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

2. Objectives Find the zeros of a quadratic function from its graph.
Find the axis of symmetry and the vertex of a parabola.

3. Vocabulary
quadratic function
parabola
vertex
minimum
maximum

4. The function y = x2 is shown in the graph
The function y = x2 is shown in the graph. Notice that the graph is not linear. This function is a quadratic function. A quadratic function is any function that can be written in the standard form y = ax2 + bx + c, where a, b, and c are real numbers and a ≠ 0. The function y = x2 can be written as y = 1x2 + 0x + 0, where a = 1,
b = 0, and c = 0.

7. 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.

8. Once you have found the axis of symmetry, you can use it to identify the vertex.

9. 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
= – 3 = 0
y = 32 –2(3) – 3
= 9 – 6 – 3 = 0
y = x2 – 2x – 3
Check 
The zeros appear to be –1 and 3.

10. 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 – = 0 
The zero appears to be –4.

11. 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.

12. Check It Out! Example 1a
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.

13. Check It Out! Example 1b
Find the zeros of the quadratic function from its graph. Check your answer.
y = x2 – 6x + 9
y = (3)2 – 6(3) + 9
= 9 – = 0
y = x2 – 6x + 9
Check 
The zero appears to be 3.

14. 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.

15. Check It Out! Example 2
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.

16. Check It Out! Example 3
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

17. 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 = 6
Identify a and b.
Substitute –3 for a and 6 for b.
The x-coordinate of the vertex is 1.

18. 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).

19. Check It Out! Example 4
Find the vertex.
y = x2 – 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.

20. Check It Out! Example 4 Continued
Find the vertex.
y = x2 – 4x – 10
Step 2 Find the corresponding y-coordinate.
y = x2 – 4x – 10
Use the function rule.
= (2)2 – 4(2) – 10
Substitute 2 for x.
= 4 – 8 – 10
= –14
Step 3 Write the ordered pair.
The vertex is (2, –14).

21. Example 5: Application
The graph of f(x) = –0.06x x 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 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.

22. 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) = –0.06x x
= –0.06(5) (5)
Substitute 5 for x.
= 11.76
Since the height of each support is m, the sailboat cannot pass under the bridge.

23. Check It Out! Example 5
The height of a small rise in a roller coaster track is modeled by f(x) = –0.07x x , 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.

24. Check It Out! Example 5 Continued
Step 2 Find the corresponding y-coordinate.
f(x) = –0.07x x
Use the function rule.
= –0.07(3) (3)
Substitute 3 for x.
= 7 ft
The height of the rise is 7 ft.