Axis of Symmetry of a Parabola

Picture of Axis of symmetry of a parabola

Every parabola has an axis of symmetry which is the line that runs down its 'center'. This line divides the graph into two perfect halves.

Explore how the graph and equation relate to the axis of symmetry, by using our interactive program below.

Identify Axis of Symmetry From Graph

Problem 1

In Graph 2, what is the equation of the axis of symmetry ?

It is the line $$ x = 2 $$

Problem 2

In Graph 3, what is the equation of the axis of symmetry ?

It is the line $$ x = -1 $$

Axis of Symmetry Formula

There are two different formulas that you can use to find the axis of symmetry. One formula works when the parabola's equation is in vertex form and the other works when the parabola's equation is in standard form.

If your equation is in vertex form, then the axis of is:
x= h in the general vertex form equation y = (x-h)2 + k

If your equation is in standard form, then the formula for the axis of symmetry is:
$$ x = \frac {-b}{ 2a} $$ from the general standard form equation $$ y = ax^2 +bx + c $$

Practice finding axis

of symmetry from Vertex Form Equation

Problem 3

What is the following parabola's axis of symmetry of $$ y = (x + 3)^2 + 4$$

Since this equation is in vertex form,use the formula x = h

The line $$ x = -3 $$ is this parabola's axis of symmetry.

Problem 4

What is the axis of symmetry of the parabola $$ y = (x - 3)^2 + 4 $$

Since this equation is in vertex form,use the formula x = h
The axis of symmetry is the line $$ x = 3$$

Problem 5

What is the axis of symmetry of $$ y = (x - 1)^2 + 1 $$

Practice

finding axis of symmetry from Vertex Form Equation

Problem 6

What is the following parabola's axis of symmetry of $$ y =x^2 - 2x - 3 $$

Since this equation is in standard form, use the formula for standard form equation $$ x = \frac{ -b}{ 2a} $$

Answer: the axis of symmetry is the line $$ x = 1 $$

Problem 7

What is the following parabola's axis of symmetry of $$y =x^2 - 4x + 5 $$

Since this equation is in standard form, use the formula for standard form equation x = -b/2a

Answer: the axis of symmetry is the line x = 2

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