9 3 Functions 9 3 Transforming Functions Warm

9 -3 Functions 9 -3 Transforming Functions Warm Up Lesson Presentation Lesson Quiz Holt Algebra 22

9 -3 Transforming Functions 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? 45 if 0 ≤ d ≤ 200 C(d) = 45 + 0. 1(d – 200) if d > 200 Holt Algebra 2

9 -3 Transforming Functions Objectives Transform functions. Recognize transformations of functions. Holt Algebra 2

9 -3 Transforming Functions 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. Holt Algebra 2

9 -3 Transforming Functions Holt Algebra 2

9 -3 Transforming Functions Caution Horizontal translations change both the rules and the intervals of piecewise functions. Vertical translations change only the rules. Holt Algebra 2

9 -3 Transforming Functions Example 1: Transforming Piecewise Functions – Given f(x) = 1 2 x x 2 if x < 0 if x ≥ 0 write the rule g(x), a vertical stretch by a factor of 3. Holt Algebra 2

9 -3 Transforming Functions Example 1 Continued Each piece of f(x) must be vertically stretched by a factor of 3. Replace every y in the function by 3 y, and simplify. 3(– g(x) = 3 f(x) = = Holt Algebra 2 3( 1 2 – 1 2 x) if x < 0 x 2) if x ≥ 0 3 2 x if x < 0 3 2 x 2 if x ≥ 0

9 -3 Transforming Functions Example 1 Continued Check Graph both functions to support your answer. Holt Algebra 2

9 -3 Transforming Functions Check It Out! Example 1 x 2 if x ≤ 0 Given f(x) = write the rule x – 3 if x > 0 for g(x), a horizontal stretch of f(x) by a factor of 2. Each piece of f(x) must be stretched by a factor of 2 units. 1 2 x if x ≤ 0 1 2 g(x) = f( x) = 1 b x – 3 if x > 0 = Holt Algebra 2 2 x 2 if x ≤ 0 – 3 if x > 0

9 -3 Transforming Functions Check It Out! Example 1 Continued Check Graph both functions to support your answer. Holt Algebra 2

9 -3 Transforming Functions 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. Holt Algebra 2

9 -3 Transforming Functions Holt Algebra 2

9 -3 Transforming Functions Holt Algebra 2

9 -3 Transforming Functions Example 2 A: Identifying Intercepts Identify the x- and y-intercepts of f(x). Without graphing g(x), identify its x- and yintercepts. f(x) =– 2 x – 4 ; g(x) = Find the intercepts of the original function. f(0) = – 2(0) – 4 = – 4 0 = – 2 x – 4 f(0) = – 4 – 2 = x Holt Algebra 2

9 -3 Transforming Functions Example 2 A 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. Holt Algebra 2

9 -3 Transforming Functions Example 2 B: Identify Intercepts f(x) = x 2 – 1; g(x) = f(–x) From the graph of f(x), the y-intercept is – 1, and the x-intercepts are – 1 and 1. Note that g(x) is a horizontal Check A graph supports your answer. 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. Holt Algebra 2

9 -3 Transforming Functions Check It Out! Example 2 a Identify the x- and y-intercepts of f(x). Without graphing g(x), identify its x- and yintercepts. f(x) = 2 3 x + 4 and g(x) = –f(x) Find the intercepts of the original functions. f(0) = 2 (0) + 4 3 f(0) = 4 Holt Algebra 2 0= 2 3 – 6 = x x+4

9 -3 Transforming Functions Check It Out! Example 2 a 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. Holt Algebra 2

9 -3 Transforming Functions Check It Out! Example 2 b f(x) = x 2 – 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 1 Check A graph supports compression by a factor of. 3 So the x-intercept of g(x) will be ± 3. The y-intercept of 1 g(x) will be (– 9) = – 3. 3 Holt Algebra 2 your answer

9 -3 Transforming Functions Remember! The factor for horizontal stretches and compressions is the reciprocal of the coefficient in the equation. 1 3 2 Holt Algebra 2 = 2 3

9 -3 Transforming Functions Example 3: Combining Transformations 1 Given f(x) = (x – 2)2 and g(x) = 2 f(x) – 3 and 3 graph g(x). Step 1 Graph f(x). The graph of f(x) has y-intercept 4 (0, ) and x-intercept (2, 0). 3 Holt Algebra 2

9 -3 Transforming Functions 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 8 will remain 2, but the y-intercept will be. 3 The second transformation is a vertical translation of 3 units down. Use a table to shift each identified point down 3 units. Holt Algebra 2

9 -3 Transforming Functions Example 3 Continued Intercept Points (2, 0) Shifted (2, – 3) 8 ) 3 1 (0, – ) 3 (0, Step 3 Graph the final result. Holt Algebra 2

