# 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.[1] There are two basic methods.

### Method 1 Method 1 of 2: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.[2] 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.[3]
• 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 Method 2 of 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.[4]
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.

## Community Q&A

Search
• Question
What is the axis of symmetry of f for f(x)=-2|x+3|-7?
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
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
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.
• Question
How do I find the vertex?
Donagan
• Question
How do you find the axis of symmetry for a hyperbola?
A hyperbola has two axes of symmetry. One of them is the line passing through both foci. The other is the perpendicular bisector of the foci.
• Question
How do I find a quadratic equation given 2 points and a maximum (max y coordinate)?
Use the vertex form for the quadratic function: y = a(x-h)^2 + k. The value of k is the y-coordinate of the vertex which was given to you as the max, so substitute that in first. Then use the other two (x,y) pairs to get two equations in two unknowns, a and h. You can solve the system by solving one equation for a and substituting into the other. But since the equations are quadratic in h, you won't get a unique solution. One solution corresponds to a steep parabola with vertex between the other two given points; the other is a shallow parabola with a vertex is further out.
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## Tips

• The lengths of your x- and y- axes should allow the overall shape of the graph to be clearly visible.
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• Some polynomials are not symmetrical. For example, y = 3x has no axis of symmetry.
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• 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.”
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wikiHow is a “wiki,” similar to Wikipedia, which means that many of our articles are co-written by multiple authors. To create this article, volunteer authors worked to edit and improve it over time. This article has been viewed 83,367 times.
Co-authors: 4
Updated: July 10, 2019
Views: 83,367
Categories: Algebra
Article SummaryX

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!
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