1. Transforming Functions
9-3
Transforming Functions
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 2

2. A rental car costs $45 per day plus $0.10 for every mile over 200.
Warm Up
A rental car costs $45 per day plus $0.10 for every mile over 200.
1. Find the cost of renting the car for a day and driving 250 miles.
$50
2. Write a function of d, the number of miles driven in a day, to describe the cost of renting the car for one day?
C(d) =
if 0 ≤ d ≤ 200
(d – 200) if d > 200

3. Objectives Transform functions.
Recognize transformations of functions.

4. In previous lessons, you learned how to transform several types of functions. You can transform piecewise functions by applying transformations to each piece independently. Recall the rules for transforming functions given in the table.

6. Horizontal translations change both the rules and the intervals of piecewise functions. Vertical translations change only the rules.
Caution

7. Example 1: Transforming Piecewise Functions2
– x if x < 0
Given f(x) = write the
rule g(x), a vertical stretch by a factor of 3. 1 2
x if x ≥ 0

8. Example 1 Continued
Each piece of f(x) must be vertically stretched by a factor of 3. Replace every y in the function by 3y, and simplify.
3(– x) if x < 0 1 2
g(x) = 3f(x) =
3( x2) if x ≥ 0 1 2
– x if x < 0 3 2 =
x if x ≥ 0 3 2

9. Example 1 Continued
Check Graph both functions to support your answer.

10. Given f(x) = write the rule
Check It Out! Example 1
Given f(x) = write the rule
for g(x), a horizontal stretch of f(x) by a factor of 2.
x if x ≤ 0
x – 3 if x > 0
Each piece of f(x) must be stretched by a factor of 2 units.
x if x ≤ 0 1 2
g(x) = f( x) = 1 b
x – 3 if x > 0 1 2
if x ≤ 0 x2 2
– if x > 0 x =

11. Check It Out! Example 1 Continued
Check Graph both functions to support your answer.

12. When functions are transformed, the intercepts may or may not change
When functions are transformed, the intercepts may or may not change. By identifying the transformations, you can determine the intercepts, which can help you graph a transformed function.

15. Example 2A: Identifying Intercepts
Identify the x- and y-intercepts of f(x). Without graphing g(x), identify its x- and y-intercepts.
f(x) =–2x – 4 ; g(x) =
Find the intercepts of the original function.
f(0) = –2(0) – 4 = – 4
0 = –2x – 4
f(0) = –4
–2 = x

16. Example 2A Continued
The y-intercept is –4, and the x-intercept is –2. Note that g(x) is a horizontal stretch of f(x) by a factor of 2. So the y-intercept of g(x) is also –4. The x-intercept is 2(–2), or –4.
Check A graph supports your answer.

17. Example 2B: Identify Intercepts
f(x) = x2 – 1; g(x) = f(–x)
From the graph of f(x), the y-intercept is –1, and the x-intercepts are –1 and 1.
Check A graph supports your answer.
Note that g(x) is a horizontal reflection across the y-axis. So the x-intercepts of g(x) will be –1(–1) and –1(1), or 1 and –1. The y-intercept is unchanged at –1.

18. f(x) = x + 4 and g(x) = –f(x)
Check It Out! Example 2a
Identify the x- and y-intercepts of f(x). Without graphing g(x), identify its x- and y-intercepts.
f(x) = x + 4 and g(x) = –f(x) 2 3
Find the intercepts of the original functions.
f(0) = (0) + 4 2 3
0 = x + 4 2 3
f(0) = 4
–6 = x

19. Check It Out! Example 2a Continued
The y-intercept is 4, and the x-intercept is –6. Note that g(x) is a reflection of f(x) across the x-axis. So the x-intercept of g(x) is also –6. The y-intercept is –1(4), or –4.
Check A graph supports your answer.

20. Note that g(x) is a vertical compression by a factor of .
Check It Out! Example 2b
f(x) = x2 – 9 and g(x) = f(x)
From the graph of f(x), the y-intercept is –9, and the x-intercepts are ±3.
Note that g(x) is a vertical compression by a factor of 1 3
Check A graph supports your answer
So the x-intercept of g(x) will be ±3. The y-intercept of g(x) will be (–9) = –3. 1 3

21. The factor for horizontal stretches and compressions is the reciprocal of the coefficient in the equation.
Remember! 2 3 1 =

22. Example 3: Combining Transformations1 3
Given f(x) = (x – 2)2 and g(x) = 2f(x) – 3 and graph g(x).
Step 1 Graph f(x). The graph of f(x) has y-intercept (0, ) and x-intercept (2, 0). 4 3

23. Step 2 Analyze each transformation one at a time.
Example 3 Continued
Step 2 Analyze each transformation one at a time.
The first transformation is a vertical stretch by a factor of 2. After the vertical stretch, the x-intercept will remain 2, but the y-intercept will be 8 3
The second transformation is a vertical translation of 3 units down. Use a table to shift each identified point down 3 units.

24. Step 3 Graph the final result.
Example 3 Continued
Intercept Points
(2, 0)
(0, )
Shifted
(2, –3)
(0, – ) 8 3 1
Step 3 Graph the final result.

