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Axis of Symmetry of a Parabola

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The dictionary meaning of symmetry is “such proportion and balance that is both harmonious and attractive.” However, it has a more exact definition in mathematics, and it generally refers to an object that is stable under certain transformations, such as rotation, mirroring or translation. Symmetrical forms or figures are things that can have a line drawn through them so that the representations on both sides of the line mirror each other. Look at the triangle below, which when divided by a line segment, gets parted into two identical shapes, which are like mirror reflections of each other.

Axis of Symmetry

A line that splits an item into two equal halves, providing a mirror-like replica of either side of the object, is known as an axis of symmetry. The word symmetry connotes a sense of equilibrium. Symmetry may be used in a variety of circumstances and scenarios. Symmetry is a fundamental idea in geometry that divides a shape into two halves, each of which is a perfect reflection of the other, as seen in the diagram below. Different forms have various symmetry lines. A square has four symmetry lines, a rectangle has two symmetry lines, a circle has infinite symmetry lines, and a parallelogram has none. A regular polygon with n sides has n symmetry axes. 

The axes of symmetry of a pentagon are shown below:

Axis of Symmetry of a Parabola

A parabola has just one symmetry line. The straight line that splits a parabola into two symmetrical pieces is the axis of symmetry. There are four different types of parabolas. It 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.

Formula of Axis of Symmetry of a Parabola

For a parabola with an equation of the form ax2 + bx + c, the axis of symmetry can be calculated using the following formula:

x = −b/2a

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

Derivation of the Formula

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 = \frac{0+[-\frac{b}{a}]}{2}

⇒ x = -b/2a

Hence proved.

Sample Problems

Question 1. Find the axis of symmetry of the parabola y = x2 − 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 = x2 + 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



Last Updated : 10 Jan, 2024
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