how to find the axis of symmetry

Axis of Symmetry

The axis of symmetry is an imaginary straight line that divides a shape into two identical parts, thereby creating one part as the mirror image of the other part. When folded along the axis of the symmetry, the two parts get superimposed. The straight line is called the line of symmetry/the mirror line. This line can be vertical, horizontal, or slanting. We can see this axis of symmetry even in nature such as flowers, riverbanks, buildings, leaves, and so on. We can observe the axis of symmetry in Taj Mahal, the iconic marble structure in India.

1.	What is Axis of Symmetry?
2.	Axis of Symmetry of a Parabola
3.	Axis of Symmetry Equation
4.	Axis of Symmetry Formula
5.	Axis of Symmetry Formula Examples
6.	Derivation of the Axis of Symmetry
7.	Identification of the Axis of Symmetry
8.	FAQs on Axis of Symmetry

What is Axis of Symmetry?

The axis of symmetry is a straight line that makes the shape of the object symmetrical. The axis of symmetry creates the exact reflections on each of its sides. If we fold and unfold an object along the axis of symmetry, the two sides are identical. The axis of symmetry can be either horizontal, vertical, or lateral. Different shapes have different lines of symmetry. A square has four lines of symmetry, a rectangle has 2 lines of symmetry, a circle has infinite lines of symmetry and a parallelogram has no line of symmetry. A regular polygon of 'n' sides has 'n' axes of symmetry.

axis of symmetry

Axis of Symmetry of a Parabola

A Parabola has one line of symmetry. The axis of the symmetry is the straight line that divides a parabola into two symmetrical parts. The parabola can be in four forms. It can be either horizontal or vertical, facing left or right. The axis of symmetry determines the form of the parabola. If the axis of symmetry is vertical, then the parabola is vertical. If the axis of symmetry is horizontal, then the parabola is horizontal.

axis of symmetry of a parabola

Axis of Symmetry Equation

The vertex is the point where the axis of symmetry intersects the parabola. If the parabola opens up or down, the axis of symmetry is vertical. The equation of the axis of symmetry is the equation of the vertical line that passes through the x-intercept. If the parabola opens right or left, the axis of symmetry is horizontal. The equation of the axis of symmetry is the horizontal line that passes through the y-intercept.

Axis of Symmetry Formula

The axis of symmetry formula helps to find the axis of symmetry of parabola, applied on quadratic equations where the standard form of the equation and the line of symmetry are used. A line that divides or bifurcates any object into two equal halves, both halves of which are mirror images of each other is called the axis of symmetry. This line of axis dividing the objects could be any one of the three types that are: horizontal (x-axis), vertical (y-axis), or inclined axis. In geometry, a line of symmetry implies a line that cuts a geometric shape into two equal halves making it look like a mirror image.

The equation of the axis of symmetry can be represented in two forms:

Standard form
Vertex form

Standard form

The quadratic equation in standard form is, y = ax²+ b x+c

where a, b are the coefficients of "x" and c is the constant form.

Here, the axis of symmetry formula is: x = - b/2a

Vertex form

The quadratic equation in vertex form is, y=a (x-h)²+ k

where (h, k) is the vertex of the parabola. In vertex form, we can say, x = h since the axis of symmetry and vertex lies on the same line: h = - b/2a

Derivation of the Axis of Symmetry for Parabola

The axis of symmetry always passes through the vertex of the parabola. Thus identification of the vertex helps us to calculate the position of the axis of symmetry. Axis of symmetry formula for a parabola is, x = -b/2a. Let us derive the equation of the axis of symmetry.

The quadratic equation of a parabola is, y = a x²+ b x+ c

where (a, b) is the co-efficient of x and 'c' is the constant term.

The constant term 'c' does not affect the parabola.Therefore, let us consider, y = a x²+b x.

Assume y= 0 and x= 0, which implies, a x²+b x=0

x(ax+b)=0

x = 0 and (ax+b)=0

x = -b/a

Therefore, the two values of x= (0,-b/a)

The mid-point formula is x=\(\frac{x_{1}+x_{2}}{2}\)

x= \(\dfrac{0+\dfrac{-b}{a}}{2}\)

Therefore x = -b/2a

If the axis of symmetry lies on the x-axis, we take the y-intercept. y = -b/2a

Axis of Symmetry Formula Examples

Example 1: Using the axis of symmetry formula, find the axis of symmetry of the quadratic equation y = x² - 4x + 3.

Solution:

Given: y = x² - 4x + 3

Using axis of symmetry formula,

x = -b/2a

x = -(-4)/2(1)

x = 4/2

= 2

Therefore, axis of symmetry of equation y = x² - 4x + 3 is x = 2.

Example 2: Find the axis of symmetry of a parabola y = 4x².

Solution:

Using axis of symmetry formula,

x = -b/2a

x = -(0)/2(4)

= 0

Therefore, axis of symmetry of equation y = 4x² is x = 0.

Identification of the Axis of Symmetry

Let us identify the axis of symmetry for the given parabola using the formula learned in the previous section.

1) Consider equation y = x²- 3x + 4. Comparing this with the equation of the standard form of the parabola (y = a x²+ b x+ c), we have

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

This is a vertical parabola. Thus it has a vertical axis of symmetry. We know the equation of the axis of symmetry is at x-axis.

