Holt McDougal Algebra Characteristics of Quadratic Functions Warm Up Find the x-intercept of each linear function. 1. y = 2x – 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

Holt McDougal Algebra 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.

Holt McDougal Algebra 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 x + 9. Step 1. Find the values of a and b. y = –3x x + 9 a = –3, b = 10 Step 2. Use the formula. The axis of symmetry is

Holt McDougal Algebra 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.

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

Holt McDougal Algebra 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).

Holt McDougal Algebra 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.

Holt McDougal Algebra 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

Holt McDougal Algebra 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

Holt McDougal Algebra 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

Holt McDougal Algebra 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 x , 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.

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