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How To Find the Axis of Symmetry of a Parabola

The axis of symmetry is the vertical line that goes through the vertex of a parabola so the left and right sides of the parabola are symmetric. To simplify, this line splits the graph of a quadratic equation into two mirror images.

In this tutorial, we will show you how to find the axis of symmetry by looking at the quadratic equation itself.

Equation of the Axis of Symmetry of a Parabola

The equation for the axis of symmetry of a parabola can be expressed as:

how to find axis of symmetry: Equation for the axis of symmetry of a parabola

Remember that every quadratic function can be written in the standard form how to find axis of symmetry: Quadratic function written in standard form. The graph of a quadratic function is called a parabola, where every point on that parabola represents an x and a y that solves the quadratic function.

The vertex of a quadratic function is the highest or lowest point on the graph. The coordinate of the vertex of the parabola, then, is the x and y solution for the lowest or highest point of the parabola.

The vertex of the red parabola is (-2, -1) and the vertex of the blue parabola is (0, -2).

how to find axis of symmetry: Graph showing the vertexes of two parabolas

Calculating the Axis of Symmetry of a Parabola

Again, the axis of symmetry of the parabola is the line on the graph that passes through the vertex of the parabola and splits the graph into two symmetrical sides.

It is expressed as:

how to find axis of symmetry: Equation for the axis of symmetry of a parabola

And when you put the quadratic function in standard form, it's how to find axis of symmetry: Quadratic function written in standard form.

For example, we can put in the quadratic equation for the red parabola in its standard form, how to find axis of symmetry: Quadratic equation of the red parabola in standard form, where a = 1, b = 4, and c = 3. The green line is the axis of symmetry.

parabola example equation

Or x = -2 after you substitute in the values for a and b.

Here’s how this formula looks on the graph. Note where the green line is and how it divides the parabola.

Graph showing the axis of symmetry of a parabola

Finding the Vertex of a Parabola

To find the actual coordinates for the vertex of the parabola, simply substitute the x value into the polynomial expression to find the corresponding y value. Remember, each point on the quadratic graph is a solution to the equation.

When we continue with the previous example, we know that x = -2.

We substitute that value for x in the original quadratic function.

Formula on how to find the vertex of a parabola

Formula on how to find the vertex of a parabola

Formula on how to find the vertex of a parabola

Solving it gives us y = -1. We now know that the vertex of the parabola is the coordinate (-2, -1). Finding the vertex of a parabola couldn’t be easier.

How To Find Axis of Symmetry

Here's what you need to remember: Whether you’re after the axis of symmetry or the full coordinates of the vertex of the parabola, use this formula to start graphing a quadratic equation.

how to find axis of symmetry: Equation for the axis of symmetry of a parabola

Solving for x gives you the axis of symmetry. This line of symmetry will intersect with the parabola at its vertex, where x is the coordinate you just calculated and y is the coordinate when you substitute x back into the quadratic equation, how to find axis of symmetry: Quadratic function written in standard form.

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