1. 9-2 Characteristics of Quadratic Functions Warm Up Warm Up Lesson Presentation Lesson Presentation California Standards California StandardsPreview

2. 9-2 Characteristics of Quadratic Functions Warm Up Find the x-intercept of each linear function. ( Hint: y = 0 at x-intercept; set y = 0, then solve for x) 1. y = 2x – 3 2. y = 3x + 6 Evaluate each quadratic function for the given input values. 3. y = –3x 2 + x – 2, when x = 2 4. y = x 2 + 2x + 3, when x = –1 –2 –12 2

3. 9-2 Characteristics of Quadratic Functions 21.0 Students graph quadratic functions and know that their roots are the x-intercepts. Also covered: 23.0 California Standards

4. 9-2 Characteristics of Quadratic Functions zero of a function axis of symmetry Vocabulary

5. 9-2 Characteristics of Quadratic Functions 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. So a zero of a function is the same as an x-intercept. Since a graph intersects the x-axis at the point or points containing an x-intercept, these intersections are also at the zeros of the function. A quadratic function may have one, two, or no zeros.

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

7. 9-2 Characteristics of Quadratic Functions Additional Example 1B: Finding Zeros of Quadratic Functions From Graphs Find the zeros of the quadratic function from its graph. Check your answer. y = x 2 + 8x + 16 y = (–4) 2 + 8(–4) + 16 = 16 – 32 + 16 = 0 y = x 2 + 8x + 16 Check The zero appears to be –4.

8. 9-2 Characteristics of Quadratic Functions Notice that if a parabola has only one zero, the zero is the x-coordinate of the vertex. Helpful Hint

9. 9-2 Characteristics of Quadratic Functions Additional Example 1C: Finding Zeros of Quadratic Functions From Graphs Find the zeros of the quadratic function from its graph. Check your answer. y = –2x 2 – 2 The graph does not cross the x-axis, so this function has no zeros.

10. 9-2 Characteristics of Quadratic Functions Check It Out! Example 1a Find the zeros of the quadratic function from its graph. Check your answer. y = –4x 2 – 2 The graph does not cross the x-axis, so this function has no zeros.

11. 9-2 Characteristics of Quadratic Functions Check It Out! Example 1b Find the zeros of the quadratic function from its graph. Check your answer. y = x 2 – 6x + 9 The zero appears to be 3. y = (3) 2 – 6(3) + 9 = 9 – 18 + 9 = 0 y = x 2 – 6x + 9 Check

12. 9-2 Characteristics of Quadratic Functions The 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. Recall that a vertical line is in the form x = n

13. 9-2 Characteristics of Quadratic Functions

14. 9-2 Characteristics of Quadratic Functions Additional 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. Find the average of the zeros. The axis of symmetry is x = 2.5. B.

15. 9-2 Characteristics of Quadratic Functions Check It Out! Example 2 Find the axis of symmetry of each parabola. (–3, 0) Identify the x-coordinate of the vertex. The axis of symmetry is x = –3. a. b. Find the average of the zeros. The axis of symmetry is x = 1.

16. 9-2 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.

17. 9-2 Characteristics of Quadratic Functions Additional Example 3: Finding the Axis of Symmetry by Using the Formula Find the axis of symmetry of the graph of y = – 3x 2 + 10x + 9. Step 1. Find the values of a and b. y = –3x 2 + 10x + 9 a = –3, b = 10 Step 2. Use the formula. The axis of symmetry is

18. 9-2 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.

19. 9-2 Characteristics of Quadratic Functions Once you have found the axis of symmetry, you can use it to identify the vertex.

20. 9-2 Characteristics of Quadratic Functions Additional 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 of the vertex. 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).

21. 9-2 Characteristics of Quadratic Functions Additional 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.

22. 9-2 Characteristics of Quadratic Functions Additional Example 4B Continued Find the vertex. Step 2 Find the corresponding y-coordinate of the vertex. 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

23. 9-2 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

24. 9-2 Characteristics of Quadratic Functions Find the vertex. Step 2 Find the corresponding y-coordinate of the vertex. 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

25. 9-2 Characteristics of Quadratic Functions Additional Example 5: Application The graph of f(x) = –0.06x 2 + 0.6x + 10.26 can be used to model the height in meters of an arch support for a bridge, where the x- axis represents the water level and x represents the 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.

26. 9-2 Characteristics of Quadratic Functions Additional Example 5 Continued Step 1 Find the x-coordinate of the vertex. 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 of the vertex. = –0.06(5) 2 + 0.6(5) + 10.26 f(x) = –0.06x 2 + 0.6x + 10.26 = 11.76 Use the function rule. Substitute 5 for x. Since the height of the support is 11.76 m, the sailboat cannot pass under the bridge.

27. 9-2 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) = – 0.07x 2 + 0.42x + 6.37, where x is the distance in feet from a supported pole at ground level. Find the height of the rise. Step 1 Find the x-coordinate of the vertex. a = –0.07, b= 0.42 Identify a and b. Substitute –0.07 for a and 0.42 for b.

28. 9-2 Characteristics of Quadratic Functions Check It Out! Example 5 Continued Step 2 Find the corresponding y-coordinate of the vertex. = –0.07(3) 2 + 0.42(3) + 6.37 f(x) = –0.07x 2 + 0.42x + 6.37 = 7 ft Use the function rule. Substitute 3 for x. The height of the rise is 7 ft.

29. 9-2 Characteristics of Quadratic Functions Lesson Quiz: Part I 1. Find the zeros and the axis of symmetry. 2. Find the axis of symmetry and the vertex of the graph of y = 3x 2 + 12x + 8. zeros: –6, 2; x = –2 x = –2; (–2, –4)

30. 9-2 Characteristics of Quadratic Functions Lesson Quiz: Part II 25 feet 3. The graph of f(x) = –0.01x 2 + 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.