Symmetry in a parabola

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:

Symmetry 1

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:

B3 - The formula to find X a vertex using two symmetric points

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

Detailed explanation

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Test yourself on symmetry!

einstein

Given the expression of the quadratic function

The symmetrical axis of the expression must be found.

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

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Let's look at an example

First method: Solve for X at the vertex based on the formula.

Given the function $x^2+8x+5$
Let's plug it into the formula and we will get:
$x=-\frac{8}{2\cdot1}=-\frac{8}{1}=-8$
From this it follows that, the axis of symmetry is $X=-8$

Second method: Find the axis of symmetry based on symmetrical points.

Given the points $(2,2)$ , $(6,2)$

Let's plug them into the formula and we will get:
$x=\frac{6+2}{2}=\frac{8}{2}=4$
The axis of symmetry is $X=4$

If you are interested in this article, you might also be interested in the following articles:

Symmetry in Trapezoids

Rotational Symmetry in Parallelograms

Symmetry of the Rhombus

In the Tutorela blog, you will find a variety of articles on mathematics.

Examples and exercises with solutions on symmetry

Exercise #1

What is the axis of symmetry of the equation?

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

Video Solution

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

Answer

$x=5$

Exercise #2

Given the expression of the quadratic function

The symmetrical axis of the expression must be found.

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

Video Solution

Answer

$x=0$

Exercise #3

Given the expression of the quadratic function

The symmetrical axis of the expression must be found.

$f(x)=7x^2$

Video Solution

Answer

$x=0$

Exercise #4

Given the expression of the quadratic function

Finding the symmetry point of the function

$f(x)=2x^2$

Video Solution

Answer

$(0,0)$

Exercise #5

Given the expression of the quadratic function

Finding the symmetry point of the function

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

Video Solution

Answer

$(0,10)$

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Test your knowledge

Question 1

Given the expression of the quadratic function

The symmetrical axis of the expression must be found.

$\( f(x)=7x^2 \)$

Question 2

Given the expression of the quadratic function

Finding the symmetry point of the function

$\( f(x)=2x^2 \)$

Question 3

Given the expression of the quadratic function

Finding the symmetry point of the function

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

Start practice

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