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Preview Warm Up California Standards Lesson Presentation

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**Warm Up What three terms come next? 1. 9, 12, 15, 18, . . .**

2. –8, –3, 2, 7, … 3. 9, 10, 12, 15, 19, … 21, 24, 27 12, 17, 22 24, 30, 37

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Preparation for AF3.3 Graph linear functions, noting that the vertical change (change in y-value) per unit of horizontal change (change in x-value) is always the same and know that the ratio (“rise over run”) is called the slope of a graph. California Standards

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Vocabulary function input output vertical line test

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**A function is a set of ordered pairs (x, y) so that each x-value corresponds to exactly one y-value.**

Some function can be described by a rule written in words, such as “double a number and then add nine to the result,” or by an equation with two variables. One variable (often x) represents the input, and the other variable (often y) represents the output. Function Rule y = 2x + 9 Output variable Input variable

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When a function can be written as an equation, the input is the value substituted into the function rule. The output is the result of that substitution.

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**Additional Example 1: Finding Output Values**

Find the output for each input: –2, 0, 2. A. y = 3 – x 2 Make a table. Input Rule Output x 3 – x2 y Substitute –2 for x. Then simplify. –2 3 – (–2)2 –1 Substitute 0 for x. Then simplify. 3 – (0)2 3 Substitute 2 for x. Then simplify. 2 3 – (2)2 –1

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**The input values of a function are also called the domain**

The input values of a function are also called the domain. The output values of a function are also called the range. Reading Math

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**Additional Example 1: Completing a Function Table**

Find the output for each input: – 2, 0, 2. B. y = –2x - 3 Input Rule Output Make a table. x –2x – 3 y Substitute –2 for x and simplify. –2 –2(–2) – 3 1 Substitute 0 for x and simplify. –2(0) – 3 –3 Substitute 2 for x and simplify. 2 –2(2) – 3 –7

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Check It Out! Example 1 Find the output for each input: –2 , 0, 5 A. y = 5x + 3 Input Rule Output Make a table. x 5x + 3 y Substitute –2 for x. Then simplify. –2 5(–2) + 3 –7 Substitute 0 for x. Then simplify. 5(0) + 3 3 Substitute 5 for x. Then simplify. 5 5(5) + 3 28

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Check It Out! Example 1 Find the output for each input: –2, 0, 5. B. y = 3x2 Input Rule Output Make a table. x 3x2 y Substitute –2 for x and simplify. –2 3(–2)2 12 Substitute 0 for x and simplify. 3(0)2 Substitute 5 for x and simplify. 5 3(5)2 75

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Because a function has exactly one output for each input, you can use the vertical line test to determine whether a graph is a function. If no vertical line intersects the graph at more than one point, then the graph is a function. One way to perform the vertical line test is to pass a vertical line across a graph.

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**Additional Example 2: Identifying Functions**

Determine if the relationship represents a function. A. x y 2 3 4 5 6 The input x = 2 has two outputs, y = 3 and y = 6. The input x = 3 also has more than one output. The relationship is not a function.

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**Additional Example 2: Identifying Functions**

Determine if the relationship represents a function. B. x y –1 2 –4 5 – 7 8 –10 Each input has only one output value. The relationship is a function.

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**Additional Example 2: Identifying Functions**

Determine if the relationship represents a function. C. Pass a vertical line across the graph. Many vertical lines intersect the graph at two points. The relationship is not a function.

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**Additional Example 2: Identifying Functions**

Determine if the relationship represents a function. D. x y 2 -2 -4 4 Pass a vertical line across the graph. No vertical lines intersect the graph at more than one point. The relationship is a function.

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Check It Out! Example 2 Determine if the relationship represents a function. A. x y 1 2 3 Each input x has only one output y. The relationship is a function.

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Check It Out! Example 2 Determine if the relationship represents a function. B. x y 2 1 3 4 5 Each input has only one output value. The relationship is a function.

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**Determine if the relationship represents a function. C.**

Check It Out! Example 2 Determine if the relationship represents a function. C. x y Pass a vertical line across the graph. No vertical lines intersect the graph at more than one point. 2 -2 2 The relationship is a function. -2

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**Determine if the relationship represents a function. D.**

Check It Out! Example 2 Determine if the relationship represents a function. D. x y 2 -2 4 -4 Pass a vertical line across the graph. No vertical lines intersect the graph at more than one point. The relationship is a function.

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Lesson Quiz: Part I Find the output for each input: –2, 0, 2. 1. y = 3x + 3 2. y = –x2 –3, 3, 9 –4, 0, –4

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**Determine if each relationship represents a function. 3.**

Lesson Quiz: Part Il Determine if each relationship represents a function. 3. 4. no x y 2 -2 -4 4 no

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