9 -3 Transforming Functions Check It Out! Example 3 Given f(x) = 2 x – 4 and g(x)= – 1 f(x), graph 2 g(x). Step 1 Graph f(x). The graph of f(x) has y-intercept (0, – 3) and x-intercept (2, 0). Holt Algebra 2

9 -3 Transforming Functions Check It Out! Example 3 Continued Step 2 Analyze each transformation one at a time. The first transformation is a vertical compression 1 by. After the vertical compression, the x 2 intercept is (2, 0) but the y-intercept is (0, – 3). The second transformation is a reflection across the x-axis. The x-intercept remains (2, 0) but the yintercept will be (0, 1. 5). Holt Algebra 2

9 -3 Transforming Functions Check It Out! Example 3 Continued Step 3 Holt Algebra 2 Graph the final result.

9 -3 Transforming Functions 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) = Holt Algebra 2 9 w if 0 < w < 3 27 + 7. 5(w– 3) if 3 ≤ w < 6 49. 5 + 6(w– 6) if w ≥ 6

9 -3 Transforming Functions Example 4 Continued Orders placed directly through the Web site are 1 discounted by , but a shipping fee of $2. 50 3 is added. Write a pricing function for orders placed through the Web site. Holt Algebra 2

9 -3 Transforming Functions Example 4 Continued 1 Understand the Problem The new price function will include two changes, a 1 discount by and an additional shipping fee of 3 $2. 50. The discount is equivalent to multiplying all 2 of the parts of the function by. This will be a 3 2 vertical compression by a factor of. The 3 shipping price will be a vertical translation of 2. 5 units up. Holt Algebra 2

9 -3 Transforming Functions Example 4 Continued 2 Make a Plan Perform each transformation, one at a time, and then write the new rule. Holt Algebra 2

9 -3 Transforming Functions Example 4 Continued 3 Solve First find the prices after the discount for Web purchases. pdiscount(w) = 2 3 p(w) = 2 3 (9 w) 2 3 (27 + 7. 5(w – 3)) if 3 ≤ w < 6 2 3 Holt Algebra 2 if 0 < w < 3 (49. 5 + 6(w – 6)) if w ≥ 6 Multiply all parts of the function by 2. 3

9 -3 Transforming Functions Example 4 Continued 3 Solve Continued = Holt Algebra 2 6 w if 0 < w < 3 18 + 5(w – 3) if 3 ≤ w < 6 33 + 4(w – 6) if w ≥ 6

9 -3 Transforming Functions Example 4 Continued 3 Solve Continued Then find the prices after applying the shipping fees. Pw(w) = pdiscount(w) + 2. 5 = = Holt Algebra 2 6 w + 2. 5 if 0 < w < 3 18 + 5(w – 3) + 2. 5 if 3 ≤ w < 6 33 + 4(w – 6) + 2. 5 if w ≥ 6 6 w + 2. 5 if 0 < w < 3 20. 5 + 5(w – 3) if 3 ≤ w < 6 35. 5 + 4(w – 6) if w ≥ 6

9 -3 Transforming Functions Example 4 Continued 4 Look Back Check your answer by trying a few values. For 9 pounds of coffee, the original fee would have been 1 $67. 50. A discount plus $2. 50 would amount to 3 $47. 50. Evaluate the function for x = 9 to check. Pw(9) = {35. 5 + 4(9 – 6) = 47. 50 Continue by checking each piece of the function. Holt Algebra 2

9 -3 Transforming Functions 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. f(x) = Holt Algebra 2 5 x if 0 < x < 12 7. 5 x if x ≥ 12

9 -3 Transforming Functions 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. Holt Algebra 2

9 -3 Transforming Functions Check It Out! Example 4 Continued 3 Solve First find the increase. (1. 2)5 x fincrease(x) = (1. 2)f(x) = if 0 < x < 12 (1. 2)7. 5 x if x ≥ 12 Then find the online fees. fonline(x) = f(x + 0. 50) = Holt Algebra 2 6. 50 x if 0 < x < 12 9. 50 x if x ≥ 12 Multiply all parts of the function by 1. 2.

9 -3 Transforming Functions Check It Out! Example 4 Continued 4 Look Back Check your answer by trying a few values. For 1 child age 10 and 1 adult age 35, the original fee would have been $12. 50. 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) + 9. 50(1) = 16 Continue by checking each piece of the function. Holt Algebra 2

9 -3 Transforming Functions Lesson Quiz: Part I Consider the functions 1 2 f(x) = x 2 – 2, g(x) = 4 f(x), and h(x) = f( 1 x) + 3. 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). Holt Algebra 2

9 -3 Transforming Functions 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. 4. 50 if 0 ≤ a < 12 p(a) = 7. 00 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. p(a) = Holt Algebra 2 5. 50 if 0 ≤ a < 12 8. 00 if 12 ≤ a < 55 6. 00 if a ≥ 55