How to find axis of symmetry from a graph

To find an axis of symmetry, start by checking the degree or largest exponential value of the polynomial. If the degree of your polynomial is 2, you can find the axis of symmetry by plugging the numbers directly into the axis of symmetry formula. Solve the formula and the answer you get is the x-intercept of the axis of symmetry. If the degree of the polynomial is higher than 2, you will need to find the axis of symmetry by using a graph. For tips on solving graphically, read on!

Table of Contents Show

Equation of the Axis of Symmetry of a Parabola
Calculating the Axis of Symmetry of a Parabola
Finding the Vertex of a Parabola
How To Find Axis of Symmetry
What is the equation to find the axis of symmetry?
How do you find the vertex and axis of symmetry of a function?
How do you find the axis of symmetry of a problem?

Did this summary help you?YesNo

Thanks to all authors for creating a page that has been read 89,824 times.

If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

All parabolas have exactly one axis of symmetry (unlike a circle, which has infinitely many axes of symmetry). If the vertex of a parabola is \((k,l)\), then its axis of symmetry has equation \(x=k\).

Detailed description of diagram

We can find a simple formula for the value of \(k\) in terms of the coefficients of the quadratic. As usual, we complete the square:

\begin{align*} y &= ax^2 + bx + c\\ &= a\Big[x^2 + \dfrac{b}{a}x + \dfrac{c}{a}\Big]\\ &= a\Big[\Big(x+\dfrac{b}{2a}\Big)^2 + \dfrac{c}{a} - \Big(\dfrac{b}{2a}\Big)^2\Big]. \end{align*}

We can now see that the \(x\)-coordinate of the vertex is \(-\dfrac{b}{2a}\). Thus the equation of the axis of symmetry is

\[ x = -\dfrac{b}{2a}. \]

We could also find a formula for the \(y\)-coordinate of the vertex, but it is easier simply to substitute the \(x\)-coordinate of the vertex into the original equation \(y=ax^2+bx+c\).

Example

Sketch the parabola \(y=2x^2+8x+19\) by finding the vertex and the \(y\)-intercept. Also state the equation of the axis of symmetry. Does the parabola have any \(x\)-intercepts?

Solution

Here \(a=2\), \(b=8\) and \(c=19\). So the axis of symmetry has equation \(x=-\dfrac{b}{2a}=-\dfrac{8}{4}=-2\). We substitute \(x=-2\) into the equation to find \(y = 2\times (-2)^2 + 8\times (-2) + 19 = 11\), and so the vertex is at \((-2,11)\). Finally, putting \(x=0\) we see that the \(y\)-intercept is 19.

The axis of symmetry is the vertical line that goes through the vertex of a quadratic equation. There's even a formula to help find it! In this tutorial, you'll see how to find the axis of symmetry for a given quadratic equation.

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:

Remember that every quadratic function can be written in the 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).

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:

And when you put the quadratic function in standard form, it's

For example, we can put in the quadratic equation for the red parabola in its standard form,

, where a = 1, b = 4, and c = 3. The green line is the axis of symmetry.

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.

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.

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.

What is the equation to find the axis of symmetry?

The symmetry cuts any geometric shape into two equal halves. The axis of symmetry formula is given as, for a quadratic equation with standard form as y = ax² + bx + c, is: x = -b/2a. If the parabola is in vertex form y = a(x-h)² + k, then the formula is x = h.

How do you find the vertex and axis of symmetry of a function?

The Vertex Form of a quadratic function is given by: f(x)=a(x−h)2+k , where (h,k) is the Vertex of the parabola. x=h is the axis of symmetry. Use completing the square method to convert f(x) into Vertex Form.

How do you find the axis of symmetry of a problem?

Axis of Symmetry Formula When a parabola is graphed in standard form, we can use the formula x=−b2a in order to locate the axis of symmetry. Remember that when an equation is in standard form it appears as y=ax2+bx+c. We can take the values of a and b and plug them into the formula x=−b2a in order to solve for x.