1. Holt McDougal Algebra 1 8-3 Graphing Quadratic Functions Graph a quadratic function in the form y = ax 2 + bx + c. Objective

2. Holt McDougal Algebra 1 8-3 Graphing Quadratic Functions PAGE 538 Recall that a y-intercept is the y-coordinate of the point where a graph intersects the y-axis. The x-coordinate of this point is always 0. For a quadratic function written in the form y = ax 2 + bx + c, when x = 0, y = c. So the y-intercept of a quadratic function is c.

3. Holt McDougal Algebra 1 8-3 Graphing Quadratic Functions Example 1: Graphing a Quadratic Function Graph y = 3x 2 – 6x + 1. Step 1 Find the axis of symmetry. = 1 The axis of symmetry is x = 1. Simplify. Use x =. Substitute 3 for a and –6 for b. Step 2 Find the vertex. y = 3x 2 – 6x + 1 = 3(1) 2 – 6(1) + 1 = 3 – 6 + 1 = –2= –2 The vertex is (1, –2). The x-coordinate of the vertex is 1. Substitute 1 for x. Simplify. The y-coordinate is –2.

4. Holt McDougal Algebra 1 8-3 Graphing Quadratic Functions Example 1 Continued Step 3 Find the y-intercept. y = 3x 2 – 6x + 1 The y-intercept is 1; the graph passes through (0, 1). Identify c.

5. Holt McDougal Algebra 1 8-3 Graphing Quadratic Functions Step 4 Find two more points on the same side of the axis of symmetry as the point containing the y-intercept. Since the axis of symmetry is x = 1, choose x- values less than 1. Let x = –1. y = 3(–1) 2 – 6(–1) + 1 = 3 + 6 + 1 = 10 Let x = –2. y = 3(–2) 2 – 6(–2) + 1 = 12 + 12 + 1 = 25 Substitute x-coordinates. Simplify. Two other points are (–1, 10) and (–2, 25). Example 1 Continued

6. Holt McDougal Algebra 1 8-3 Graphing Quadratic Functions Graph y = 3x 2 – 6x + 1. Step 5 Graph the axis of symmetry, the vertex, the point containing the y-intercept, and two other points. Step 6 Reflect the points across the axis of symmetry. Connect the points with a smooth curve. Example 1 Continued x = 1 (–2, 25) (–1, 10) (0, 1) (1, –2) x = 1 (–1, 10) (0, 1) (1, –2) (–2, 25)

7. Holt McDougal Algebra 1 8-3 Graphing Quadratic Functions Because a parabola is symmetrical, each point is the same number of units away from the axis of symmetry as its reflected point. Helpful Hint

8. Holt McDougal Algebra 1 8-3 Graphing Quadratic Functions Check It Out! Example 1b Graph the quadratic function. y + 6x = x 2 + 9 Step 1 Find the axis of symmetry. Simplify. Use x =. Substitute 1 for a and –6 for b. The axis of symmetry is x = 3. = 3= 3 y = x 2 – 6x + 9 Rewrite in standard form.

9. Holt McDougal Algebra 1 8-3 Graphing Quadratic Functions Step 2 Find the vertex. Simplify. Check It Out! Example 1b Continued = 9 – 18 + 9 = 0= 0 The vertex is (3, 0). The x-coordinate of the vertex is 3. Substitute 3 for x. The y-coordinate is 0. y = x 2 – 6x + 9 y = 3 2 – 6(3) + 9

10. Holt McDougal Algebra 1 8-3 Graphing Quadratic Functions Step 3 Find the y-intercept. y = x 2 – 6x + 9 The y-intercept is 9; the graph passes through (0, 9). Identify c. Check It Out! Example 1b Continued

