Axis of Symmetry of a Parabola

The axis of symmetry is an imaginary line that divides a shape into two perfect mirror-image halves. Different shapes have different numbers of symmetry axes: a square has four, a rectangle has two, a circle has infinitely many, and a parallelogram has none. In general, a regular polygon with n sides has n axes of symmetry.

The axes of symmetry of a pentagon are shown below:

Parabola-Axis of Symmetry

• The axis of symmetry of a parabola is a line that divides the parabola into two mirror-image halves. This line passes through the vertex of the parabola and is perpendicular to the directrix.
• A parabola has just one symmetry line.
• Axis of symmetry of a parabola might be horizontal or vertical, and it can face left or right. The parabola's shape is determined by its symmetry axis. The parabola is vertical when its axis of symmetry is vertical and vice versa.

Equation of Axis of Symmetry

For a parabola with an equation of the form ax²² + bx + c, the axis of symmetry is

x = −b/2a

where a and b are the coefficients of x² and x, respectively, and c is the constant.

Derivation of Equation of Axis of Symmetry

The vertex of parabola is the only point from where the axis of symmetry passes. A vertical parabola's quadratic equation is y = ax² + bx + c

The parabola is unaffected by the constant term 'c.'

Consider the equation y = ax² + bx.

The axis of symmetry is the midpoint of its two x-intercepts. To find the x-intercept, substitute y = 0.

⇒ x(ax + b) = 0

⇒ x = 0 or, x = -b/a

Using the mid- point formula, we have:

⇒ x = 

⇒ x = -b/2a

Hence proved.

Sample Problems

Question 1. Find the axis of symmetry of the parabola y = x²² − 4x + 8.

Solution:

Given: y = x² − 4x + 8

Compare the given equation to the standard form ax² + bx + c.

⇒ a = 1, b = −4, c = 8

Axis of symmetry = −b/2a

= −(−4)/2(1)

⇒x = 2

Question 2. Find the axis of symmetry of the parabola y = 4x².

Solution:

Given: y = 4x²

Compare the given equation to the standard form ax² + bx + c.

⇒ a = 4, b = 0, c = 0

Axis of symmetry = −b/2a

= 0/2(4)

⇒ x = 0

Question 3. Find the axis of symmetry of the parabola y = 7x².

Solution:

Given: y = 7x²

Compare the given equation to the standard form ax² + bx + c.

⇒ a = 7, b = 0, c = 0

Axis of symmetry = −b/2a

= 0/2(7)

⇒ x = 0

Question 4. Find the axis of symmetry of a parabola y = x²² + 8x − 3.

Solution:

Given: y = x² + 8x − 3

Compare the given equation to the standard form ax² + bx + c.

⇒ a = 1, b = 8, c = –3

Axis of symmetry = −b/2a

= –8/2(1)
⇒ x = –4

Question 5. Find the axis of symmetry of the parabola y = 2x² + 12x.

Solution:

Given: y = 2x² + 12x

Compare the given equation to the standard form ax² + bx + c.
⇒ a = 2, b = 12, c = 0

Axis of symmetry = −b/2a
= −12/2(2)
⇒ x = −3

Question 6. Find the axis of symmetry of the parabola y = 3x² − 6x + 5.

Solution:

Given: y = x² − 6x + 5

Compare the given equation to the standard form ax² + bx + c.
⇒ a = 1, b = −6, c = 5

Axis of symmetry = −b/2a
= −(−6)/2(3)
⇒ x = 1

Question 7. Find the axis of symmetry of the parabola y = 9x².

Solution:

Given: y = 9x²

Compare the given equation to the standard form ax² + bx + c.
⇒ a = 9, b = 0, c = 0

Axis of symmetry = −b/2a
= 0/2(9)
⇒ x = 0

Practice Problems

Problem 1: Find the axis of symmetry for the parabola given by the equation y = 2x + 4x + 1.

Problem 2: Determine the axis of symmetry for the parabola described by y = −3x + 6x − 2.

Problem 3: Identify the axis of symmetry for the quadratic function y = x² − 8x + 15.

Problem 4: What is the axis of symmetry for the parabola y = −x + 10x − 25?

Problem 5: Find the axis of symmetry for the equation y = 4x − 16x + 7.