How to find the Axis of Symmetry of a Quadratic Function

What is a Quadratic Function

A polynomial function of second degree is called a quadratic function. Formally,  f(x)=ax2+bx+c is a quadratic function, where a,b and c are real constant and a≠0 for all values of x. The graph of a quadratic function is a parabola.

How to find the Axis of Symmetry of a Quadratic Function |

How to find the axis of symmetry of a Quadratic Function 

Any quadratic function shows lateral symmetry across y axis or a line parallel to it. The axis of symmetry of a quadratic function can be found as follows:

f(x)=ax2+bx+c where a,b,c,x∈R and a≠0

Writing x terms as a full square we have,

How to find the Axis of Symmetry of a Quadratic Function |

By rearranging the terms of the above equation

How to find the Axis of Symmetry of a Quadratic Function|

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 Axis of Symmetry of a Quadratic Function | clickspay.ruThese values are located,

How to find the Axis of Symmetry of a Quadratic Function | 


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

x=-b/2a is the equation of the axis of symmetry for a given quadratic function in the form f(x)=ax2+bx+c

How to find the axis of symmetry of a Quadratic Function – Examples

  • A quadratic function is given by f(x)=4x2+x+1. Find the symmetric axis.


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

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

By simplifying the expression we have f(x)=2x2-5x-4x+10=2x2-9x+10

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

x=- (-9)/(2×2)=9/4

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