11. Holt McDougal Algebra 1 8-3 Graphing Quadratic Functions Step 4 Find two more points on the same side of the axis of symmetry as the point containing the y- intercept. Since the axis of symmetry is x = 3, choose x-values less than 3. Let x = 2 y = 1(2) 2 – 6(2) + 9 = 4 – 12 + 9 = 1 Let x = 1 y = 1(1) 2 – 6(1) + 9 = 1 – 6 + 9 = 4 Substitute x-coordinates. Simplify. Two other points are (2, 1) and (1, 4). Check It Out! Example 1b Continued

12. Holt McDougal Algebra 1 8-3 Graphing Quadratic Functions Step 5 Graph the axis of symmetry, the vertex, the point containing the y-intercept, and two other points. Step 6 Reflect the points across the axis of symmetry. Connect the points with a smooth curve. y = x 2 – 6x + 9 Check It Out! Example 1b Continued x = 3 (3, 0) (0, 9) (2, 1) (1, 4) (0, 9) (1, 4) (2, 1) x = 3 (3, 0)

13. Holt McDougal Algebra 1 8-3 Graphing Quadratic Functions Check It Out! Example 2 As Molly dives into her pool, her height in feet above the water can be modeled by the function f(x) = –16x 2 + 16x + 12, where x is the time in seconds after she begins diving. Find the maximum height of her dive and the time it takes Molly to reach this height. Then find how long it takes her to reach the pool.

14. Holt McDougal Algebra 1 8-3 Graphing Quadratic Functions Step 1 Find the axis of symmetry. Use x =. Substitute –16 for a and 16 for b. Simplify. The axis of symmetry is x = 0.5. Check It Out! Example 2 Continued

15. Holt McDougal Algebra 1 8-3 Graphing Quadratic Functions Step 2 Find the vertex. f(x) = –16x 2 + 16x + 12 = –16(0.5) 2 + 16(0.5) +12 = –16(0.25) + 8 +12 = –4 + 20 = 16 The vertex is (0.5, 16). Simplify. The y-coordinate is 9. The x-coordinate of the vertex is 0.5. Substitute 0.5 for x. Check It Out! Example 2 Continued

16. Holt McDougal Algebra 1 8-3 Graphing Quadratic Functions Step 3 Find the y-intercept. Identify c. f(x) = –16x 2 + 16x + 12 The y-intercept is 12; the graph passes through (0, 12). Check It Out! Example 2 Continued

17. Holt McDougal Algebra 1 8-3 Graphing Quadratic Functions Step 4 Find another point on the same side of the axis of symmetry as the point containing the y-intercept. Since the axis of symmetry is x = 0.5, choose an x-value that is less than 0.5. Let x = 0.25 f(x) = –16(0.25) 2 + 16(0.25) + 12 = –1 + 4 + 12 = 15 Another point is (0.25, 15). Substitute 0.25 for x. Simplify. Check It Out! Example 2 Continued

18. Holt McDougal Algebra 1 8-3 Graphing Quadratic Functions Step 5 Graph the axis of symmetry, the vertex, the point containing the y-intercept, and the other point. Then reflect the points across the axis of symmetry. Connect the points with a smooth curve. Check It Out! Example 2 Continued

19. Holt McDougal Algebra 1 8-3 Graphing Quadratic Functions The vertex is (0.5, 16). So at 0.5 seconds, Molly's dive has reached its maximum height of 16 feet. The graph shows the zeros of the function are –0.5 and 1.5 seconds. At –0.5 seconds the dive has not begun, and at 1.5 seconds she reaches the pool. Molly reaches the pool in 1.5 seconds. Check It Out! Example 2 Continued

20. Holt McDougal Algebra 1 8-3 Graphing Quadratic Functions Look Back 4 Check by substituting (0.5, 16) and (1.5, 0) into the function. 16 = 16 Check It Out! Example 2 Continued 0 = 0 16 = –16(0.5) 2 + 16(0.5) + 12 ? 16 = –4 + 8 +12 ? 0 = –16(1.5) 2 + 16(1.5) +12 ? 0 = –36 + 24 + 12 ?