Understanding Function Transformations: Multiplying Variables

1. Introduction Transformations can be made to functions in two distinct ways: by transforming the core variable of the function (multiplying the independent variable by k), and by transforming the function as a whole (multiplying the dependent variable by k). Previously, we saw how adding some constant k to the variable of a function or to the entire function affected the graph of the function. In this lesson, we will see how multiplying by a constant affects the graph of a function. Given f(x) and a constant k, we will observe the transformations f(k • x) and k • f(x). 5.8.2: Replacing f(x) with k• f(x) and f(k• x)

2. Key Concepts Graphing and Points of Interest In the graph of a function, there are key points of interest that define the graph and represent the characteristics of the function. When a function is transformed, the key points of the graph define the transformation. The key points in the graph of a quadratic equation are the vertex and the roots, or x-intercepts. 5.8.2: Replacing f(x) with k• f(x) and f(k• x)

3. Key Concepts, continued Multiplying the Dependent Variable by a Constant, k: k • f(x) In general, multiplying a function by a constant will stretch or shrink (compress) the graph of f vertically. If k > 1, the graph of f(x) will stretch vertically by a factor of k (so the parabola will appear narrower). A vertical stretch pulls the parabola and stretches it away from the x-axis. If 0 < k < 1, the graph of f(x)will shrink or compress vertically by a factor of k (so the parabola will appear wider). 5.8.2: Replacing f(x) with k• f(x) and f(k• x)

4. Key Concepts, continued A vertical compression squeezes the parabola toward the x-axis. If k < 0, the parabola will be first stretched or compressed and then reflected over the x-axis. The x-intercepts (roots) will remain the same, as will the x-coordinate of the vertex (the axis of symmetry). While k • f(x) = f(k • x) can be true, generally k • f(x) ≠ f(k • x). 5.8.2: Replacing f(x) with k• f(x) and f(k• x)

5. Key Concepts, continued 5.8.2: Replacing f(x) with k• f(x) and f(k• x)

6. Key Concepts, continued 5.8.2: Replacing f(x) with k• f(x) and f(k• x)

7. Key Concepts, continued 5.8.2: Replacing f(x) with k• f(x) and f(k• x)

8. Key Concepts, continued Multiplying the Independent Variable by a Constant, k: f(k • x) In general, multiplying the independent variable in a function by a constant will stretch or shrink the graph of f horizontally. If k > 1, the graph of f(x) will shrink or compress horizontally by a factor of (so the parabola will appear narrower). A horizontal compression squeezes the parabola toward the y-axis. 5.8.2: Replacing f(x) with k• f(x) and f(k• x)

9. Key Concepts, continued If 0 < k < 1, the graph of f(x) will stretch horizontally by a factor of (so the parabola will appear wider). A horizontal stretch pulls the parabola and stretches it away from the y-axis. If k < 0, the graph is first horizontally stretched or compressed and then reflected over the y-axis. The y-intercept remains the same, as does the y-coordinate of the vertex. 5.8.2: Replacing f(x) with k• f(x) and f(k• x)

10. Key Concepts, continued When a constant k is multiplied by the variable x of a function f(x), the interval of the intercepts of the function is increased or decreased depending on the value of k. The roots of the equation ax2 + bx + c = 0 are given by the quadratic formula, Remember that in the standard form of an equation, ax2 + bx + c, the only variable is x; a, b, and c represent constants. 5.8.2: Replacing f(x) with k• f(x) and f(k• x)

11. Key Concepts, continued If we were to multiply x in the equation ax2 + bx + c by a constant k, we would arrive at the following: Use the quadratic formula to find the roots of , as shown on the following slide. 5.8.2: Replacing f(x) with k• f(x) and f(k• x)

12. Key Concepts, continued Quadratic equation Substitute ak2 for a and bkfor b based on the equation in standard form. Simplify. 5.8.2: Replacing f(x) with k• f(x) and f(k• x)

13. Key Concepts, continued 5.8.2: Replacing f(x) with k• f(x) and f(k• x)

14. Key Concepts, continued 5.8.2: Replacing f(x) with k• f(x) and f(k• x)

15. Key Concepts, continued 5.8.2: Replacing f(x) with k• f(x) and f(k• x)

16. Common Errors/Misconceptions thinking that multiplying the dependent variable by a constant k yields the same equation as multiplying the independent variable by the same constant k in all cases (i.e., that k • f(x) is always equal to f(k • x), but this is not always true) forgetting to substitute all values of x with k • x when working with the transformation f(k • x) forgetting to square the constant k when substituting f(k • x) into ax2 + bx + c confusing horizontal with vertical transformations and vice versa confusing stretches and compressions 5.8.2: Replacing f(x) with k• f(x) and f(k• x)

