Axis of Symmetry Formula

We get the Axis of symmetry formula by taking the axis of symmetry of a parabola in the form of the equation that can be represented in two forms i.e:

1. Standard form

2. Vertex form

For Standard form, the quadratic equation is, y= ax² + bx + c

Where a, b are the coefficients of variable "x" while c is the constant form.

For the standard form of quadratic equation, the axis of symmetry formula is: - x= -b/2a

On the other hand, For Vertex form, the quadratic equation is, y= a(x-h)² + k

Where (h, k) represents the vertex of the parabola.

In vertex form, we can assert, x=h because the vertex and axis of symmetry lie on the same line.

For vertex form, the axis of symmetry formula is:- h= -b/2a

Axis of Symmetry of A Parabola

Every parabola contains an axis of symmetry which is actually a line segment that divides the graph into two perfect halves.

Derive Formula For Axis of Symmetry of A Parabola

The formula for the Axis of symmetry for a parabola is: x= -b/2a

We need to find why the axis of symmetry has the above-stated formula.

Here's how it is.

By the explanation of the axis of symmetry for a parabola, it is a vertical line that splits the parabola into two similar halves.

The axis of symmetry always crosses across the vertex of the parabola.

The quadratic equation of a parabola is, y= ax² + bx + c

Where

a and b = coefficient of x

'c' = constant term.

The quadratic equation of a parabola is, y= ax² + bx + c equals to y= ax² + bx because the constant term 'c' does not impose any impact upon the parabola.

Thus, we take,  

y= ax² + bx

Now suppose that, y=0 and x=0, that suggests,

ax² + bx = 0

x (ax² + b) = 0

(ax + b) = 0

Because that taking x on the other side

X = -b/a

Thereby, the two values of x are

0 and –b/a

The midpoint formula is as follows:-

X= x₁ + x₂ / 2

X= 0 + (-b/a) /2

Hence, the quadratic formula axis of symmetry is proved as:-

x= -b/2a

Solved Example

Example:

A math teacher calls his student to construct the axis of symmetry in a graphical representation along the x-axis. How will you help the student draw the axis of symmetry?

Hint: a = 3, b = -12 & c = 5

Solution:

We know the parabola axis of symmetry formula, x= -b/2a

To find the axis of symmetry of the parabola, plug in the value of a and b, we get

x=−2(3)

Therefore, x=2

[Image will be uploaded soon]