1. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 1 Chapter 3 Quadratic Functions and Equations

2. 2 Copyright © 2014, 2010, 2006 Pearson Education, Inc. Transformations of Graphs ♦ Graph functions using vertical and horizontal shifts ♦ Graph functions using stretching and shrinking ♦ Graph functions using reflections ♦ Combine transformations ♦ Model data with transformations (optional) 3.5

3. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 3 Vertical and Horizontal Shifts We use these two graphs to demonstrate shifts, or translations, in the xy-plane.

4. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 4 Vertical Shifts A graph is shifted up or down. The shape of the graph is not changed—only its position. Every point moves upward 2.

5. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 5 Horizontal Shifts A graph is shifted right: replace x with (x – 2) Every point moves right 2.

6. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 6 Horizontal Shifts A graph is shifted left: replace x with (x + 3), Every point moves left 3.

7. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 7 Vertical and Horizontal Shifts Let f be a function, and let c be a positive number.

8. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 8 Combining Shifts Shifts can be combined to translate a graph of y = f(x) both vertically and horizontally. Shift the graph of y = |x| to the right 2 units and downward 4 units. y = |x| y = |x – 2| y = |x – 2|  4

9. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 9 Example: Combining vertical and horizontal shifts Complete the following. (a) Write an equation that shifts the graph of f(x) = x 2 left 2 units. Graph your equation. (b) Write an equation that shifts the graph of f(x) = x 2 left 2 units and downward 3 units. Graph your equation. Solution To shift the graph left 2 units, replace x with x + 2.

10. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 10 Example: Combining vertical and horizontal shifts (b) Write an equation that shifts the graph of f(x) = x 2 left 2 units and downward 3 units. Graph your equation. Solution To shift the graph left 2 units, and downward 3 units, we subtract 3 from the equation found in part (a).

11. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 11 Vertical Stretching and Shrinking If the point (x, y) lies on the graph of y = f(x), then the point (x, cy) lies on the graph of y = cf(x). If c > 1, the graph of y = cf(x) is a vertical stretching of the graph of y = f(x), whereas if 0 < c < 1 the graph of y = cf(x) is a vertical shrinking of the graph of y = f(x).

12. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 12 Vertical Stretching and Shrinking

13. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 13 Horizontal Stretching and Shrinking If the point (x, y) lies on the graph of y = f(x), then the point (x/c, y) lies on the graph of y = f(cx). If c > 1, the graph of y = f(cx) is a horizontal shrinking of the graph of y = f(x), whereas if 0 < c < 1 the graph of y = f(cx) is a horizontal stretching of the graph of y = f(x).

14. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 14 Horizontal Stretching and Shrinking

15. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 15 Example: Stretching and shrinking of a graph Use the graph of y = f(x) to sketch the graph of each equation. a) y = 3f(x) b)

16. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 16 Example: Stretching and shrinking of a graph Solution a) y = 3f(x) Vertical stretching Multiply each y-coordinate on the graph by 3. (  1, –2  3) = (  1, –6) (0, 1  3) = (0, 3) (2, –1  3) = (2, –3)

17. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 17 Example: Stretching and shrinking of a graph Solution continued b) Horizontal stretching Multiply each x-coordinate on the graph by 2 or divide by ½. (  1  2, –2) = (  2, –2) (0  2, 1) = (0, 1) (2  2, –1) = (4, –1)

18. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 18 Reflection of Graphs Across the x- and y-Axes 1. The graph of y = –f(x) is a reflection of the graph of y = f(x) across the x-axis. 2. The graph of y = f(–x) is a reflection of the graph of y = f(x) across the y-axis.

19. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 19 Reflection of Graphs Across the x- and y-axes

20. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 20 Example: Reflecting graphs of functions For the representation of f, graph the reflection across the x-axis and across the y- axis. The graph of f is a line graph determined by the table.

21. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 21 Example: Reflecting graphs of functions Solution Here’s the graph of y = f(x).

22. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 22 Example: Reflecting graphs of functions Solution continued To graph the reflection of f across the x-axis, start by making a table of values for y = –f(x) by negating each y-value in the table for f(x).

23. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 23 Example: Reflecting graphs of functions Solution continued To graph the reflection of f across the y-axis, start by making a table of values for y = f(–x) by negating each x-value in the table for f(x).

24. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 24 Combining Transformations Transformations of graphs can be combined to create new graphs. For example the graph of y =  2(x – 1) 2 + 3 can be obtained by performing four transformations on the graph of y = x 2.

25. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 25 Combining Transformations 1. Shift the graph 1 unit right: y = (x – 1) 2 2. Vertically stretch the graph by factor of 2: y = 2(x – 1) 2 3. Reflect the graph across the x-axis: y =  2(x – 1) 2 4. Shift the graph upward 3 units: y =  2(x – 1) 2 + 3

26. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 26 Combining Transformations continued y =  2(x – 1) 2 + 3 Shift to the left 1 unit. Shift upward 3 units. Reflect across the x-axis. Stretch vertically by a factor of 2

27. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 27 Combining Transformations The graphs of the four transformations.

28. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 28 Combining Transformations The graphs of the four transformations.

29. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 29 Example: Combining transformations of graphs Describe how the graph of each equation can be obtained by transforming the graph of Then graph the equation.

30. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 30 Example: Combining transformations of graphs Solution Vertically shrink the graph by factor of 1/2 then reflect it across the x-axis.

31. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 31 Example: Combining transformations of graphs Solution continued Reflect it across the y-axis. Shift left 2 units. Shift down 1 unit.

32. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 32 Modeling Data with Transformations Transformations of the graph of y = x 2 can be used to model some types of nonlinear data. By shifting, stretching, and shrinking this graph, we can transform it into a portion of a parabola that has the desired shape and location. In the next example we demonstrate this technique by modeling numbers of Wal-Mart employees.

33. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 33 Example: Modeling data with a quadratic function The table lists numbers of Wal-Mart employees in millions for selected years.

34. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 34 Example: Modeling data with a quadratic function (a) Make a scatterplot of the data. (b) Use transformations of graphs to determine f(x) =a(x – h) 2 + k so that f(x) models the data. Graph y = f(x) together with the data. (c) Use f(x) to estimate the number of Wal-Mart employees in 2010. Compare it with the actual value of 2.1 million employees.

35. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 35 Example: Modeling data with a quadratic function Solution Here’s a calculator display of a scatterplot of the data.

36. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 36 Example: Modeling data with a quadratic function Solution continued It’s a parabola opening up so a > 0. Vertex (minimum number of employees) could be (1987, 0.20): translate graph right 1987 units and up 0.20 unit. f(x) = a(x – 1987) + 0.20 To determine a, graph the data for different values of a: First graph a = 0.001 and a = 0.01.

37. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 37 Example: Modeling data with a quadratic function Solution continued From this graph we see the value of a is between 0.001 and 0.01.

38. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 38 Example: Modeling data with a quadratic function Solution continued Experimenting yields a value of a near 0.005. So f(x) = 0.005(x – 1987) + 0.20