Axis of Symmetry Calculator

This calculator finds the axis of symmetry for a parabola. The axis of symmetry is the vertical line that passes through the vertex, dividing the parabola into two mirror images. Enter your quadratic equation in one of the three forms below to find the solution.

How to Find the Axis of Symmetry

The method for finding the axis of symmetry depends on the form of your quadratic equation. The axis of symmetry is always a vertical line given by the equation $x = h$, where $h$ is the x-coordinate of the vertex.

1. Standard Form: $y = ax^2 + bx + c$

This is the most common form. The x-coordinate of the vertex, and thus the axis of symmetry, is found using the formula:

You must have $a \neq 0$, or it is not a parabola.

Example: For $y = 2x^2 + 8x - 5$, where $a=2$, $b=8$, and $c=-5$.

$x = \frac{-8}{2(2)} = \frac{-8}{4} = -2$. The axis of symmetry is $x = -2$.

2. Vertex Form: $y = a(x - h)^2 + k$

This form gives you the vertex $(h, k)$ directly. The axis of symmetry is simply the x-value of the vertex, $h$.

Example: For $y = 3(x - 4)^2 + 1$, the vertex is $(4, 1)$.

The axis of symmetry is $x = 4$. (Note: if the equation is $y = 3(x + 4)^2 + 1$, this is the same as $y = 3(x - (-4))^2 + 1$, so $h = -4$.)

3. Intercept Form (or Factored Form): $y = a(x - p)(x - q)$

This form gives you the two x-intercepts (where the parabola crosses the x-axis) at $x = p$ and $x = q$. The axis of symmetry is the vertical line exactly halfway between these two points.

Example: For $y = -2(x - 1)(x - 7)$, the intercepts are $p=1$ and $q=7$.

$x = \frac{1 + 7}{2} = \frac{8}{2} = 4$. The axis of symmetry is $x = 4$.

Frequently Asked Questions (FAQ)