1 / 52

Describing Functions using Graphs

Learn how to describe the relationships between functions by observing their graphs and applying transformations. Topics include translations, compressions, and vertical/horizontal stretches.

mkrista
Download Presentation

Describing Functions using Graphs

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Use the graph of y = x2to describe the graph of the related function y = 0.5x2. A.The parent graph is translated up 0.5 units. B.The parent graph is compressed horizontally by a factor of 0.5. C.The parent graph is compressed vertically by a factor of 0.5. D.The parent graph is translated down 0.5 units. 5–Minute Check 1

  2. Use the graph of y = x2to describe the graph of the related function y = 0.5x2. A.The parent graph is translated up 0.5 units. B.The parent graph is compressed horizontally by a factor of 0.5. C.The parent graph is compressed vertically by a factor of 0.5. D.The parent graph is translated down 0.5 units. 5–Minute Check 1

  3. Use the graph of y = x2to describe the graph of the related function y = (x – 4)2– 3. A.The parent graph is translated left 3 units and up 4 units. B.The parent graph is translated right 3 units and down 4 units. C.The parent graph is translated left 4 units and down 3 units. D.The parent graph is translated right 4 units and down 3 units. 5–Minute Check 2

  4. Use the graph of y = x2to describe the graph of the related function y = (x – 4)2– 3. A.The parent graph is translated left 3 units and up 4 units. B.The parent graph is translated right 3 units and down 4 units. C.The parent graph is translated left 4 units and down 3 units. D.The parent graph is translated right 4 units and down 3 units. 5–Minute Check 2

  5. A. B. C. D. 5–Minute Check 3

  6. A. B. C. D. 5–Minute Check 3

  7. Identify the parent function f(x) if and describe how the graphs of g(x) and f(x) are related. A.f(x) = x; g(x) is f(x) translated left 4 units. B.f(x) = |x|; g(x) is f(x) translated right 4 units. C.g(x) is f(x) translated right 4 units. D.g(x) is f(x) translated left 4 units. 5–Minute Check 4

  8. Identify the parent function f(x) if and describe how the graphs of g(x) and f(x) are related. A.f(x) = x; g(x) is f(x) translated left 4 units. B.f(x) = |x|; g(x) is f(x) translated right 4 units. C.g(x) is f(x) translated right 4 units. D.g(x) is f(x) translated left 4 units. 5–Minute Check 4

  9. Key Concept 1

  10. The domain of f and g are both so the domain of (f + g) is Operations with Functions A. Given f(x) = x2 – 2x, g(x) = 3x – 4, and h(x) = –2x2 + 1, find the function and domain for (f + g)(x). (f + g)(x) = f(x) + g(x) Definition of sum oftwo functions = (x2 – 2x) + (3x – 4) f(x) = x2 – 2x; g(x) = 3x – 4 = x2 + x – 4 Simplify. Answer: Example 1

  11. The domain of f and g are both so the domain of (f + g) is Answer: Operations with Functions A. Given f(x) = x2 – 2x, g(x) = 3x – 4, and h(x) = –2x2 + 1, find the function and domain for (f + g)(x). (f + g)(x) = f(x) + g(x) Definition of sum oftwo functions = (x2 – 2x) + (3x – 4) f(x) = x2 – 2x; g(x) = 3x – 4 = x2 + x – 4 Simplify. Example 1

  12. The domain of f and h are both so the domain of (f – h) is Operations with Functions B. Given f(x) = x2 – 2x, g(x) = 3x – 4, and h(x) = –2x2 + 1, find the function and domain for (f – h)(x). (f – h)(x) = f(x) – h(x) Definition of difference of two functions = (x2 – 2x) – (–2x2 + 1) f(x) = x2 – 2x; h(x) = –2x2 + 1 = 3x2 – 2x – 1 Simplify. Answer: Example 1

  13. The domain of f and h are both so the domain of (f – h) is Answer: Operations with Functions B. Given f(x) = x2 – 2x, g(x) = 3x – 4, and h(x) = –2x2 + 1, find the function and domain for (f – h)(x). (f – h)(x) = f(x) – h(x) Definition of difference of two functions = (x2 – 2x) – (–2x2 + 1) f(x) = x2 – 2x; h(x) = –2x2 + 1 = 3x2 – 2x – 1 Simplify. Example 1

