How to find the symmetry axis of a quadratic function

What is a quadratic function?

A polynomial function of the second degree is called a quadratic function. Formally, f (x) = ax 2 + bx + c is a quadratic function, where a, b and c are real constants and a ≠ 0 for all values ​​of x. The graph of a quadratic function is a parabola.

How to find the symmetry axis of a quadratic function | euresisjournal.org

How to find the symmetry axis of a quadratic function

Every quadratic function shows lateral symmetry along the y-axis or a line parallel to it. The axis of symmetry of a quadratic function can be determined as follows:

f (x) = ax 2 + bx + c where a, b, c, x∈R and a ≠ 0

We have written x terms as a full square,

How to find the symmetry axis of a quadratic function | euresisjournal.org

By rearranging the terms of the above equation

How to find the symmetry axis of a quadratic function | euresisjournal.org

This implies that for every possible value f (x) there are two corresponding x-values. This can be clearly seen in the diagram below.

How to find the symmetry axis of a quadratic function | euresisjournal.org These values ​​are located

How to find the symmetry axis of a quadratic function | euresisjournal.org

Distance to the left and right of the value -b / 2a. In other words, the -b / 2a value is always the midpoint of a line connecting the corresponding x-values ​​(points) for any given f (x).

Hence, x = -b / 2a is the equation of the axis of symmetry for a given quadratic function in the form f (x) = ax 2 + bx + c

How to find the axis of symmetry of a quadratic function - examples

  • A quadratic function is given by f (x) = 4x 2 + x + 1. Find the axis of symmetry.

x = -b / 2a = -1 / (2 × 4) = - 1/8

Therefore the equation of the axis of symmetry is x = -1 / 8

  • A quadratic function is given by the expression f (x) = (x-2) (2x-5)

By simplifying the expression, we have f (x) = 2x 2 -5x-4x + 10 = 2x 2 -9x + 10

We can deduce that a = 2 and b = -9. Hence we can get the axis of symmetry as

x = - (-9) / (2 × 2) = 9/4

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