1. Holt McDougal Algebra 1 8-2 Characteristics of Quadratic Functions Warm Up Find the x-intercept of each linear function. 1. y = 2x – 32. 3. y = 3x + 6 Evaluate each quadratic function for the given input values. 4. y = –3x 2 + x – 2, when x = 2 5. y = x 2 + 2x + 3, when x = –1 –2 –12 2

2. Holt McDougal Algebra 1 8-2 Characteristics of Quadratic Functions If a function has no zeros or they are difficult to identify from a graph, you can use a formula to find the axis of symmetry. The formula works for all quadratic functions.

3. Holt McDougal Algebra 1 8-2 Characteristics of Quadratic Functions Example 3: Finding the Axis of Symmetry by Using the Formula Find the axis of symmetry of the graph of y = – 3x 2 + 10x + 9. Step 1. Find the values of a and b. y = –3x 2 + 10x + 9 a = –3, b = 10 Step 2. Use the formula. The axis of symmetry is

4. Holt McDougal Algebra 1 8-2 Characteristics of Quadratic Functions Check It Out! Example 3 Find the axis of symmetry of the graph of y = 2x 2 + x + 3. Step 1. Find the values of a and b. y = 2x 2 + 1x + 3 a = 2, b = 1 Step 2. Use the formula. The axis of symmetry is.

5. Holt McDougal Algebra 1 8-2 Characteristics of Quadratic Functions Once you have found the axis of symmetry, you can use it to identify the vertex.

6. Holt McDougal Algebra 1 8-2 Characteristics of Quadratic Functions Example 4A: Finding the Vertex of a Parabola Find the vertex. y = 0.25x 2 + 2x + 3 Step 1 Find the x-coordinate of the vertex. The zeros are –6 and –2. Step 2 Find the corresponding y-coordinate. y = 0.25x 2 + 2x + 3 = 0.25(–4) 2 + 2(–4) + 3 = –1 Step 3 Write the ordered pair. (–4, –1) Use the function rule. Substitute –4 for x. The vertex is (–4, –1).

7. Holt McDougal Algebra 1 8-2 Characteristics of Quadratic Functions Example 4B: Finding the Vertex of a Parabola Find the vertex. y = –3x 2 + 6x – 7 Step 1 Find the x-coordinate of the vertex. a = –3, b = 6 Identify a and b. Substitute –3 for a and 6 for b. The x-coordinate of the vertex is 1.

8. Holt McDougal Algebra 1 8-2 Characteristics of Quadratic Functions Example 4B Continued Find the vertex. Step 2 Find the corresponding y-coordinate. y = –3x 2 + 6x – 7 = –3(1) 2 + 6(1) – 7 = –3 + 6 – 7 = –4 Use the function rule. Substitute 1 for x. Step 3 Write the ordered pair. The vertex is (1, –4). y = –3x 2 + 6x – 7

9. Holt McDougal Algebra 1 8-2 Characteristics of Quadratic Functions Find the vertex. y = x 2 – 4x – 10 Step 1 Find the x-coordinate of the vertex. a = 1, b = –4 Identify a and b. Substitute 1 for a and –4 for b. The x-coordinate of the vertex is 2. Check It Out! Example 4

10. Holt McDougal Algebra 1 8-2 Characteristics of Quadratic Functions Find the vertex. Step 2 Find the corresponding y-coordinate. y = x 2 – 4x – 10 = (2) 2 – 4(2) – 10 = 4 – 8 – 10 = –14 Use the function rule. Substitute 2 for x. Step 3 Write the ordered pair. The vertex is (2, –14). y = x 2 – 4x – 10 Check It Out! Example 4 Continued

11. Holt McDougal Algebra 1 8-2 Characteristics of Quadratic Functions Check It Out! Example 5 The height of a small rise in a roller coaster track is modeled by f(x) = – 0.07x 2 + 0.42x + 6.37, where x is the distance in feet from a supported pole at ground level. Find the greatest height of the rise. Step 1 Find the x-coordinate. a = – 0.07, b= 0.42 Identify a and b. Substitute –0.07 for a and 0.42 for b.

12. Holt McDougal Algebra 1 8-2 Characteristics of Quadratic Functions Check It Out! Example 5 Continued Step 2 Find the corresponding y-coordinate. = –0.07(3) 2 + 0.42(3) + 6.37 f(x) = –0.07x 2 + 0.42x + 6.37 = 7 ft Use the function rule. Substitute 3 for x. The height of the rise is 7 ft.