  14. The domain of f and g are both so the domain of (f ● g) is Operations with Functions C. Given f(x) = x2 – 2x, g(x) = 3x – 4, and h(x) = –2x2 + 1, find the function and domain for (f ● g)(x). (f ● g)(x) = f(x) ● g(x) Definition of product of two functions = (x2 – 2x)(3x – 4) f(x) = x2 – 2x; g(x) = 3x – 4 = 3x3 – 10x2 + 8x Simplify. Answer: Example 1

  15. The domain of f and g are both so the domain of (f ● g) is Answer: Operations with Functions C. Given f(x) = x2 – 2x, g(x) = 3x – 4, and h(x) = –2x2 + 1, find the function and domain for (f ● g)(x). (f ● g)(x) = f(x) ● g(x) Definition of product of two functions = (x2 – 2x)(3x – 4) f(x) = x2 – 2x; g(x) = 3x – 4 = 3x3 – 10x2 + 8x Simplify. Example 1

  16. D. Given f(x) = x2 – 2x, g(x) = 3x – 4, and h(x) = –2x2 + 1, find the function and domain for Operations with Functions Definition of quotient of two functions f(x) = x2 – 2x; h(x) = –2x2 + 1 Example 1

  17. The domains of h and f are both (–∞, ∞), but x = 0 or x = 2 yields a zero in the denominator of . So, the domain of (–∞, 0) È (0, 2) È (2, ∞). Operations with Functions Answer: Example 1

  18. The domains of h and f are both (–∞, ∞), but x = 0 or x = 2 yields a zero in the denominator of . So, the domain of (–∞, 0) È (0, 2) È (2, ∞). Answer: D = (–∞, 0) È (0, 2) È (2, ∞) Operations with Functions Example 1

  19. Find (f + g)(x), (f – g)(x), (f ● g)(x), and for f(x) = x2 + x, g(x) = x – 3. State the domain of each new function. Example 1

  20. A. B. C. D. Example 1

  21. A. B. C. D. Example 1

  22. Key Concept 2

  23. = 2(x2 + 6x + 9) – 1 Expand (x +3)2 = 2x2 + 12x + 17 Simplify. = f(x + 3) Replace g(x) with x + 3 = 2(x + 3)2 – 1 Substitute x + 3 for x in f(x). Compose Two Functions A. Given f(x) = 2x2 – 1 and g(x) = x + 3, find [f ○ g](x). Answer: Example 2

  24. = 2(x2 + 6x + 9) – 1 Expand (x +3)2 = 2x2 + 12x + 17 Simplify. = f(x + 3) Replace g(x) with x + 3 = 2(x + 3)2 – 1 Substitute x + 3 for x in f(x). Compose Two Functions A. Given f(x) = 2x2 – 1 and g(x) = x + 3, find [f ○ g](x). Answer: [f ○ g](x) = 2x2 + 12x + 17 Example 2

  25. = (2x2 – 1) + 3 = 2x2 + 2 Substitute 2x2 – 1 for x in g(x). Simplify Compose Two Functions B. Given f(x) = 2x2 – 1 and g(x) = x + 3, find [g ○ f](x). Answer: Example 2

  26. = (2x2 – 1) + 3 = 2x2 + 2 Substitute 2x2 – 1 for x in g(x). Simplify Compose Two Functions B. Given f(x) = 2x2 – 1 and g(x) = x + 3, find [g ○ f](x). Answer: [g ○ f](x) = 2x2 + 2 Example 2

  27. Compose Two Functions C. Given f(x) = 2x2 – 1 and g(x) = x + 3, find [f ○ g](2). Evaluate the expression you wrote in part A for x = 2. [f ○ g](2) = 2(2)2 + 12(2) + 17 Substitute 2 for x. = 49 Simplify. Answer: Example 2

  28. Compose Two Functions C. Given f(x) = 2x2 – 1 and g(x) = x + 3, find [f ○ g](2). Evaluate the expression you wrote in part A for x = 2. [f ○ g](2) = 2(2)2 + 12(2) + 17 Substitute 2 for x. = 49 Simplify. Answer: [f ○ g](2) = 49 Example 2

  29. Find for f(x) = 2x – 3 and g(x) = 4 + x2. A. 2x2 + 11; 4x2 – 12x + 13; 23 B. 2x2 + 11; 4x2 – 12x + 5; 23 C. 2x2 + 5; 4x2 – 12x + 5; 23 D. 2x2 + 5; 4x2 – 12x + 13; 23 Example 2

  30. Find for f(x) = 2x – 3 and g(x) = 4 + x2. A. 2x2 + 11; 4x2 – 12x + 13; 23 B. 2x2 + 11; 4x2 – 12x + 5; 23 C. 2x2 + 5; 4x2 – 12x + 5; 23 D. 2x2 + 5; 4x2 – 12x + 13; 23 Example 2

