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# 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 this in the Taj Mahal, the iconic marble structure in India.

## 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. It can be either horizontal, vertical, or lateral. If we fold and unfold an object along the axis of symmetry, the two sides are identical. 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 Definition

The axis of symmetry is an imaginary line that divides a figure into two identical parts such that each part is a mirror reflection of one another. When the figure is folded along the axis of symmetry, the two identical parts superimpose.

## 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 (opens up/down).
- If it is horizontal, then the parabola is horizontal (opens left/right).

The axis of symmetry which is horizontal has zero slope, and the axis of symmetry which is vertical has an undefined slope.

## Axis of Symmetry Equation

The vertex is the point where the axis of symmetry intersects the parabola. This is the key point to determine its equation. If the parabola opens up or down, the axis of symmetry is vertical and in this case, its equation is the vertical line that passes through its vertex. If the parabola opens right or left, the axis of symmetry is horizontal and its equation is the horizontal line that passes through its vertex. i.e.,

- The axis of symmetry equation of a parabola whose vertex is (h, k) and opens up/down is x = h.
- The axis of symmetry equation of a parabola whose vertex is (h, k) and opens left/right is y = k.

## Axis of Symmetry Formula

The axis of symmetry formula is 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 line.

The equation of the axis of symmetry can be represented when a parabola is in two forms:

- Standard form
- Vertex form

### Standard form

The quadratic equation in standard form* *is, y = ax^{2}+ b x+c

where a, b, and c are real numbers.

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.

Here, the axis of symmetry formula is **x = h.**

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 = ax^{2 }+ bx + c (up/down parabola).

The constant term ‘c’ does not affect the parabola.Therefore, let us consider, y = ax^{2 }+ 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 and (ax+b)=0

x = 0 and x = -b/a

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

x= [0 + (-b/a)] / 2

Therefore x = -b/2a

**Note: **If the parabola is left/right open, then find the midpoint of y-intercepts.

## Find Axis of Symmetry

**Example 1: **Find the axis of symmetry of the quadratic equation y = x^{2} – 4x + 3.

**Solution:**

Given: y = x^{2} – 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 = x^{2} – 4x + 3 is x = 2.

**Example 2:** Find the axis of symmetry of a parabola y = 4x^{2}.

**Solution:**

Using axis of symmetry formula,

x = -b/2a

x = -(0)/2(4)

x = 0

Therefore, axis of symmetry of parabola y = 4x^{2} 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 = x^{2}– 3x + 4. Comparing this with the equation of the standard form of the parabola (y = ax^{2 }+ bx + 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 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 = x^{2}– 3x + 4.

2) Let us consider another example. x = 4y^{2}+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.

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 are 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 (3, 4) and (9, 4) are points on a parabola, 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 is x = 6.

**Example: **If the axis of symmetry of the equation y = qx^{2} – 32x – 10 is 8, then find the value of q.

**Solution:** Given,

y = qx^{2} – 32x – 10

Axis of symmetry is x = 8

Using the formula:

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.

**Important Notes on Axis of Symmetry**

- An axis of symmetry is an imaginary line that divides a figure into two identical parts that are mirror images of one another.
- For parabola y = ax
^{2}+ b x+c, the axis of symmetry is given by x = -b/2a - A regular polygon of ‘n’ sides has ‘n’ axes of symmetry.

## Axis of Symmetry Examples

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

**Solution:**

Here the vertex = (6, -6)

The axis of symmetry intersects the parabola at its vertex and it is a vertical line since the parabola opens up.

Hence its equation is:

x = 6

**Answer:** x = 6.

**Example 2:** Which line divides the parabola 3x^{2}-12 x + 5 = 0 into two equal parts? Graph it.

**Solution:**

Comparing with the standard form of the quadratic equation ax^{2} + bx + c, we have a = 3, b = -12 & c = 5

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

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

x = 2

Let us graph it now.

**Answer:** x = 2 is drawn.

**Example 3:** Find the equation of the line that divides the given parabola into two equal parts.

**Solution:**

Here the vertex = (-4,0).

The axis of symmetry intersects the parabola at its vertex and is a horizontal line in this case as the parabola opens sideways.

Hence, its equation is y = 0

**Answer:** y = 0.

**Axis of Symmetry Equation Definitions and Examples**

**Introduction**

In mathematics, the term “axis of symmetry” refers to an imaginary line that runs through the center of a shape or object and divides it into two equal halves. An axis of symmetry can be either horizontal, vertical, or diagonal. There are a few different equations that can be used to find the axis of symmetry for a given shape or object. In this blog post, we will go over a few of these equations and provide some examples to help you better understand how they work.

**What is Axis of Symmetry?**

The axis of symmetry is an imaginary line that bisects a two-dimensional figure. It is the line about which a figure is symmetrical. The equation for the axis of symmetry of a parabola is y = x^2. The axis of symmetry is the y-axis.

**Axis of Symmetry of a Parabola**

A parabola is a two-dimensional curve that is the result of a quadratic equation. The axis of symmetry of a parabola is the line that bisects the curve and is perpendicular to the directrix. The axis of symmetry divides the parabola into two mirror images.

The equation for a parabola can be written in standard form, which is y = ax^2 + bx + c. The axis of symmetry is located at x = -b/2a. To find the axis of symmetry, plug in the values for a, b, and c into the equation.

**Axis of Symmetry Equation**

The axis of symmetry equation is a mathematical formula used to determine the location of the axis of symmetry for a given function. The axis of symmetry is the line that divides a function into two mirror image halves. The axis of symmetry equation is used to find the x-coordinate of the point on the graph where the line of symmetry intersects the x-axis.

