How to Find an Axis of Symmetry

Last Updated: December 30, 2022 References

The graph of a polynomial or function reveals many characteristics that would not be clear without a visual representation. One of these characteristics is the axis of symmetry: a vertical line on a graph that splits the graph into two symmetrical mirror images. Finding the axis of symmetry for a given polynomial is fairly simple.^{[1]
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Research source} There are two basic methods.

Method 1

Method 1 of 2:

Finding the Axis of Symmetry for Polynomials with a Degree of 2

1
Check the degree of your polynomial. The degree (or “order”) of a polynomial is simply the largest exponent value in the expression.^{[2]
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Research source} If the degree of your polynomial is 2 (there is no exponent larger than x²), you can find the axis of symmetry using this method. If the degree of the polynomial is higher than 2, use Method 2.
- To illustrate, take, as an example, the polynomial 2x² + 3x – 1. This highest exponent present is the x², so it is a 2nd order polynomial, and you can use this first method to find the axis of symmetry.
2
Plug your numbers into the axis of symmetry formula. To calculate the axis of symmetry for a 2nd order polynomial in the form ax² + bx +c (a parabola), use the basic formula x = -b / 2a.^{[3]
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- In the example above, a = 2 b = 3, and c = -1. Insert these values into your formula, and you will get:
  x = -3 / 2(2) = -3/4.
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3
Write down the equation of the axis of symmetry. The value you calculated with your axis of symmetry formula is the x-intercept of the axis of symmetry.^{[4]
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- In the example above, the axis of symmetry is -3/4.

Method 2

Method 2 of 2:

Finding the Axis of Symmetry Graphically

1
Check the degree of your polynomial. The degree (or “order”) of a polynomial is simply the largest exponent value in the expression. If the degree of your polynomial is 2 (there is no exponent larger than x²), you can find the axis of symmetry using the formula method above. If the degree of the polynomial is higher than 2, use this graphical method.
2
Draw the x- and y- axes. Make two lines in the shape of a plus sign. The horizontal line is your x-axis; the vertical line is your y-axis.
3
Number your graph. Mark both axes with numbers at equal intervals. Spacing should be uniform on both axes.
4
Calculate y = f(x) for every x. Take your polynomial or function and calculate values of f(x) by putting all values of x into it.
5
Make a graph point for each pair. You now have pairs of y = f(x) for every x on the axis. For each (x, y) pair, make a point on the graph – vertically on the x-axis and horizontally on the y-axis.
6
Draw the graph of the polynomial. Once you have marked all the graph points, you can connect your dots smoothly to reveal a continuous graph of your polynomial.^{[5]
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7
Look for the axis of symmetry. Inspect your graph carefully. Look for a point on the axis such that when a line is passed through it, the graph splits into two equal, mirrored halves.^{[6]
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8
Note the axis of symmetry. If you can find a point – call it “b” – on the x-axis that splits the graph into two mirrored halves, then that point, b, is your axis of symmetry.^{[7]
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Community Q&A

Question

What is the axis of symmetry of f for f(x)=-2|x+3|-7?

Community Answer

The axis of symmetry is x=-3, because the vertex is at (-3,7). It is an absolute value graph that faces down.
Question

What is the axis of symmetry in x = -2(x - 3) + 5?

Donagan

Top Answerer

Because this graph consists of a straight line, it does not have an axis of symmetry. Axes of symmetry occur with parabolic graphs representing quadratic equations ("second-degree" polynomials).
Question

What is the axis of symmetry of f(x) = -x^2 - 6x + 4?

Donagan

Top Answerer

As explained in the above article, the axis of symmetry of a second-degree polynomial in the form of ax² + bx + c is given by the formula x = -b/2a, which in this case is x = -(-6) / 2(1) = 6/2 = 3. x=3.