1. 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. 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

2. 9-2
Characteristics of Quadratic Functions
Holt Algebra 1

3. 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.

4. 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.

5. 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.

6. 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.

7. 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.

9. 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.

11. 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

13. 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.

14. 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).

15. 3.

16. 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)

17. 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

18. 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)