is a nonlinear function. If an equation has exponents that are not 1, then it is a nonlinear function. “f of x”

Example: If f(x) = 3x + 1 and g(x) = 2x2 – 5x – 6, find each value: g(-1) – 4[g(3)] HW: Pg. 52 – 53 #20-25, 27, 28, 33-38

Review Evaluate: Simplify each of the following: 3 + (5 – 2) – 32 35 ÷ 5 ∙ 22 + 1 Simplify each of the following: 3(2d + 6) (6h – 1)5 Express the following relation as a table, graph, and mapping. Determine the domain and range. {(3, 1), (-1, 4), (-1, 7)}

Review Determine whether each relation is a function: If f(x) = 4x + 5, find each of the following: f(3) f(-5) + 1 Domain Range -3 -1 1 3 7 -10 12 42 {(0,2), (3,5), (0,-1), (-2,4)}

1-8: Interpreting Graphs of Functions Students will . . . interpret intercepts, and symmetry of graphs and functions. interpret positive, negative, increasing, and decreasing behavior, extrema, and end behavior of graphs of functions.

The intercepts of a graph are the points where the graph crosses an axis. The point where the graph crosses the y-axis is the y-intercept. The point where the graph crosses the x-axis is the x-intercept. Example: The graph shows the height y of an object as a function of time x. Is the function linear or nonlinear? Estimate the intercepts of the graph. What do these intercepts mean?

Example: The graph shows the temperature of a medical sample thawed at a controlled rate. Identify the function as linear or nonlinear. Estimate and interpret the intercepts.

A graph has line symmetry if it can be divided along a line into two parts that match exactly. The line can be the y-axis or any other vertical line. Example: An object is launched. The graph shows the height y of the object as a function of time x. Does this graph have symmetry? What is the line of symmetry? What does this symmetry mean?

A function is positive where its graph is above the x-axis. A function is negative where its graph is below the x-axis. positive negative

A function is increasing where the graph goes up. A function is decreasing where the graph goes down.

High or low points of the graph are called extrema. Point A is a relative minimum since it has the lowest y-coordinate of the points near it. Point B is a relative maximum since it has the highest y-coordinate of the points near it. These can be found where the graph changes back and forth between increasing and decreasing.

End behavior describes the values of a function at the positive and negative extremes in its domain. As x decreases, y increases. As x increases, y decreases.

Example: The graph shows the value of a vehicle over time Example: The graph shows the value of a vehicle over time. Identify the function as linear or nonlinear. Estimate and interpret the intercepts, any symmetry, where the function is positive, negative, increasing, and decreasing, the x- coordinates of any relative extrema, and the end behavior of the graph. HW: Pg. 59 #4 – 8 even, 11, 12, 18, 20