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## Quadratic Functions

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**Chapter 9**Quadratic Functions**Section 9.1**Graphing Quadratic Functions in Vertex Form**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**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**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**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**Example**Compare the graph of Solution Translating Graphs Vertical Translation (Up-Down Shifts)**Example**Compare the graph of with the graph of Solution Translating Graphs Horizontal Translation (Left-Right Shifts)**Example**Compare the graph of with the graph of Solution Translating Graphs Horizontal Translation (Left-Right Shifts)**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**Example**Sketch the graph of Solution Translating Graphs Sketching the Graph of a Quadratic Function**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**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**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**Domain and Range of a Quadratic Function**Solution Continued Sketching the Graph of a Quadratic Function**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**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**Property**Solution Continued • Solution is: • Check solution using graphing calculator Find a Quadratic Model in Vertex Form Sketching the Graph of a Quadratic Function**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**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**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