Warm Up 1. y = 2x – 3 2. 3. y = 3x + 6 4. y = –3x2 + x – 2, when x = 2 Find the x-intercept of each linear function. 1. y = 2x – 3 2. 3. y = 3x + 6 Evaluate each quadratic function for the given input values. 4. y = –3x2 + x – 2, when x = 2 5. y = x2 + 2x + 3, when x = –1 –2 –12 2

9-2 Characteristics of Quadratic Functions Holt Algebra 1

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.

Example 1C: Finding Zeros of Quadratic Functions From Graphs 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.

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.

Example 2: Finding the Axis of Symmetry by Using Zeros 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 3: 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 4B: 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 = 10 Identify a and b. Substitute –3 for a and 6 for b. The x-coordinate of the vertex is 1.

Example 4B 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).

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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: –6, 2; x = –2 x = –2; (–2, –4)

Lesson Quiz: Part II 3. The graph of f(x) = –0.01x2 + x can be used to model the height in feet of a curved arch support for a bridge, where the x-axis represents the water level and x represents the distance in feet from where the arch support enters the water. Find the height of the highest point of the bridge. 25 feet

Warm-Up(Add to HW) 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: –6, 2; x = –2 x = –2; (–2, –4)