We know that x = -b/2a is the equation of the axis of symmetry.

x = -(-3)/2(1) = 1.5

x = 1.5 is the axis of symmetry of the parabola y = x²- 3x + 4.

2) Let us consider another example. x = 4y²+5y+3.

Comparing with the standard form of the quadratic equation, we get a = 4, b = 5, and c = 3. This parabola is horizontal and the axis of symmetry is horizontal too.

Thus we know the equation of the axis of symmetry is at the y-axis. We know that y = -b/2a is the equation of the axis of symmetry.

y = -b/2a

y = -5/2(4)}

y = -0.625

3) If two points (x,y) at the same distance from the vertex of the parabola are given, then we determine the equation of the axis of symmetry by finding the midpoint of those points. Suppose the two points are (3, 4) and (9, 4), then the vertex passes through the intercept which forms the midpoint of these given points. Thus x = (3+9)/2 = 12/2 = 6. Therefore, the equation of the axis of symmetry passes through (6,0) which is the required equation of the vertical line/ the axis of symmetry.

Example: If the axis of symmetry of the equation y = qx² – 32x – 10 is 8, then find the value of q

Solution: Given,

y = qx² – 32x – 10

Axis of symmetry = x = 8

Applying the axis of symmetry formula in the standard form of the quadratic equation:

x = -b/2a

where a = q, b = -32 and x = 8

8 = -(-32)/2 × q

8 = 32/2q

16q = 32

q = 2

Therefore, the value of q = 2

☛ Also Check:

Lines of symmetry in a rectangle
Lines of symmetry in a parallelogram

Axis of Symmetry Examples

Example 1: Find the equation of the axis of symmetry of the given parabola.

Solution:

The axis of symmetry intersects the parabola at its vertex.

Here the vertex = (-6,6)

Thus the equation of the vertical axis of symmetry is at the x-axis, where y = 0.(y-intercept)

x = 6

Answer: The equation of the given axis of symmetry is x = 6.
Example 2: Find the axis of symmetry of the parabola 3x²-12 x + 5 = 0 and graph it.

Solution:

We need to draw the axis of symmetry in the graph along the x-axis.

Comparing with the standard form of the quadratic equation a x²+b x+ c, we have a = 3, b = -12 & c = 5

We know that, x = -b/2a

x = -(-12)/2(3)

x = 2

Thus the axis of symmetry x = 2 is drawn.
Example 3: Find the equation of the axis of symmetry of the given parabola.

Solution: The axis of symmetry intersects the parabola at its vertex.

Here the vertex = (-4,0)

Thus the equation of the horizontal axis of symmetry is at the y-axis, where y = 0. (x-intercept)

y = 0

Answer: The equation of the given axis of symmetry is y = 0.

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Practice Questions on Axis of Symmetry

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FAQs on Axis of Symmetry

What is The Axis of Symmetry?

The axis of symmetry is an imaginary straight line that divides the shape into two identical parts or that makes the shape symmetrical.

What is the Axis of Symmetry Formula?

The axis of symmetry formula helps in finding out the axis of symmetry of any given graph. The axis of symmetry formula using the standard form of the quadratic equation as well as the vertex form. The symmetry cuts any geometric shape into two equal halves. The axis of symmetry formula is given as, for a quadratic equation with standard form as y = ax² + bx + c, is:
Axis of Symmetry Formula, x = -b/2a

What is the Formula to Calculate the Axis of Symmetry for Standard Form?

The formula used to find the axis of symmetry for a quadratic equation with standard form as y = ax² + bx + c, is:

Axis of Symmetry Formula, x = -b/2a

What is the Axis of Symmetry Formula for Vertex Form?

The quadratic equation is represented in the vertex form as: y = a(x−h)² + k , where (h, k) is the vertex of the parabola. Since the axis of symmetry and the vertex form lie on the same line, hence the formula is:

Axis of symmetry formula ⇒ x = h.

Using the Axis of Symmetry Formula, Find the Axis of Symmetry of the Quadratic Equation y = 5x² - 10x + 3

We can apply the axis of symmetry formula for standard form, to find the axis of symmetry of the given quadratic equation.
Given: y = 5x² - 10x + 3
Using axis of symmetry formula,
x = -b/2a
x = -(-10)/2(5)
x = 10/10
= 1
Therefore, axis of symmetry of equation y = 5x² - 10x + 3 is x = 1.

What is Axis of Symmetry of a Parabola?

The axis of the symmetry is the straight line that divides a parabola into two symmetrical parts. The axis of symmetry passes through the vertex of the parabola. The axis of symmetry of a parabola can be horizontal or vertical.

How Do You Find The Axis of Symmetry Using The Vertex Form of Equation?

The quadratic equation in the vertex form is y = a(x-h)²+k

The axis of symmetry is where the vertex intersects the parabola at the point denoted by the vertex(h,k). h is the x coordinate. and in the vertex form, x = h and h =-b/2a where b and a are the coefficients in the standard form of the equation, y = ax² + bx + c.

How Do You Find The Axis of Symmetry Using The Standard Form of Equation?

The quadratic equation in standard form is y=a x²+b x+c

The axis of the symmetry formula is x =-b/2a, after comparing the values of a, b, and c with the standard form.

What is the Axis of Symmetry on a Graph?

The horizontal or the vertical line on the graph that passes through the vertex of the parabola forms the axis of symmetry of a parabola.