# How to Find an Axis of Symmetry

Two Methods:Finding the Axis of Symmetry for Polynomials with a Degree of 2Finding the Axis of Symmetry Graphically

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.

## Steps

### Method 1 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. If the degree of your polynomial is 2 (there is no exponent larger than x^{2}), 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 2x
^{2}+ 3x – 1. This highest exponent present is the x^{2}, so it is a 2nd order polynomial, and you can use this first method to find the axis of symmetry.

- To illustrate, take, as an example, the polynomial 2x
- 2
**Plug your numbers into the axis of symmetry formula.**To calculate the axis of symmetry for a 2nd order polynomial in the form ax^{2}+ 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.

- In the example above, a = 2 b = 3, and c = -1. Insert these values into your formula, and you will get:
- 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
**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 x^{2}), 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. - 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
**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