17. Guided Practice Example 1 Consider the function f(x) = x2, its graph, and the constant k = 2. What is k • f(x)? How are the graphs of f(x) and k • f(x) different? How are they the same? 5.8.2: Replacing f(x) with k• f(x) and f(k• x)

18. Guided Practice: Example 1, continued Substitute the value of k into the function. If f(x) = x2 and k = 2, then k • f(x) = 2 • f(x) = 2x2. 5.8.2: Replacing f(x) with k• f(x) and f(k• x)

19. Guided Practice: Example 1, continued Use a table of values to graph the functions. 5.8.2: Replacing f(x) with k• f(x) and f(k• x)

20. Guided Practice: Example 1, continued 5.8.2: Replacing f(x) with k• f(x) and f(k• x)

21. Guided Practice: Example 1, continued Compare the graphs. Notice the position of the vertex has not changed in the transformation of f(x). Therefore, both equations have same root, x = 0. However, notice the inner graph, 2x2, is more narrow than x2because the value of 2 • f(x) is increasing twice as fast as the value of f(x). Since k > 1, the graph of f(x) will stretch vertically by a factor of 2. The parabola appears narrower. ✔ 5.8.2: Replacing f(x) with k• f(x) and f(k• x)

22. Guided Practice: Example 1, continued 5.8.2: Replacing f(x) with k• f(x) and f(k• x)

23. Guided Practice Example 2 Consider the function f(x) = x2 – 81, its graph, and the constant k = 3. What is f(k • x)? How do the vertices and the zeros of f(x) and f(k • x) compare? 5.8.2: Replacing f(x) with k• f(x) and f(k• x)

24. Guided Practice: Example 2, continued Substitute the value of k into the function. If f(x) = x2 – 81 and k = 3, then f(k • x) = f(3x) = (3x)2– 81 = 9x2– 81. 5.8.2: Replacing f(x) with k• f(x) and f(k• x)

25. Guided Practice: Example 2, continued Use the zeros and the vertex of f(x) to graph the function. To find the zeros of f(x), set the function equal to 0 and factor. 5.8.2: Replacing f(x) with k• f(x) and f(k• x)

26. Guided Practice: Example 2, continued The zeros are –9 and 9. 5.8.2: Replacing f(x) with k• f(x) and f(k• x)

27. Guided Practice: Example 2, continued The vertex of f(x) is (0, –81). This can be seen as the translation of the parent function f(x) = x2. The parent function is translated down 81 units. 5.8.2: Replacing f(x) with k• f(x) and f(k• x)

28. Guided Practice: Example 2, continued Using the zeros and the vertex of the transformed function, graph the new function on the same coordinate plane. Set the transformed function equal to 0 and factor to find the zeros. 5.8.2: Replacing f(x) with k• f(x) and f(k• x)

29. Guided Practice: Example 2, continued The zeros are –3 and 3. 5.8.2: Replacing f(x) with k• f(x) and f(k• x)

30. Guided Practice: Example 2, continued The vertex is the same as the original function, (0, –81). This again can be seen as the translation of 9x2 down 81 units. 5.8.2: Replacing f(x) with k• f(x) and f(k• x)

31. Guided Practice: Example 2, continued Compare the graphs. Notice the position of the vertex has not changed in the transformation of f(x). However, notice the inner graph, f(3x) = 9x2 – 81, is narrower than f(x). Specifically, the roots are x = –9 and 9 for f(x) and x = –3 and 3 for f(3x). This is because the roots of the function f(k • x) are times the roots of f(x) in a quadratic equation. 5.8.2: Replacing f(x) with k• f(x) and f(k• x)

32. Guided Practice: Example 2, continued Since k > 1, the graph of f(x) will shrink horizontally by a factor of , so the parabola appears narrower. ✔ 5.8.2: Replacing f(x) with k• f(x) and f(k• x)

33. Guided Practice: Example 2, continued 5.8.2: Replacing f(x) with k• f(x) and f(k• x)