How to Find an Axis of Symmetry

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Learn to calculate the axes of symmetry for algebraic expressions
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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

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    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 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.
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    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. [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.
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    The value you calculated with your axis of symmetry formula is the x-intercept of the axis of symmetry. [4]
    • In the example above, the axis of symmetry is -3/4.
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Method 2
Method 2 of 2:

Finding the Axis of Symmetry Graphically

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    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.
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    Make two lines in the shape of a plus sign. The horizontal line is your x-axis; the vertical line is your y-axis.
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    Mark both axes with numbers at equal intervals. Spacing should be uniform on both axes.
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    Take your polynomial or function and calculate values of f(x) by putting all values of x into it.
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    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.
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    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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    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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    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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  • 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.
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  • Question
    What is the axis of symmetry in x = -2(x - 3) + 5?
    Donagan
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    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).
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  • Question
    What is the axis of symmetry of f(x) = -x^2 - 6x + 4?
    Donagan
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    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.
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      • 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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      References

      1. ↑ https://www.purplemath.com/modules/symmetry3.htm
      2. ↑ https://www.mathsisfun.com/algebra/degree-expression.html
      3. ↑ https://virtualnerd.com/algebra-1/quadratic-equations-functions/graphing/graph-basics/axis-symmetry-example
      4. ↑ https://virtualnerd.com/algebra-1/quadratic-equations-functions/graphing/graph-basics/axis-symmetry-example
      5. ↑ https://www.learnalberta.ca/content/memg/Division04/Axis%20of%20Symmetry%20(Parabola)/index.html
      6. ↑ https://www.khanacademy.org/math/algebra2/polynomial-functions/introduction-to-symmetry-of-functions/v/recognizing-odd-and-even-functions
      7. ↑ https://www.learnalberta.ca/content/memg/Division04/Axis%20of%20Symmetry%20(Parabola)/index.html

      About This Article

      Article Summary X

      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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