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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.

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SECTION 1.5

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  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

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