# How to Find an Axis of Symmetry

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. There are two basic methods.

### Method 1 Finding the Axis of Symmetry for Polynomials with a Degree of 2

1. 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 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. 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.
• 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.
3. 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.
• In the example above, the axis of symmetry is -3/4.

### Method 2 Finding the Axis of Symmetry Graphically

1. 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. 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. 3
Number your graph. Mark both axes with numbers at equal intervals. Spacing should be uniform on both axes.
4. 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. 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. 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.
7. 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.
8. 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.

## Tips

• The lengths of your x- and y- axes should allow the overall shape of the graph to be clearly visible.
• Some polynomials are not symmetrical. For example, y = 3x has no axis of symmetry.
• The symmetry of a polynomial can be classified into even or odd symmetry. Any graph that has an axis of symmetry on the y-axis has an “even” symmetry; any graph that has an axis of symmetry on the x-axis is “odd.”

## Article Info

Categories: Algebra