1 / 26

Transforming Polynomial Functions: Understanding Reflections, Compressions, and Translations

This lesson focuses on transforming polynomial functions through various operations, including vertical/horizontal translations, reflections across axes, and compressions or stretches. Using examples like f(x) = x³ - 6 and transformations such as g(x) = f(x) - 2 or g(x) = -f(x), students will explore how to graph these transformations and understand their effects on the polynomial's shape. Problem sets are provided for guided practice, reinforcing the learning objectives and preparing students for further topics, such as exponential functions.

thane-whitney
Télécharger la présentation

Transforming Polynomial Functions: Understanding Reflections, Compressions, and Translations

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. Chapter 3 3-8 transforming polynomial functions

  2. SAT Problem of the day • Lines l and m are perpendicular lines that intersect at the origin. If line l passes through the point (2,-1), then line m must pass through which of the following points? • A) (0,2) • B)(1,3) • C)(2,1) • D)(3,6) • E)(4,0)

  3. solution • Right Answer: D

  4. Objectives • Transform polynomial functions.

  5. Transforming polynomial functions • You can perform the same transformations on polynomial functions that you performed on quadratic and linear functions.

  6. Transforming Polynomial functions

  7. Example#1 • Translating polynomial • For f(x) = x3 – 6, write the rule for each function and sketch its graph. • g(x) = f(x) –2 • Solution: • To graph g(x) = f(x) – 2, translate the graph of f(x) 2 units down. • This is a vertical translation.

  8. Example#1 continue

  9. Example#2 • For f(x) = x3 – 6, write the rule for each function and sketch its graph. • h(x) = f(x + 3) • Solution: • To graph h(x) = f(x + 3), translate the graph 3 units to the left. • This is a horizontal translation.

  10. Example#2 continue

  11. Example#3 • For f(x) = x3 + 4, write the rule for each function and sketch its graph. • g(x) = f(x) –5 • Solution: • To graph g(x) = f(x) – 5, translate the graph of f(x) 5 units down. • This is a vertical translation.

  12. Example#3 continue

  13. Student guided practice • Do problems 1 and 4 in your book page 207

  14. Reflecting polynomial functions Example#4 • Let f(x) = x3 + 5x2 – 8x + 1. Write a function g that performs each transformation. • Reflect f(x) across the x-axis. • Solution : • g(x) = –f(x) • g(x) = –(x3 + 5x2 – 8x + 1) • g(x) = –x3 – 5x2 + 8x – 1

  15. Example#5 • Let f(x) = x3 + 5x2 – 8x + 1. Write a function g that performs each transformation. • Reflect f(x) across the y-axis. • Solution: • g(x) = f(–x) • g(x) = (–x)3 + 5(–x)2 – 8(–x) + 1 • g(x) = –x3 + 5x2 + 8x + 1

  16. Student guided practice Do problems 5 and 6 in your book page 207

  17. Do compressions/stretches • Let f(x) = 2x4 – 6x2 + 1. Graph f and g on the same coordinate plane. Describe g as a transformation of f. • Solution: • g(x) = 1/2f(x) • g(x) = 1/2 (2x4 – 6x2 + 1) • g(x) = x4 – 3x2+ 1/2 • g(x) is a vertical compression of f(x).

  18. Example continue

  19. Example • Let f(x) = 2x4 – 6x2 + 1. Graph f and g on the same coordinate plane. Describe g as a transformation of f. • g(x) = f( 1/3 x) • Solution: • g(x) = 2( 1/3x)4– 6(1/3x)2+ 1 • g(x) = 2/81x4– 2/3 x2+ 1 • g(x) is a horizontal stretch of f(x).

  20. Student guided practice • Do problems 7-9

  21. Combining transformations • Write a function that transforms f(x) = 6x3 – 3 in each of the following ways. Support your solution by using a graphing calculator. • Compress vertically by a factor of 1/3 , and shift 2 units right. • Solution: • g(x) = 1/3f(x – 2) • g(x) = 1/3(6(x – 2)3 – 3) • g(x) = 2(x – 2)3 – 1

  22. Write a function that transforms f(x) = 6x3 – 3 in each of the following ways. Support your solution by using a graphing calculator. • Reflect across the y-axis and shift 2 units down. • Solution: • g(x) = f(–x)– 2 • g(x) = (6(–x)3– 3) – 2 • g(x) = –6x3 – 5

  23. Student guided practice • Do problems 10-12 pg. 207

  24. Homework!! • Do problems 14-20 page 207 and 208 in your book

  25. Closure • Today we learn about transforming polynomial • Next class we are going to learn about Exponential functions , growth, and decay

More Related