1 / 23

SECTION 1.5

SECTION 1.5. GRAPHING TECHNIQUES; TRANSFORMATIONS. TRANSFORMATIONS. Recall our “library” of functions. Here we will learn techniques for graphing a function which is “related” to one we already know how to graph. HORIZONTAL SHIFTS. On the same screen, graph each of the following functions:

max
Download Presentation

SECTION 1.5

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. SECTION 1.5 • GRAPHING TECHNIQUES; • TRANSFORMATIONS

  2. TRANSFORMATIONS Recall our “library” of functions. Here we will learn techniques for graphing a function which is “related” to one we already know how to graph.

  3. HORIZONTAL SHIFTS On the same screen, graph each of the following functions: Y1 = x2 Y2 = (x - 1)2 Y3 = (x - 3)2 Y4 = (x + 2)2

  4. COMPARING y = x 2 and y = (x - 2) 2 If we named the first function f(x), we could denote the second one by f(x - 2). In general, we can refer to any horizontal shift of a function f(x) by using the notation f(x - h)

  5. y = f(x - 2) y = f(x + 3) • When h is positive (that is, when there is a value being subtracted from x) the shift is to the right. • When h is negative (that is, when there is a value being added to x) the shift is to the left.

  6. VERTICAL SHIFTS In general, we can refer to any vertical shift of a function f(x) by using the notation: f(x) + k

  7. y = f(x) + 4 y = f(x) - 1 When k is positive, the shift is upward. When k is negative, the shift is downward.

  8. EXAMPLE: The figure shows the graph of f(x). Sketch the graphs of f(x + 1) and f(x) - 1. - 2 - 1 1 2

  9. y = f(x + 1) - 3 - 2 - 1 1 2

  10. y = f(x) - 1

  11. VERTICAL STRETCHES • When we compare the graph of y = x2 to the graph of y = 2x2, we find the second one is more narrow than the first. • This is called a vertical stretch. All the y-values are being doubled.

  12. VERTICAL SHRINKS • When we compare the graph of y = x2 to the graph of y = .5x2, we find the second one is wider than the first. • This is called a vertical shrink. All the y-values are being halved.

  13. In general, we can denote vertical stretches and shrinks to a function f(x) in the following way: For ïa ï > 1, stretch y = af(x) For 0 < ïa ï < 1, shrink

  14. EXAMPLE: • Sketch the graphs of y = 3f(x), y = .5f(x), and y = - .5f(x)

  15. y = 3f (x) - 2 - 1 1 2

  16. y = .5f (x)

  17. y = - .5f (x)

  18. HORIZONTAL STRETCHES AND SHRINKS In general, we can denote horizontal stretches and shrinks to a function f(x) in the following way: For c > 1, shrink y = f(cx) For 0 < c < 1, stretch

  19. EXAMPLE: • Sketch the graphs of y = f(2x) and y = f(.5x)

  20. y = f (2x)

  21. y = f (.5x)

  22. EXAMPLE: Given f(x) = x 3 - 4x, explain the transformations that will occur to the graph of the function for f(2x) + 3 The graph will be compressed horizontally and shifted 3 units up. Graph it!

  23. CONCLUSION OF SECTION 1.5

More Related