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
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,
By rearranging the terms of the above equation
This implies that for every possible value f (x) there are two corresponding x-values. This can be clearly seen in the diagram below.
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