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Polynomials & Functions

Parabolas: Standard Form & Axis of Symmetry

Standard form: y = ax² + bx + c. The axis of symmetry is x = −b/(2a). The vertex x-coordinate is found with this formula, then plug back in to find y.

✨ The axis of symmetry formula x = −b/2a is your key — memorize it and the rest follows!

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Worked Examples

Follow along with these step-by-step examples. Take your time — there's no rush!

1Example 1

Problem

Find the axis of symmetry of y = 2x² − 8x + 3

Step-by-Step Solution

1

Identify a = 2, b = −8.

2

Axis of symmetry: x = −b/(2a) = −(−8)/(2×2) = 8/4 = 2.

3

Axis of symmetry is x = 2.

Answer

x = 2

2Example 2

Problem

Find the vertex of y = x² − 4x + 1

Step-by-Step Solution

1

a = 1, b = −4. Axis: x = −(−4)/(2×1) = 4/2 = 2.

2

Plug x = 2 back in: y = (2)² − 4(2) + 1 = 4 − 8 + 1 = −3.

3

Vertex = (2, −3)

Answer

Vertex at (2, −3)

3Example 3

Problem

Does y = −3x² + 6x − 1 open up or down?

Step-by-Step Solution

1

Look at the coefficient of x²: a = −3.

2

Since a is negative, the parabola opens DOWNWARD.

3

The vertex is a maximum point.

Answer

Opens downward (a < 0)

4Example 4

Problem

Find the vertex of y = x² − 6x + 8

Step-by-Step Solution

1

a = 1, b = −6. Axis: x = −(−6)/(2×1) = 6/2 = 3.

2

Plug x = 3 back in: y = (3)² − 6(3) + 8 = 9 − 18 + 8 = −1.

3

Vertex = (3, −1)

Answer

Vertex at (3, −1)

5Example 5

Problem

Find the axis of symmetry of y = −x² + 4x + 5

Step-by-Step Solution

1

a = −1, b = 4.

2

Axis: x = −4 / (2 × −1) = −4 / −2 = 2.

3

Axis of symmetry is x = 2.

Answer

x = 2

Your Turn — Practice Problems

Try all 5 problems on your own first. Write out your work — that's how it sticks!

💡 Tip: Don't peek at the answers until you've genuinely tried each one.

1

Find axis of symmetry: y = x² − 6x + 5

2

Find vertex of y = x² + 2x − 3

3

Does y = 4x² − 8x + 1 open up or down?

4

Find axis of symmetry: y = 2x² + 4x − 6

5

Find vertex of y = x² − 2x + 5

6

Find axis of symmetry: y = x² + 8x + 3

7

Does y = −2x² + 4x − 1 open up or down?

8

Find vertex of y = x² − 4x + 4

9

Find axis of symmetry: y = 3x² − 12x + 1

10

Is the vertex a max or min for y = −x² + 6x?

11

Find vertex of y = x² + 6x + 9

12

Find axis of symmetry: y = x² − 10x + 21

Finished all 12? Give yourself a pat on the back — then check your work!

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