How to Find the Line of Symmetry in a Quadratic Equation

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Quadratic equations have between one and three terms, one of which always incorporates x^2. When graphed, quadratic equations produce a U-shaped curve known as a parabola. The line of symmetry is an imaginary line which runs down the center of this parabola and cuts it into two equal halves. This line is commonly referred to as the axis of symmetry. It can be found quite quickly by using a simple algebraic formula.

Finding the Line of Symmetry Algebraically

    Rewrite the quadratic equation so the terms are in descending order. Write the squared term first, followed by the term with the next highest degree, and so on. For instance, consider the equation y = 6x - 1 + 3x^2. Arranging the terms in descending order yields y = 3x^2 + 6x - 1.

    Identify “a” and “b.” When written in descending order, quadratic equations take the form ax^2 + bx + c. Hence, “a” is the number to the left of the x^2, while “b” is the number to the left of the x. In y = 3x^2 + 6x - 1, a = 3 and b = 6.

    Insert the “a” and “b” values into the equation x = -b/(2a). Using the values from the example, you would write x = -6/(2*3).

    Simplify using the order of operations, also known as PEMDAS. First, multiply the numbers in the denominator, yielding x = -6/6 in the example. Next, perform the division. The example produces x = -1. This is the line of symmetry.

    Check your work. You may repeat each step to ensure you’ve performed the substitutions and calculations correctly. Alternatively, you may graph the equation on a graphing calculator, checking the accuracy of the line of symmetry visually.

    Tips

    • Be careful when simplifying with negatives. If the “b” term is negative in your original equation, it will become positive when substituted and simplified in the axis of symmetry formula.

      If your quadratic equation lacks a “b” term, the axis of symmetry is automatically x = 0.

      The “c” term is irrelevant when finding the axis of symmetry.

References

About the Author

Based in western New York, Amy Harris began writing for Demand Media and Great Lakes Brewing News in 2010. Harris holds a Bachelor of Science in Mathematics from Penn State University; she taught high school math for several years and has also worked in the field of instructional design.

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