To find the axis of symmetry equation, first determine whether the function is even or odd. An even function is one that has a graph that is symmetrical about the y-axis, while an odd function has a graph that is symmetrical about the origin. Even functions have an axis of symmetry that passes through the y-axis at the point where y = 0, while odd functions have an axis of symmetry that passes through the origin at the point where x = 0 and y = 0.

Once you have determined whether the function is even or odd, use the following equation to find the axis of symmetry:

For even functions: x = -b/2a

For odd functions: x = 0

**Axis of Symmetry Formula**

The axis of symmetry is an important concept in geometry, and the formula for finding the axis of symmetry of a line or curve is relatively simple. In general, the axis of symmetry is the line that divides a figure into two halves that are mirror images of each other. So, if you were to fold a figure along its axis of symmetry, the two halves would match up perfectly.

There are a few different ways to find the axis of symmetry mathematically. For a linear equation, the axis of symmetry is simply the line x = -b/2a. So, if you have an equation in the form y = mx + b, the axis of symmetry would be x = -b/2m. For a quadratic equation, theaxis of symmetry is usually given by x = -b/2a. However, it’s important to note that this only works for equations where a ? 0. If a = 0, then theaxis of symmetry is simply y = -c/2b.

It’s also worth mentioning that some figures have more than one axis of symmetry. For example, a square has four axes of symmetry that intersect at its center point. Circles also have multiple axes of symmetry that pass through their centers.

**Derivation of the Axis of Symmetry for Parabola**

A parabola is a two-dimensional figure with mirror symmetry. The axis of symmetry is the line that divides the parabola into two equal halves. The axis of symmetry is perpendicular to the directrix and passes through the focus.

The focus is the point on the parabola where all the rays of light converge. The directrix is a line that is perpendicular to the axis of symmetry and passes through the focus.

To find the equation for the axis of symmetry, we need to find the x-coordinate of the focus. This can be done by using the equation for a parabola:

y = ax^2 + bx + c

where a, b, and c are constants. We can use this equation to find the x-coordinate of the focus by setting y = 0:

0 = ax^2 + bx + c

This is a quadratic equation that can be solved using the quadratic formula:

**Find Axis of Symmetry**

An axis of symmetry is an imaginary line that passes through the center of a shape and divides it into two equal halves. The axis of symmetry can be either vertical, horizontal, or diagonal. To find the axis of symmetry of a shape, you need to know the coordinates of the vertices (the corners) of the shape.

To find the axis of symmetry for a shape with vertices at (x1, y1), (x2, y2), and (x3, y3), use the following equation:

Axis of Symmetry = x1 + x2 + x3 / 3

For example, if you have a triangle with vertices at (-1, 2), (3, 4), and (5, -6), then the axis of symmetry would be:

Axis of Symmetry = -1 + 3 + 5 / 3 = 3

**Identification of the Axis of Symmetry**

To identify the axis of symmetry of a parabola, one must first find the equation of the parabola. This can be done by finding the vertex and using it to Plug and Chug, or by using the Quadratic Formula. Once the equation is found, set y = 0 and solve for x. This will give you the x-coordinate of the axis of symmetry.

**Conclusion**

The axis of symmetry is an important concept in math and science, and it has a wide range of applications. We hope that this article has helped you to better understand what an axis of symmetry is and how to find one. With practice, you should be able to quickly identify the axis of symmetry for any given equation. As always, if you have any questions or comments, please feel free to reach out to us!

## FAQs on Axis of Symmetry

### What is Axis of Symmetry in Algebra?

The axis of symmetry is an imaginary line that divides a figure into two identical parts such that each part is a mirror reflection of one another. A regular polygon of ‘n’ sides has ‘n’ axes of symmetry.

### What is The Axis of Symmetry Definition?

The **axis of symmetry** is an imaginary straight line that divides the shape into two identical parts or that makes the shape symmetrical. For example, a square has 4 and a rectangle has 2 axes of symmetry.

### What is the Axis of Symmetry Formula?

The axis of symmetry formula uses 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 = ax^{2} + bx + c, is: x = -b/2a. If the parabola is in vertex form y = a(x-h)^{2} + k, then the formula is x = h.

### 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 = ax^{2} + bx + c, is: 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, the formula is x = h.

### Find the Axis of Symmetry of the Quadratic Equation y = 5x^{2} – 10x + 3.

Given: y = 5x^{2} – 10x + 3

Using axis of symmetry formula,

x = -b/2a

x = -(-10)/2(5)

x = 10/10

x = 1

Therefore, axis of symmetry of equation y = 5x^{2} – 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. It 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 = ax^{2} + bx + c.

### 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. In the case of any other graph, the axis of symmetry is the equation of a line that divides the figure into two equal parts where one is the mirror image of the other.

### Is the Axis of Symmetry the Same as the Line of Symmetry?

Yes, the line of symmetry and the axis of symmetry are the same. They are imaginary lines that divide a figure into two identical parts and each part is a mirror reflection of one another. When the figure is folded along this line, the two parts superimpose.

## Axis of Symmetry of a Parabola

The graph of a quadratic function is a parabola. The axis of symmetry of a parabola is a vertical line that divides the parabola into two congruent halves. The axis of symmetry always passes through the vertex of the parabola . The x -coordinate of the vertex is the equation of the axis of symmetry of the parabola.

For a quadratic function in standard form, y=ax^{2}+bx+c , the axis of symmetry is a vertical line

**Example 1:**

Find the axis of symmetry of the parabola shown.

The x -coordinate of the vertex is the equation of the axis of symmetry of the parabola.

The vertex of the parabola is (2,1) .

So, the axis of symmetry is the line x=2 .

**Example 2:**

Find the axis of symmetry of the graph of y=x^{2}−6x+5 using the formula.

For a quadratic function in standard form, y=ax^{2}+bx+c , the axis of symmetry is a vertical line

Therefore, the axis of symmetry is x=3.