6.5 COMBINING TRANSFORMATIONS 1

Multiple Inside Changes Example 2 (a) Rewrite the function y = f(2x − 6) in the form y = f(B(x − h)). (b) Use the result to describe the graph of y = f(2x − 6) as the result of first applying a horizontal stretch or compression to the graph of f and then applying a horizontal shift. What is the stretch/compression factor? What is the shift? Solution (a) We have f(2x − 6) = f(2(x − 3)). (b) Thus, the graph of y = f(2x − 6) can be obtained from the graph of y = f(x) by first horizontally compressing it by a factor of ½ and then shifting horizontally right by h = 3 units. 2

Graphical Demonstration of Example 2: f(2x – 6) Multiple Inside Changes Final y = f(2(x – 3))=f(2x – 6) Original y = f(x) Intermediate y = f(2x) 3

Multiple Outside Changes Example 3 (a) The graph of a function is obtained from the graph of y = f(x) by first stretching vertically by a factor of 3 and then shifting upward by 2. Give a formula for the function in terms of f. (b) The graph of a function is obtained from the graph of y = f(x) by first shifting upward by 2 and then stretching vertically by a factor of 3. Give a formula for the function in terms of f. (c) Are the results of (a) and (b) the same? Solution Original y = f(x) Intermediate y = 3 f(x) Intermediate y = f(x) + 2 Final y = 3 f(x ) + 2 Final y = 3(f(x ) + 2) = 3f(x) + 6 (c) 3f(x) + 2 ≠ 3f(x) + 6 (a) (b) 4

Ordering Horizontal and Vertical Transformations For nonzero constants A, B, h and k, the graph of the function y = A f(B (x − h)) + k is obtained by applying the transformations to the graph of f(x) in the following order: Horizontal stretch/compression by a factor of 1/|B| Horizontal shift by h units Vertical stretch/compression by a factor of |A| Vertical shift by k units If A < 0, follow the vertical stretch/compression by a reflection about the x-axis. If B < 0, follow the horizontal stretch/compression by a reflection about the y-axis. For nonzero constants A, B, h and k, the graph of the function y = A f(B (x − h)) + k is obtained by applying the transformations to the graph of f(x) in the following order: Horizontal stretch/compression by a factor of 1/|B| Horizontal shift by h units Vertical stretch/compression by a factor of |A| Vertical shift by k units If A < 0, follow the vertical stretch/compression by a reflection about the x-axis. If B < 0, follow the horizontal stretch/compression by a reflection about the y-axis. 5

Multiple Transformations Example 5 (a) Let y = 5(x + 2) Determine the values of A, B, h, and k when y is put in the form y = Af(B(x − h)) + k with f(x) = x 2. List the transformations applied to f(x) = x 2 to give y = 5(x + 2) (b) Sketch a graph of y = 5(x + 2) 2 + 7, labeling the vertex. Solution (a)If f(x) = x 2, we see y = 5(x + 2) = 5f(x + 2) + 7, so A = 5, B = 1, h = −2, and k = 7. Based on the values of these constants, we carry out these steps: Horizontal shift 2 to the left of the graph of f(x) = x 2. Vertical stretch of the resulting graph by a factor of 5. Vertical shift of the resulting graph up by 7. Since B = 1, there is no horizontal compression, stretch, or reflection. y=x2y=x2 y = 5(x + 2) (0,0) (-2,7) (b) 6