8-2 Characteristics of Quadratic Functions Warm Up Lesson Presentation Lesson Quiz Holt McDougal Algebra 1 Holt Algebra 1

Warm Up 1. y = 2x – 3 2. 3. y = –3x2 + x – 2, when x = 2 Find the x-intercept of each linear function. 1. y = 2x – 3 2. Evaluate each quadratic function for the given input values. 3. y = –3x2 + x – 2, when x = 2

ESSENTIAL QUESTIONS 1. How do you find the zeros of a quadratic function from its graph? 2. How do you find the axis of symmetry and the vertex of a parabola?

Vocabulary zero of a function axis of symmetry

Recall that an x-intercept of a function is a value of x when y = 0. A zero of a function is an x-value that makes the function equal to 0.

Example 1A: Finding Zeros of Quadratic Functions From Graphs Find the zeros of the quadratic function from its graph. Check your answer. y = x2 – 2x – 3 y = (–1)2 – 2(–1) – 3 = 1 + 2 – 3 = 0 y = 32 –2(3) – 3 = 9 – 6 – 3 = 0 y = x2 – 2x – 3 Check  The zeros appear to be –1 and 3.

Example 1B: Finding Zeros of Quadratic Functions From Graphs Find the zeros of the quadratic function from its graph. Check your answer. y = x2 + 8x + 16 Check y = x2 + 8x + 16 y = (–4)2 + 8(–4) + 16 = 16 – 32 + 16 = 0  The zero appears to be –4.

Notice that if a parabola has only one zero, the zero is the x-coordinate of the vertex. Helpful Hint

Example Find the zeros of the quadratic function from its graph. Check your answer. y = x2 – 6x + 9 y = (3)2 – 6(3) + 9 = 9 – 18 + 9 = 0 y = x2 – 6x + 9 Check  The zero appears to be 3.

Example Find the zeros of the quadratic function from its graph. Check your answer. y = –2x2 – 2 The graph does not cross the x-axis, so there are no zeros of this function.

Example Find the zeros of the quadratic function from its graph. Check your answer. y = –4x2 – 2 The graph does not cross the x-axis, so there are no zeros of this function.

A vertical line that divides a parabola into two symmetrical halves is the axis of symmetry. The axis of symmetry always passes through the vertex of the parabola. You can use the zeros to find the axis of symmetry.

Example Find the axis of symmetry of each parabola. A. (–1, 0) Identify the x-coordinate of the vertex. The axis of symmetry is x = –1. B. Find the average of the zeros. The axis of symmetry is x = 2.5.

Example Find the axis of symmetry of each parabola. a. (–3, 0) Identify the x-coordinate of the vertex. The axis of symmetry is x = –3. b. Find the average of the zeros. The axis of symmetry is x = 1.

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.

Example: Finding the Axis of Symmetry by Using the Formula Find the axis of symmetry of the graph of y = –3x2 + 10x + 9. Step 1. Find the values of a and b. Step 2. Use the formula. y = –3x2 + 10x + 9 a = –3, b = 10 The axis of symmetry is

Example Find the axis of symmetry of the graph of y = 2x2 + x + 3. Step 1. Find the values of a and b. Step 2. Use the formula. y = 2x2 + 1x + 3 a = 2, b = 1 The axis of symmetry is .

Example: Finding the Vertex of a Parabola Find the vertex. y = –3x2 + 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.

Example: Continued Find the vertex. y = –3x2 + 6x – 7 Step 2 Find the corresponding y-coordinate. y = –3x2 + 6x – 7 Use the function rule. = –3(1)2 + 6(1) – 7 Substitute 1 for x. = –3 + 6 – 7 = –4 Step 3 Write the ordered pair. The vertex is (1, –4).

Lesson Quiz: Part I 1. Find the zeros and the axis of symmetry of the parabola. 2. Find the axis of symmetry and the vertex of the graph of y = 3x2 + 12x + 8. zeros: ( , ); x = ___ (axis of symmetry) Vertex ( , ) , x = __ (axis of symmetry)