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

2. The highest or lowest point on a parabola is the vertex
The highest or lowest point on a parabola is the vertex. If a parabola opens upward, the vertex is the lowest point. If a parabola opens downward, the vertex is the highest point.

3. Example 4: Identifying the Vertex and the Minimum or Maximum
Identify the vertex of each parabola. Then give the minimum or maximum value of the function. A. B.
The vertex is (–3, 2), and the minimum is 2.
The vertex is (2, 5), and the maximum is 5.

4. Check It Out! Example 4
Identify the vertex of each parabola. Then give the minimum or maximum value of the function. a. b.
The vertex is (–2, 5) and the maximum is 5.
The vertex is (3, –1), and the minimum is –1.

5. 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 are –1 and 3.

6. 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 is –4.

7. 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.

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.

10. 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.

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

12. 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

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. 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.

16. 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).