Axis of Symmetry Calculator

Determine the axis of symmetry for a parabola equation with the calculator. The axis of symmetry calculator shows the complete steps involved in calculating the vertex that divides the parabola into two equal parts. Also, the tool shows a graph that better helps to understand the behavior of a symmetrical axis in a plane.

What Is the Axis of Symmetry?

“Axis of symmetry is a line that passes through the parabola and divides it into two halves”

The symmetrical axis is also known as the line of symmetry that creates a mirror image like the reflection on each side of the parabola.

AOS Graph:

Axis of Symmetry Formula:

The formula to calculate the axis of symmetry for parabola equation is: Axis of Symmetry Equation: \(f(x) = ax^2 + bx + c\)

Axis of Symmetry = \(X = -b / 2a\)

Steps To Find AOS:

• Reduce the given expression into a standard quadratic function form such as:

\(ax^{2} + b x + c\)

• Next, pick the values of a and b (a being the coefficient of \(x^2\) and b being the coefficient of x)
• Now put in the values in the axis of symmetry equation which is:

\(x=\dfrac{-b}{2a}\)

• The above formula denotes that the AOS is a vertical line passing through the following point in the 2D plane:

\(\left(\dfrac{-b}{2a}, 0\right)\)

Example:

Calculate the axis of symmetry of the graph of \( y = (x + 5)^2 – 4 \) by using the formula.

Solution:

Step # 01:

Expand the given function to make a perfect quadratic function \(f(x) = (x + 5) 2 − 4\) \(f(x) = x^2 + 10x + 21\)

Step # 02:

Figure out the values of a and b in the quadratic function \(f(x) = x^2 + 10x + 21\) \(a=1,b=10\)

Step # 03:

Put in the values in the axis of symmetry equation to determine its value

\(X = −b / 2a\)

\(x = −(10) / 2(1)\)

\(x = −10 / 2\)

\(x = −5 / 1\)

\(x = − 5\)

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

Alternative Approach To Calculate AOS:

In this case, we will discuss the way you can find the axis of symmetry if you are given:

• Quadratic equation
• Its roots

Steps:

• Suppose you have the roots of a quadratic equation as i and j (If you have only one root, you have to consider a second root for the same value)
• Now, you need to input the roots into the Midpoint formula to calculate the axis of symmetry

\(\dfrac{i + j}{2}\)

Example:

If the given roots of the quadratic function are 3 and 7, calculate the axis of symmetry.

Solution:

Here we have:

\(\dfrac{i + j}{2}\)

\(=\dfrac{3 + 7}{2}\)

\(=\dfrac{10}{2}\) \(=5\)

Result:

Axis of symmetry = (5, 0)