# Symmetry - Examples, Exercises and Solutions

The axis of symmetry in a parabola is the axis that passes through its vertex in such a way that if we folded the right side over the left side, both sides would appear joined.
Let's see it in an illustration:

To find the axis of symmetry, we must locate the value of $X$ of the vertex of the parabola or do it through the parabola's vertex formula or with the help of two symmetric points on the parabola.

## Vertex Formula of the Parabola

$X=\frac{-b}{2a}$

## Formula for two symmetric points:

$X_{Vertex}=\frac{The~value~of~X~at~the~first~point~+~The~value~of~X~at~the~second~point}{2}$

## Examples with solutions for Symmetry

### Exercise #1

What is the axis of symmetry of the equation?

$y=(x-5)^2+15$

### Step-by-Step Solution

The first step in solving the equation you presented:

y=(x-5)²+15

is to expand the parentheses:

y=x²-10x+25+15

y=x²-10x+40

From here, we can use the formula to find the X-coordinate of the vertex:

-b/2a

Let's substitute the values from the equation:

-(-10)/2*1 =

10/2=5

The axis of symmetry of the parabola is X=5

$x=5$

### Exercise #2

Given the expression of the quadratic function

Finding the symmetry point of the function

$f(x)=-5x^2+10$

### Video Solution

$(0,10)$

### Exercise #3

Given the expression of the quadratic function

Finding the symmetry point of the function

$f(x)=2x^2$

### Video Solution

$(0,0)$

### Exercise #4

Given the expression of the quadratic function

Finding the symmetry point of the function

$f(x)=\frac{1}{2}x^2$

### Video Solution

$(0,0)$

### Exercise #5

Given the expression of the quadratic function

The symmetrical axis of the expression must be found.

$f(x)=-3x^2+3$

### Video Solution

$x=0$

### Exercise #6

Given the expression of the quadratic function

The symmetrical axis of the expression must be found.

$f(x)=7x^2$

### Video Solution

$x=0$

### Exercise #7

Given the expression of the quadratic function

Finding the symmetry point of the function

$f(x)=3x^2+6x$

### Video Solution

$(-1,-3)$

### Exercise #8

Given the expression of the quadratic function

Finding the symmetry point of the function

$f(x)=3-5x^2$

### Video Solution

$(0,3)$

### Exercise #9

Given the expression of the quadratic function

Finding the symmetry point of the function

$f(x)=3+3x^2$

### Video Solution

$(0,3)$

### Exercise #10

A quadratic equation is graphed below.

What is the axis of symmetry for the graph $f(x)=3x^2+2$?

### Video Solution

$x=0$

### Exercise #11

Given the expression of the quadratic function

The symmetrical axis of the expression must be found.

$f(x)=4x^2+6$

### Video Solution

$x=0$

### Exercise #12

Given the expression of the quadratic function

Finding the symmetry point of the function

$f(x)=5x-x^2$

### Video Solution

$(2\frac{1}{2},6\frac{1}{4})$

### Exercise #13

Given the expression of the quadratic function

Finding the symmetry point of the function

$f(x)=-3x^2+12$

### Video Solution

$(0,12)$

### Exercise #14

Given the expression of the quadratic function

Finding the symmetry point of the function

$f(x)=-4x^2+8x+3$

### Video Solution

$(1,7)$

### Exercise #15

Given the expression of the quadratic function

Finding the symmetry point of the function

$f(x)=-6x^2+24x$

### Video Solution

$(2,24)$