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**8-2**Characteristics of Quadratic Functions Warm Up Lesson Presentation Lesson Quiz Holt McDougal Algebra 1 Holt Algebra 1**Warm Up**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.**Check**y =x2 – 2x – 3 y =(–1)2 – 2(–1) – 3 = 1 + 2 – 3 = 0 y =32 –2(3) – 3 = 9 – 6 – 3 = 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 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.**Helpful Hint**Notice that if a parabola has only one zero, the zero is the x-coordinate of the vertex.**Check**y =x2 – 6x + 9 y =(3)2 – 6(3) + 9 = 9 – 18 + 9 = 0 Example Find the zeros of the quadratic function from its graph. Check your answer. y = x2 – 6x + 9 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.**Step 2. Use the formula.**The axis of symmetry is 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. y = –3x2 + 10x + 9 a = –3, b = 10**Step 2. Use the formula.**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. y = 2x2 + 1x + 3 a = 2, b = 1**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)