How To Find An Axis Of Symmetry: 11 Steps (with Pictures) - WikiHow

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Terms of Use wikiHow is where trusted research and expert knowledge come together. Learn why people trust wikiHow How to Find an Axis of Symmetry PDF download Download Article Learn to calculate the axes of symmetry for algebraic expressions Explore this Article methods 1 Finding the Axis of Symmetry for Polynomials with a Degree of 2 2 Finding the Axis of Symmetry Graphically Other Sections Questions & Answers Tips and Warnings Related Articles References Article Summary Reviewed by Joseph Meyer

Last Updated: January 5, 2024 References

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This article was reviewed by Joseph Meyer. Joseph Meyer is a High School Math Teacher based in Pittsburgh, Pennsylvania. He is an educator at City Charter High School, where he has been teaching for over 7 years. Joseph is also the founder of Sandbox Math, an online learning community dedicated to helping students succeed in Algebra. His site is set apart by its focus on fostering genuine comprehension through step-by-step understanding (instead of just getting the correct final answer), enabling learners to identify and overcome misunderstandings and confidently take on any test they face. He received his MA in Physics from Case Western Reserve University and his BA in Physics from Baldwin Wallace University. This article has been viewed 108,940 times.

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.

Steps

Method 1 Method 1 of 2:

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

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  1. Step 1 Check the degree of your polynomial. 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. Step 2 Plug your numbers into the axis of symmetry formula. 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.
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  3. Step 3 Write down the equation of the axis of symmetry. 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]
    • 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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  1. Step 1 Check the degree of your polynomial. 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. Step 2 Draw the x- and y- axes. 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. Step 3 Number your graph. 3 Number your graph. Mark both axes with numbers at equal intervals. Spacing should be uniform on both axes.
  4. Step 4 Calculate y = f(x) for every x. 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. Step 5 Make a graph point for each pair. 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. Step 6 Draw the graph of the polynomial. 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]
  7. Step 7 Look for the axis of symmetry. 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]
  8. Step 8 Note the axis of symmetry. 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

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  • Question What is the axis of symmetry of f for f(x)=-2|x+3|-7? Community Answer 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. Thanks! We're glad this was helpful. Thank you for your feedback. If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow Yes No Not Helpful 10 Helpful 11
  • Question What is the axis of symmetry in x = -2(x - 3) + 5? Donagan 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). Thanks! We're glad this was helpful. Thank you for your feedback. If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow Yes No Not Helpful 7 Helpful 6
  • Question What is the axis of symmetry of f(x) = -x^2 - 6x + 4? Donagan 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. Thanks! We're glad this was helpful. Thank you for your feedback. If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow Yes No Not Helpful 3 Helpful 11
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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. Thanks Helpful 0 Not Helpful 1
  • Some polynomials are not symmetrical. For example, y = 3x has no axis of symmetry. Thanks Helpful 1 Not Helpful 0
  • 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.” Thanks Helpful 1 Not Helpful 0
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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

Joseph Meyer Reviewed by: Joseph Meyer Math Teacher This article was reviewed by Joseph Meyer. Joseph Meyer is a High School Math Teacher based in Pittsburgh, Pennsylvania. He is an educator at City Charter High School, where he has been teaching for over 7 years. Joseph is also the founder of Sandbox Math, an online learning community dedicated to helping students succeed in Algebra. His site is set apart by its focus on fostering genuine comprehension through step-by-step understanding (instead of just getting the correct final answer), enabling learners to identify and overcome misunderstandings and confidently take on any test they face. He received his MA in Physics from Case Western Reserve University and his BA in Physics from Baldwin Wallace University. This article has been viewed 108,940 times. 43 votes - 39% Co-authors: 4 Updated: January 5, 2024 Views: 108,940 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! Did this summary help you?YesNo

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Yes No Advertisement Cookies make wikiHow better. By continuing to use our site, you agree to our cookie policy. Joseph Meyer Reviewed by: Joseph Meyer Math Teacher 43 votes - 39% Click a star to vote 39% of people told us that this article helped them. Co-authors: 4 Updated: January 5, 2024 Views: 108,940

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