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1.2A Stretches

1.2A Stretches. The graph of y + 3 = f(x) is the graph of f ( x ) translated …  up 3 units  left 3 units  down 3 units  right 3 units. x. 2. The graph of f(x) + 4 is the graph of f ( x ) translated …

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1.2A Stretches

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  1. 1.2A Stretches The graph of y + 3 = f(x) is the graph of f(x) translated… up 3 units left 3 units down 3 units  right 3 units x 2. The graph of f(x) + 4 is the graph of f(x) translated… 4 units up  4 units left 4 units down  4 units right x 3. The graph of f(x – 7) + 6 is the graph of f(x) that has been translated..  7 units left, 6 units up 7 units left, 6 units down 7 units right, 6 units down  7 units right, 6 units up x 4. In general, the graph of f(x – h) + k, where hand k are positive, as compared to the parent function graph f(x), is translated h units left and k units up h units right and k units up h units left and k units down h units right and k units down x Math 30-1

  2. 5. When the output of a function y = f(x) is multiplied by -1, the result, y = -f(x), is a reflection of the graph in the • x – axis • y – axis • line y = x. 6. When y = f(x) is transformed to y = f(-x), then (x, y) is transformed to • (-x, -y) • (x , -y) • (-x, y) Math 30-1

  3. Vertical Stretches: The effect of parameter a. A stretch changes the shape of the graph. (Translations changed the position of a graph.) Function Transformations: Vertical Stretch |a| describes a vertical stretch about the x-axis. An Invariant point is a point on a graph that remains unchanged after a transformation. 1.2A Vertical Stretches Math 30-1

  4. Vertical Stretches In general, for any function y = f(x), the graph of the function y =af(x) has been vertically stretched about the x-axis by a factor of |a| . The point (x, y) → (x, ay). Only the y coordinates are affected. Invariant points are on the line of stretch, the x-axis. are the x-intercepts. When |a| > 1, the points on the graph move farther away from the x-axis. Vertical stretch by a factor of 3 When |a|< 1, the points on the graph move closer to the x-axis. Vertical stretch by a factor of ⅓ Math 30-1

  5. Vertical Stretching y= af(x), |a| > 1 y = 2f(x) y = f(x) Avertical stretch about the x-axis by a factor of 2. Key Points (x, 2y) (-2, 0) → (-2, 0) (-1, -14) → (-1, -7) (1/2, 2) → (1/2, 1) Invariant Points Domain and Range (-2, 0) (0, 0) (1, 0) (2, 0) Math 30-1

  6. Vertical Stretches about the x-axis y = af(x), |a| < 1 Consider Write the equation of the function after a vertical stretch about the x-axis by a factor of ½. Write the transformation in function notation. Write the coordinates of the image of the point (-7, 7) (-7, 3.5) The point (x, y) maps to List any invariant points. How are the domain and range affected? Math 30-1

  7. The graph of g(x) is a transformation of f(x). Horizontal stretch By a factor of ½ (1, 0) (2, 0)→ (x, y)→(½x, y) Invariant Point On the y-axis Is the transformation a translation? Is the transformation a vertical stretch? Math 30-1

  8. Horizontal Stretches In general, for any function y = f(x), the graph of the function y = f(bx) has been horizontally stretched by a factor of . The point (x, y) → Only the x coordinates are affected. Invariant points are on the line of stretch, the y-axis. are the y-intercepts. When |b| > 1, the points on the graph move closer to the y-axis. Horizontal stretch by a factor of ⅓ When |b|< 1, the points on the graph move farther away from the x-axis. Horizontal stretch by a factor of 4 Math 30-1

  9. Characteristics of Horizontal Stretches about the y-axis Consider Write the equation of the function after a horizontal stretch about the y-axis by a factor of 3. Write the transformation in function notation. Can it be written in any other way? Write the coordinates of the image of the point (-3, 3 ) → (-9, 3) Write the coordinates of the image of the point (x, y) List any invariant points. How are the domain and range affected? Math 30-1

  10. Is This a Horizontal or Vertical Stretch of y = f(x)? The graph y = f(x) is stretchedvertically about the x-axis by a factor of . y = f(x) Math 30-1

  11. Has the graph been stretched Horizontally or Vertically? (1, 1) Math 30-1

  12. Zeros of a Function 1. What are the zeros of the function? x = -2, 0, 3 2. Use transformations to determine the zeros of the following functions. x = -1, 1, 4 x = -2, 0, 3 x = -4, 0, 6 Math 30-1

  13. Describing the Horizontal or Vertical Stretch of a Function Stretch vertically by a factor of about the x-axis. Stretched horizontally by a factor of about the y-axis. Stretched vertically by a factor of about the x-axis. Describe the transformation in words compared to the graph of a function y = f(x). a) y = f(3x) b) 3y = f(x) c) y = f( x) d) y = 2f(x) Stretched horizontally by a factor of 2 about the y-axis. Math 30-1

  14. Consider the graph of a function y = f(x) The transformation described by y = f(2x+4) is horizontal stretch about the y-axis by a factor of ½. The translation described by y = f(2x + 4) is horizontal shift of 4 units to the left. y = f(2(x + 2)) Translations must be factored Math 30-1

  15. Stating the Equation of y = af(kx) The graph of the function y = f(x) is transformed as described. Write the new equation in the form y = af(bx). Stretched horizontal by a factor of one-third about the y-axis, and stretched vertically about the x-axis by a factor of two. y = 2f(3x) Stretched horizontally by a factor of two about the y-axisand translated four units to the left. Math 30-1

  16. Assignment Page 28 2, 5a,b, 6, 7a,c, 8, 13, 14c, d Math 30-1

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