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2. Section 9.1 Graphing Quadratic Functions in Vertex Form

3. Property Let , where a ≠ 0. f is a quadratic function, and its graph is a parabola. We say that the equation is in vertex form. Vertex Form Zero Factor Property

4. Example Compare the graph of with the graph of List the input output pairs. Solution Graphs of Quadratic Functions of the Form f (x) = ax2 Stretching a Graph Vertically

5. Example Compare the graph of with the graph of List the input output pairs. Solution Graphs of Quadratic Functions of the Form f (x) = ax2 Reflecting a Graph across the x-Axis

6. Properties For a function of the form f (x) = ax2, • The graph is a parabola with vertex (0, 0). • If |a| is a large number, then the parabola is steep. • If a is near zero, then the parabola is not steep. • If a > 0, then the parabola opens upward. • If a < 0, then the parabola opens downward. The graph of y= −ax2is the reflection across the x-axis of the graph of f (x) = ax2. Graphs of Quadratic Functions of the Form f (x) = ax2 Graphs of Quadratics of the Form f (x) = ax2

7. Example Compare the graph of Solution Translating Graphs Vertical Translation (Up-Down Shifts)

8. Example Compare the graph of with the graph of Solution Translating Graphs Horizontal Translation (Left-Right Shifts)

9. Example Compare the graph of with the graph of Solution Translating Graphs Horizontal Translation (Left-Right Shifts)

10. Process To sketch the graph of f (x) = a(x − h)2 + k, where a = 0, 1. Sketch the graph of y = ax2. 2. Translate the graph from step 1 to the right by h units if h > 0 or to the left by |h| units if h < 0. 3. Translate the graph from step 2 up by k units if k > 0 or down by |k| units if k < 0. Translating Graphs Three-Step Method of Graphing a Quadratic Function in Vertex Form

11. Example Sketch the graph of Solution Translating Graphs Sketching the Graph of a Quadratic Function

12. Property The vertex of a quadratic function in vertex form, is the point (h, k). Sketch the graph of , and find the domain and range of f. First, rewrite in vertex form (see above) Example Solution Translating Graphs Sketching the Graph of a Quadratic Function

13. Solution Continued Step 1. Sketch the graph of Step 2. Since h = –6, translate graph from step 1 to the left by 6 units. Step 3. Since k = –2, translate graph from step 2 down by 2 units. Domain and Range of a Quadratic Function Sketching the Graph of a Quadratic Function

14. Property Example Sketch the graph of , and find the domain and range of h. Step 1. Sketch the graph of Step 2. Since h = 5, translate graph from step 1 to the left by 5 units. Step 3. Since k = 3, translate the graph from step 2 up by 3 units. Solution Translating Graphs Sketching the Graph of a Quadratic Function

15. Domain and Range of a Quadratic Function Solution Continued Sketching the Graph of a Quadratic Function

16. Property Example The numbers of flights with taxi-out times of one hour or more per 1000 flights are shown in Table 11 for various years. Let f (t) be the number of flights that took one hour or more to taxi out per 1000 flights at t years since 2000. 1. Find an equation of f . 2. What is the vertex of f ? What does it mean in this situation? 3. Use f to estimate the number of flights that took one hour or more to taxi out in 2006 Find a Quadratic Model in Vertex Form Sketching the Graph of a Quadratic Function

17. Property Solution Use a graphing calculator to draw a scattergram of the data It appears to be quadratic • It’s not necessary, however it’s convenient to select the lowest point, (2, 6.8) as the vertex (h, k) • Imagine a parabola with vertex (2, 6.8) and goes through point (4, 9.7) Find a Quadratic Model in Vertex Form Sketching the Graph of a Quadratic Function

18. Property Solution Continued • Solution is: • Check solution using graphing calculator Find a Quadratic Model in Vertex Form Sketching the Graph of a Quadratic Function

19. Property Solution Continued • Vertex is (2, 6.8), which means that in 2002 there were 6.5 flights that took one hour or more to taxi out per 1000 flights • Represent 2006 by t = 6 • Evaluate f for t = 6: • About 18.6 flights took one hour or more to taxi out per 1000 flights in 2006 Find a Quadratic Model in Vertex Form Sketching the Graph of a Quadratic Function

20. Process To find a quadratic model in vertex form, given some data, 1. Create a scattergram of the data. 2. Imagine a parabola that comes close to (or contains) the data points, and select a point (h, k) to be the vertex. Although it is not necessary to select a data point, it is often convenient and satisfactory to do so. Find a Quadratic Model in Vertex Form Sketching the Graph of a Quadratic Function

21. Process 3. Select a nonvertex point (not necessarily a data point) of the parabola, substitute the point’s coordinates into the equation and solve for a. 4. Substitute the result you found for a in step 3 into Find a Quadratic Model in Vertex Form Sketching the Graph of a Quadratic Function