  31. A. Find . Find a Composite Function with a Restricted Domain Example 3

  32. To find , you must first be able to find g(x) = (x – 1)2, which can be done for all real numbers. Then you must be able to evaluate for each of these g(x)-values, which can only be done when g(x) > 1. Excluding from the domain those values for which 0 < (x – 1)2 <1, namely when 0 < x < 1, the domain of f ○ g is (–∞, 0] È [2, ∞). Now find [f ○ g](x). Find a Composite Function with a Restricted Domain Example 3

  33. Notice that is not defined for 0 < x < 2. Because the implied domain is the same as the domain determined by considering the domains of f and g, we can write the composition as for (–∞, 0] È [2, ∞). Find a Composite Function with a Restricted Domain Replace g(x) with (x – 1)2. Substitute (x – 1)2 for x in f(x). Simplify. Example 3

  34. Find a Composite Function with a Restricted Domain Answer: Example 3

  35. Answer: for (–∞, 0] È [2, ∞). Find a Composite Function with a Restricted Domain Example 3

  36. B. Find f ○ g. Find a Composite Function with a Restricted Domain Example 3

  37. To find f ○ g, you must first be able to find , which can be done for all real numbers x such that x2 1. Then you must be able to evaluate for each of these g(x)-values, which can only be done when g(x)  0. Excluding from the domain those values for which 0 >x2 – 1, namely when –1 < x< 1, the domain of f ○ g is (–∞, –1) È (1, ∞). Now find [f ○ g](x). Find a Composite Function with a Restricted Domain Example 3

  38. Find a Composite Function with a Restricted Domain Example 3

  39. Find a Composite Function with a Restricted Domain Answer: Example 3

  40. Answer: Find a Composite Function with a Restricted Domain Example 3

  41. Find f ○ g. A. D =(–∞, –1)  (–1, 1)  (1, ∞); B. D =[–1, 1]; C. D =(–∞, –1)  (–1, 1)  (1, ∞); D. D =(0, 1); Example 3

  42. Find f ○ g. A. D =(–∞, –1)  (–1, 1)  (1, ∞); B. D =[–1, 1]; C. D =(–∞, –1)  (–1, 1)  (1, ∞); D. D =(0, 1); Example 3

  43. B. Find two functions f and g such that when h(x) = 3x2 – 12x + 12. Neither function may be the identity function f(x) = x. Decompose a Composite Function h(x) = 3x2 – 12x + 12 Notice that h is factorable. = 3(x2 – 4x + 4) or 3(x – 2)2 Factor. One way to write h(x) as a composition is to let f(x) = 3x2 and g(x) = x – 2. Example 4

  44. Decompose a Composite Function Sample answer: Example 4

  45. Decompose a Composite Function Sample answer:g(x) = x – 2 and f(x) = 3x2 Example 4

  46. A. B. C. D. Example 4

  47. A. B. C. D. Example 4

  48. Compose Real-World Functions A. COMPUTER ANIMATION An animator starts with an image of a circle with a radius of 25 pixels. The animator then increases the radius by 10 pixels per second. Find functions to model the data. The length r of the radius increases at a rate of 10 pixels per second, so R(t) = 25 + 10t, where t is the time in seconds and t  0. The area of the circle is  times the square of the radius. So, the area of the circle is A(R) = R2. So, the functions are R(t) = 25 + 10t and A(R) = R2. Answer: Example 5

  49. Compose Real-World Functions A. COMPUTER ANIMATION An animator starts with an image of a circle with a radius of 25 pixels. The animator then increases the radius by 10 pixels per second. Find functions to model the data. The length r of the radius increases at a rate of 10 pixels per second, so R(t) = 25 + 10t, where t is the time in seconds and t  0. The area of the circle is  times the square of the radius. So, the area of the circle is A(R) = R2. So, the functions are R(t) = 25 + 10t and A(R) = R2. Answer:R(t) = 25 + 10t; A(R) = R2 Example 5

  50. Compose Real-World Functions B. COMPUTER ANIMATION An animator starts with an image of a circle with a radius of 25 pixels. The animator then increases the radius by 10 pixels per second. Find A ○ R. What does the function represent? A ○ R= A[R(t)] Definition of A ○ R =A(25 + 10t) Replace R(t) with 25 + 10t. = (25 + 10t)2 Substitute (25 + 10t) for R in A(R). = 100t2+ 500t + 625 Simplify. Example 5

More Related