Axis of Symmetry Calculator

Axis of Symmetry Calculator

Enter the quadratic function into the calculator to determine its axis of symmetry, along with detailed step-by-step calculations.

Axis of Symmetry Calculator

Use this axis of symmetry calculator to determine the axis of symmetry for a parabola. The calculator provides step-by-step calculations for finding the vertex and the line that divides the parabola into two equal halves. Additionally, it displays a graph to help visualize the symmetry.

What Is the Axis of Symmetry?

The axis of symmetry is a vertical line that passes through the vertex of a parabola and divides it into two mirror-image halves. It is also known as the line of symmetry.

AOS Graph:

Axis of Symmetry Calculator

Axis of Symmetry Formula

For a quadratic function in standard form:

$$ f(x) = ax^2 + bx + c $$

$$ x = -\frac{b}{2a} $$

Steps to Find the Axis of Symmetry

Write the quadratic function in standard form: $ax^2 + bx + c$.
Identify the coefficients $a$ (coefficient of $x^2$) and $b$ (coefficient of $x$).
Substitute $a$ and $b$ into the formula: $x = -\frac{b}{2a}$.
The axis of symmetry is the vertical line passing through $(-\frac{b}{2a}, 0)$.

Example 1

Find the axis of symmetry of $y = (x + 5)^2 - 4$.

Solution:

Step 1: Expand the quadratic

$f(x) = (x + 5)^2 - 4 = x^2 + 10x + 21$

Step 2: Identify coefficients

$a = 1, b = 10$

Step 3: Apply the formula

$x = -\frac{b}{2a} = -\frac{10}{2 \cdot 1} = -5$

Result: Axis of symmetry = $(-5, 0)$

Alternative Approach Using Roots

If the roots of a quadratic equation are known, the axis of symmetry can be found as the midpoint of the roots.

$$ x = \frac{r_1 + r_2}{2} $$

Example 2

If the roots of a quadratic function are $3$ and $7$, find the axis of symmetry.

Solution:

$x = \frac{3 + 7}{2} = \frac{10}{2} = 5$

Result: Axis of symmetry = $(5, 0)$