function graphs5-5z. function graphs

8/14/2019 Function Graphs5-5z. Function Graphs

1/3Copyright 2005 by Advanced Placement Strategies 139

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Math concept #11

Functions, y = f(x)

Example 1: The function graphed on the left below is defined as f(x) = x 2.The graph on the right is a function defined as g(x) = f(x + h) + k , where h andk are constants. What is the value of h + k ?

(a) -2 (b) 0 (c) 2 (d) 4 (e) 7

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First, a little review regarding graph translations (the same shape graph in a

different position on the coordinate plane).Given a function f(x), horizontal and vertical shifts (translations which we willcall g(x)) are represented by the following in which h and k are positive realnumbers:Vertical shift of k units upward: g(x) = f(x) + k Vertical shift of k units downward: g(x) = f(x) - k Horizontal shift of h units to the right: g(x) = f(x - h)Horizontal shift of h units to the left: g(x) = f(x + h)

Vertical shift of k units upward along with a horizontal shift of h units to the right: g(x) = f(x - h) + k Vertical shift of k units upward along with a horizontal shift of h units to the left: g(x) = f(x + h) + k Vertical shift of k units downward along with a horizontal shift of h units to the right: g(x) = f(x - h) - k Vertical shift of k units downward along with a horizontal shift of h units to the left: g(x) = f(x + h) - k

For example, if f(x) = x 2, and g(x) = (x - 2) 2 + 4, then g(x) is a translation of f(x)shifting f(x) two units to the right and four units upward.


2/3

Copyright 2005 by Advanced Placement Strategies 140

(d) (e)

Y = f(x)

Example 3: If the function h is defined by h(x) = ax 2 + bx + c, and aand c are positive, which of the five graphs in the answer choices for example 2 could be the graph of h .

Example 2: The graph of Y = f(x)is shown to the right. Which of thefollowing could be the graph of Y = f(x - h) + k if h and k are

positive constants.

(a) (b) (c)

Example 4: The figure below shows the graph of y = f(x) . If the func-tion g is defined by g(x) = f(4x) + 2, what is the value of g(1).

(a) 4(b) 5(c) 6(d) 7(e) 8

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x-6 -4 -2 0 2 4 6

6

4

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-2

-4

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Copyright 2005 by Advanced Placement Strategies 141

y

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1

Example 6: If h(x) = x 2 + 48, and a is positivenumber such that h(3a) = 3 h(a), what is the valueof a?

Example 7: If f (x) = (x 2/9) - 8, and f (6a) = 4a,

what is one possible value of a?

Example 9: After production the total profit p , in dollars, fromselling q basketballs is given by the function p (q) = 10q - (3q + h),where h is a constant. If 200 basketballs were sold in March for atotal profit of $1,150, what is the value of h?

Example 5: The graph to the rightshows the graph of a quadratic func-tion f whose minimum value is f (-2).If f (t ) = f (-6) = 0, which of the fol-lowing could be the value of t?

(a) -8(b) -4(c) -2(d) 0(e) 2

The following are student-produced response questions. You willenter your answer by marking the ovals in a grid like the one below.

0 0 0

7 7 7 7

1 1 1 1

2 2 2 2

3 3 3 3

4 4 4 4

5 5 5 5

6 6 6 6

8 8 8 8

. . . .

9 9 9 9

1 2 5

Example 8: If f (x) = x - 3 , and 2 f (m ) = 4,what is the value of m?

Answers 1. a 2. e 3. b 4. b 5. e 6. 4 7. -1 or 2 8. 25 9. 250

function graphs5-5z. function graphs

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