Table of Contents

Last modified on October 8th, 2024

The axis of symmetry of a parabola is a vertical line that divides a parabola into two halves. This line passes through the vertex of the parabola, which is the highest or lowest point of the graph, depending on its orientation.

**Note**: For a parabola facing upward or downward, the axis of symmetry is a vertical line (one is shown above). However, for those that open to the left or right, it is a horizontal line, as shown.

The axis of symmetry is thus the axis of the parabola.

For quadratic functions in standard form, the axes of symmetry are:

- For y = ax
^{2}+ bx + c, the axis of symmetry is:**${x=-\dfrac{b}{2a}}$** - For x = ay
^{2}+ by + c, the axis of symmetry is:**${y=-\dfrac{b}{2a}}$**

When parabolas are in vertex form, the axes of symmetry are:

- For y = a(x – h)
^{2}+ k, the axis of symmetry is:**x = h** - For x = a(y – k)
^{2}+ h, the axis of symmetry is:**y = k**

Considering the standard quadratic function y = ax^{2} + bx + c

Here, the constant term ‘c’ does not affect the parabola; thus, the equation can be written as y = ax^{2} + bx …..(i)

According to the definition, the axis of symmetry is the midpoint of its two x-intercepts.

To find the x-intercept, substituting y = 0 in the equation (i), we get

0 = ax^{2} + bx

⇒ x(ax + b) = 0

⇒ x = 0 and x = ${-\dfrac{b}{2a}}$

Now, the mid-point formula is ${x=\dfrac{\left( x_{1}+x_{2}\right) }{2}}$

⇒ ${x=\dfrac{\left( 0+\left( -\dfrac{b}{a}\right) \right) }{2}}$

⇒ ${x=-\dfrac{b}{2a}}$

This is the equation of the axis of symmetry for parabolas in the standard form y = ax^{2} + bx + c

Similarly, if the parabola opens horizontally (i.e., left/right), we can get the equation for the axis of symmetry by finding the midpoint of the y-intercepts.

**Finding the Axis of Symmetry of the Parabola y = 2x**^{2}** + 8x + 5**

Given, y = 2x^{2} + 8x + 5 …..(i)

Comparing the equation (i) of the parabola with the standard form y = ax^{2} + bx + c, we have

a = 2, b = 8, and c = 5

Using the formula, the axis of symmetry is ${x=-\dfrac{b}{2a}}$

⇒ ${x=-\dfrac{8}{2\times 2}}$

⇒ ${x=-2}$

Thus, the axis of symmetry is x = -2

By plotting the axis of symmetry of the parabola on the graph, we get

**Determine the axis of symmetry of the parabola y = 3x ^{2} **

Solution:

Given y = 3x^{2} …..(i)

As we know, the standard form of the parabola is y = ax^{2} + bx + c …..(ii)

Comparing the equations (i) and (ii), we get

a = 3, b = 0, c = 0

As we know, the axis of symmetry is x = ${-\dfrac{b}{2a}}$

Here, x = ${-\dfrac{0}{2\times 3}}$

⇒ x = ${0}$

Thus, the axis of symmetry is x = 0

**Find the axis of symmetry of the parabolic function x = y ^{2} – 4y + 3**

Solution:

Given x = y^{2} – 4y + 3 …..(i)

As we know, the standard form of the parabola is x = ay^{2} + by + c …..(ii)

Comparing the equations (i) and (ii), we get

a = 1, b = -4, and c = 3

As we know, the axis of symmetry is y = ${-\dfrac{b}{2a}}$

Here, y = ${-\dfrac{-4}{2\times 1}}$

⇒ y = ${\dfrac{4}{2}}$

⇒ y = ${2}$

Thus, the axis of symmetry is y = 2

**Find the equation of the axis of symmetry for f(x) = -x ^{2} + 4x – 1**

Solution:

Given f(x) = -x^{2} + 4x – 1 …..(i)

As we know, the standard form of the parabola is x = ay^{2} + by + c …..(ii)

Comparing the equations (i) and (ii), we get

a = -1, b = 4, c = -1

As we know, the axis of symmetry is x = ${-\dfrac{b}{2a}}$

Here, x = ${-\dfrac{4}{2\times \left( -1\right) }}$

⇒ x = ${2}$

Thus, the axis of symmetry is x = 2

**Find the equation of the axis of symmetry for y = 3(x – 5) ^{2} – 4**

Solution:

Given y = 3(x – 5)^{2} – 4 …..(i)

As we know, the vertex form of the parabola is y = a(x – h)^{2} + k …..(ii)

Comparing the equations (i) and (ii), we get

h = 5 and k = -4

Thus, the axis of symmetry is x = 5

**Determine the axis of symmetry of the parabola y = 2(x + 4) ^{2} – 3**

Solution:

Given y = 2(x + 4)^{2} – 3 …..(i)

As we know, the vertex form of the equation of the parabola is y = a(x – h)^{2} + k …..(ii)

Comparing the equations (i) and (ii), we get

h = 4 and k = -3

Thus, the axis of symmetry is x = 4

**Determine the axis of symmetry of the parabola x = 4(y + 3) ^{2} + 1**

Solution:

Given x = 4(y + 3)^{2} + 1 …..(i)

As we know, the vertex form of the parabola is x = a(y – k)^{2} + h …..(ii)

Comparing the equations (i) and (ii), we get

h = 1 and k = -3

Thus, the axis of symmetry is y = -3

Last modified on October 8th, 2024