Functions 8 2 Characteristicsofof Quadratic Functions 8 2

Functions 8 -2 Characteristicsofof. Quadratic Functions 8 -2 Characteristics Warm Up Lesson Presentation Lesson Quiz Holt Mc. Dougal Algebra 1 Algebra 11 Holt Mc. Dougal

8 -2 Characteristics of Quadratic Functions Objectives Find the zeros of a quadratic function from its graph. Find the axis of symmetry and the vertex of a parabola. Holt Mc. Dougal Algebra 1

8 -2 Characteristics of Quadratic Functions Example 1 A: Finding Zeros of Quadratic Functions From Graphs Find the zeros of the quadratic function from its graph. Check your answer. y = x 2 – 2 x – 3 Check y = x 2 – 2 x – 3 y = (– 1)2 – 2(– 1) – 3 =1 +2– 3=0 y = 32 – 2(3) – 3 =9– 6– 3=0 The zeros appear to be – 1 and 3. Holt Mc. Dougal Algebra 1

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

8 -2 Characteristics of Quadratic Functions Example 1 C: Finding Zeros of Quadratic Functions From Graphs Find the zeros of the quadratic function from its graph. Check your answer. y = – 2 x 2 – 2 The graph does not cross the x-axis, so there are no zeros of this function. Holt Mc. Dougal Algebra 1

8 -2 Characteristics of Quadratic Functions Check It Out! Example 1 a Find the zeros of the quadratic function from its graph. Check your answer. y = – 4 x 2 – 2 The graph does not cross the x-axis, so there are no zeros of this function. Holt Mc. Dougal Algebra 1

8 -2 Characteristics of Quadratic Functions Check It Out! Example 1 b Find the zeros of the quadratic function from its graph. Check your answer. y = x 2 – 6 x + 9 Check y = x 2 – 6 x + 9 y = (3)2 – 6(3) + 9 = 9 – 18 + 9 = 0 The zero appears to be 3. Holt Mc. Dougal Algebra 1

8 -2 Characteristics of Quadratic Functions Holt Mc. Dougal Algebra 1

8 -2 Characteristics of Quadratic Functions 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. Holt Mc. Dougal Algebra 1 Find the average of the zeros. The axis of symmetry is x = 2. 5.

8 -2 Characteristics of Quadratic Functions Check It Out! Example 2 Find the axis of symmetry of each parabola. a. b. Holt Mc. Dougal Algebra 1 (– 3, 0) Identify the x-coordinate of the vertex. The axis of symmetry is x = – 3. Find the average of the zeros. The axis of symmetry is x = 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. Holt Mc. Dougal Algebra 1

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

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

8 -2 Characteristics of Quadratic Functions Example 4 A: Finding the Vertex of a Parabola Find the vertex. y = 0. 25 x 2 + 2 x + 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. 25 x 2 + 2 x + 3 = 0. 25(– 4)2 + 2(– 4) + 3 = – 1 Step 3 Write the ordered pair. (– 4, – 1) The vertex is (– 4, – 1). Holt Mc. Dougal Algebra 1 Use the function rule. Substitute – 4 for x.

8 -2 Characteristics of Quadratic Functions Example 4 B: Finding the Vertex of a Parabola Find the vertex. y = – 3 x 2 + 6 x – 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 Mc. Dougal Algebra 1

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

8 -2 Characteristics of Quadratic Functions Check It Out! Example 4 Find the vertex. y = x 2 – 4 x – 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. Holt Mc. Dougal Algebra 1

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

8 -2 Characteristics of Quadratic Functions Example 5: Application The graph of f(x) = – 0. 06 x 2 + 0. 6 x + 10. 26 can be used to model the height in meters of an arch support for a bridge, where the xaxis represents the water level and x represents the horizontal distance in meters from where the arch support enters the water. Can a sailboat that is 14 meters tall pass under the bridge? Explain. The vertex represents the highest point of the arch support. Holt Mc. Dougal Algebra 1

8 -2 Characteristics of Quadratic Functions Example 5 Continued Step 1 Find the x-coordinate. a = – 0. 06, b = 0. 6 Identify a and b. Substitute – 0. 06 for a and 0. 6 for b. Step 2 Find the corresponding y-coordinate. Use the function rule. f(x) = – 0. 06 x 2 + 0. 6 x + 10. 26 Substitute 5 for x. = – 0. 06(5)2 + 0. 6(5) + 10. 26 = 11. 76 Since the height of each support is 11. 76 m, the sailboat cannot pass under the bridge. Holt Mc. Dougal 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. 07 x 2 + 0. 42 x + 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. Holt Mc. Dougal Algebra 1

8 -2 Characteristics of Quadratic Functions Check It Out! Example 5 Continued Step 2 Find the corresponding y-coordinate. f(x) = – 0. 07 x 2 + 0. 42 x + 6. 37 = – 0. 07(3)2 + 0. 42(3) + 6. 37 = 7 ft The height of the rise is 7 ft. Holt Mc. Dougal Algebra 1 Use the function rule. Substitute 3 for x.