Quadratic Functions • Topic 3 of 5

Axis of Symmetry

The axis of symmetry is the vertical line through the vertex that splits a parabola into mirror-image halves. Its equation is x = −b/(2a) from standard form, or x = h from vertex form. Equivalently, it is halfway between the two x-intercepts, so if the roots are 3 and 7 the axis is x = 5. Points equally far left and right of the axis have the same y-value. Finding the axis quickly locates the vertex and helps interpret graphs. The SAT tests it directly and as a step toward finding the vertex or matching a graph.

✅ Solved examples

1. Find the axis of symmetry of y = x² − 6x + 2.
x = −b/(2a) = 6/2 = 3.
2. Find the axis of symmetry of y = (x + 4)² − 1.
It is x = h = −4.
3. A parabola has roots 2 and 8. What is its axis of symmetry?
Midpoint of the roots: x = (2 + 8)/2 = 5.
4. Find the axis of symmetry of y = 2x² + 8x.
x = −8/(2·2) = −2.

✏️ Practice — try these, take hints as needed

1. Find the axis of symmetry of y = x² − 10x + 1.
x = −b/(2a).
10/2.
—
x = 5.
2. Find the axis of symmetry of y = (x − 7)² + 3.
x = h.
h = 7.
—
x = 7.
3. A parabola has roots −1 and 5. Axis of symmetry?
Midpoint of the roots.
(−1 + 5)/2.
—
x = 2.
4. Find the axis of symmetry of y = 3x² − 12x.
x = −b/(2a).
12/6.
—
x = 2.
5. Find the axis of symmetry of y = (x + 2)² + 9.
x = h.
h = −2.
—
x = −2.

📝 Topic test — 8 questions

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