25. Given f(x) = 2x – 4 and g(x)= – f(x), graph g(x).
Check It Out! Example 3
Given f(x) = 2x – 4 and g(x)= – f(x), graph g(x). 1 2
Step 1 Graph f(x). The graph of f(x) has y-intercept (0, –3) and x-intercept (2, 0).

26. Check It Out! Example 3 Continued
Step 2 Analyze each transformation one at a time.
The first transformation is a vertical compression by After the vertical compression, the x-intercept is (2, 0) but the y-intercept is (0, –3). 1 2
The second transformation is a reflection across the x-axis. The x-intercept remains (2, 0) but the y-intercept will be (0, 1.5).

27. Check It Out! Example 3 Continued
Step 3 Graph the final result.

28. Example 4: Problem-Solving Application
Coco’s Coffee charges different prices based on the number of pounds purchased. The pricing scale is modeled by the function below, where w is the weight in pounds purchased.
p(w) =
9w if 0 < w < 3
(w–3) if 3 ≤ w < 6
(w–6) if w ≥ 6

29. Example 4 Continued
Orders placed directly through the Web site are discounted by , but a shipping fee of $2.50 is added. Write a pricing function for orders placed through the Web site. 1 3

30. Understand the Problem
Example 4 Continued 1
Understand the Problem
The new price function will include two changes, a discount by and an additional shipping fee of $2.50. The discount is equivalent to multiplying all of the parts of the function by This will be a vertical compression by a factor of The shipping price will be a vertical translation of 2.5 units up. 2 3 1

31. Example 4 Continued 2
Make a Plan
Perform each transformation, one at a time, and then write the new rule.

32. First find the prices after the discount for Web purchases.
Example 4 Continued
Solve 3
First find the prices after the discount for Web purchases.
(9w) if 0 < w < 3 2 3
Multiply all parts of the function by 2 3
pdiscount(w) = p(w) = 2 3
( (w – 3)) if 3 ≤ w < 6 2 3
( (w – 6)) if w ≥ 6 2 3

33. Example 4 Continued 3 Solve Continued 6w if 0 < w < 3=

34. Then find the prices after applying the shipping fees.
Example 4 Continued
Solve Continued 3
Then find the prices after applying the shipping fees.
Pw(w) = pdiscount(w) =
6w if 0 < w < 3
18 + 5(w – 3) if 3 ≤ w < 6
33 + 4(w – 6) if w ≥ 6 =
6w if 0 < w < 3
(w – 3) if 3 ≤ w < 6
(w – 6) if w ≥ 6

35. Continue by checking each piece of the function.
Example 4 Continued
Look Back 4
Check your answer by trying a few values. For 9 pounds of coffee, the original fee would have been $ A discount plus $2.50 would amount to $ Evaluate the function for x = 9 to check. 1 3
Pw(9) = { (9 – 6) = 47.50 
Continue by checking each piece of the function.

36. Check It Out! Example 4
A movie theater charges $5 for children under 12 and $7.50 for anyone 12 and over. The theater decides to increase its prices by 20%. It charges an additional $0.50 fee for online ticket purchases. Write an equation for the online ticket prices.
5x if 0 < x < 12
f(x) =
7.5x if x ≥ 12

37. Understand the Problem
Check It Out! Example 4 Continued 1
Understand the Problem
Two changes, a 20% increase plus an additional fee of $0.50 for online prices. The increase is equivalent to multiplying all parts by 120% or 1.2. This is a vertical stretch by a factor of 1.2. The online fee will be a vertical translation of 0.50 units up. 2
Make a Plan
Perform each transformation, one at a time, and then write the new rule.

38. Check It Out! Example 4 Continued
Solve 3
First find the increase.
(1.2)5x if 0 < x < 12
Multiply all parts of the function by 1.2.
fincrease(x) = (1.2)f(x) =
(1.2)7.5x if x ≥ 12
Then find the online fees.
6.50x if 0 < x < 12
fonline(x) = f(x ) =
9.50x if x ≥ 12

39. Check It Out! Example 4 Continued
Look Back 4
Check your answer by trying a few values. For 1 child age 10 and 1 adult age 35, the original fee would have been $ A 20% increase plus $0.50 per ticket will be $16. Check the function x = 1 to check.
fonline(1) + fonline(1) = 6.50(1) (1) = 16 
Continue by checking each piece of the function.

40. Consider the functions
Lesson Quiz: Part I
Consider the functions 1 2
f(x) = x2 – 2, g(x) = 4f(x), and h(x) = f( x) 1 3
1. Identify the intercepts of f(x) and g(x).
f(x): x-ints. = –2 and 2, y-int. = –2
g(x): x-ints. = –2 and 2, y-int. = –8
2. Graph f(x) and h(x).

41. Lesson Quiz: Part II
3. Ticket prices to a theater are modeled by the function below, where a is a person’s age in years.
if 0 ≤ a < 12
p(a) =
if a ≥ 12
They plan to raise all prices by $1, but then they are going to offer a 25% discount to persons 55 and older. Write a function for their new prices.
if 0 ≤ a < 12
p(a) =
if 12 ≤ a < 55
if a ≥ 55