Lesson 2.1 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 h and 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

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

Vertical Stretches In general, for any function y = f(x), the graph of the function y =a f(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

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

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

The graph of g(x) is a transformation of f(x). Horizontal stretch By a factor of ½ (2, 0)→ (1, 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

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

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

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

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

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

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

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

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-axis and translated four units to the left. Math 30-1

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

Write the equation of the function after a horizontal stretch about the y-axis by a factor of 3. Check Verify by substitution Math 30-1