Symmetry - Examples, Exercises and Solutions

Question Types:
Symmetry: Axis of symmetrySymmetry: Finding a stationary point

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 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=−b2a X=\frac{-b}{2a}

Formula for two symmetric points:

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

XVertex=The value of X at the first point + The value of X at the second point2 X_{Vertex}=\frac{The~value~of~X~at~the~first~point~+~The~value~of~X~at~the~second~point}{2}

Detailed explanation

Suggested Topics to Practice in Advance

  1. The quadratic function
  2. Parabola
  3. Plotting the Quadratic Function Using Parameters a, b and c
  4. Finding the Zeros of a Parabola
  5. Positive and Negative intervals of a Quadratic Function
  6. Vertex of a parabola

Practice Symmetry

Question 1

What is the axis of symmetry of the equation?

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

Question 2

Given the expression of the quadratic function

The symmetrical axis of the expression must be found.

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

Question 3

Given the expression of the quadratic function

The symmetrical axis of the expression must be found.

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

Question 4

Given the expression of the quadratic function

Finding the symmetry point of the function

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

Question 5

Given the expression of the quadratic function

Finding the symmetry point of the function

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

Examples with solutions for Symmetry

Exercise #1

What is the axis of symmetry of the equation?

y=(x−5)2+15 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 x=5

Exercise #2

Given the expression of the quadratic function

The symmetrical axis of the expression must be found.

f(x)=−3x2+3 f(x)=-3x^2+3

Video Solution

Answer

x=0 x=0

Exercise #3

Given the expression of the quadratic function

The symmetrical axis of the expression must be found.

f(x)=7x2 f(x)=7x^2

Video Solution

Answer

x=0 x=0

Exercise #4

Given the expression of the quadratic function

Finding the symmetry point of the function

f(x)=2x2 f(x)=2x^2

Video Solution

Answer

(0,0) (0,0)

Exercise #5

Given the expression of the quadratic function

Finding the symmetry point of the function

f(x)=−5x2+10 f(x)=-5x^2+10

Video Solution

Answer

(0,10) (0,10)

Question 1

Given the expression of the quadratic function

Finding the symmetry point of the function

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

Question 2

Given the expression of the quadratic function

The symmetrical axis of the expression must be found.

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

Question 3

A quadratic equation is graphed below.

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

222

Question 4

Given the expression of the quadratic function

Finding the symmetry point of the function

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

Question 5

Given the expression of the quadratic function

Finding the symmetry point of the function

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

Exercise #6

Given the expression of the quadratic function

Finding the symmetry point of the function

f(x)=12x2 f(x)=\frac{1}{2}x^2

Video Solution

Answer

(0,0) (0,0)

Exercise #7

Given the expression of the quadratic function

The symmetrical axis of the expression must be found.

f(x)=4x2+6 f(x)=4x^2+6

Video Solution

Answer

x=0 x=0

Exercise #8

A quadratic equation is graphed below.

What is the axis of symmetry for the graph f(x)=3x2+2 f(x)=3x^2+2 ?

222

Video Solution

Answer

x=0 x=0

Exercise #9

Given the expression of the quadratic function

Finding the symmetry point of the function

f(x)=3x2+6x f(x)=3x^2+6x

Video Solution

Answer

(−1,−3) (-1,-3)

Exercise #10

Given the expression of the quadratic function

Finding the symmetry point of the function

f(x)=3−5x2 f(x)=3-5x^2

Video Solution

Answer

(0,3) (0,3)

Question 1

Given the expression of the quadratic function

Finding the symmetry point of the function

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

Question 2

Given the expression of the quadratic function

The symmetrical axis of the expression must be found.

\( f(x)=2x^2+8x+4 \)

Question 3

Given the expression of the quadratic function

The symmetrical axis of the expression must be found.

\( f(x)=-2x^2+16 \)

Question 4

Calculate the axis of symmetry of the quadratic function below:

\( f(x)=3x^2+6x-6 \)

Question 5

A quadratic function is graphed below.

What is the axis of symmetry for the graph \( f(x)=x^2+4x \)?

Exercise #11

Given the expression of the quadratic function

Finding the symmetry point of the function

f(x)=3+3x2 f(x)=3+3x^2

Video Solution

Answer

(0,3) (0,3)

Exercise #12

Given the expression of the quadratic function

The symmetrical axis of the expression must be found.

f(x)=2x2+8x+4 f(x)=2x^2+8x+4

Video Solution

Answer

x=−2 x=-2

Exercise #13

Given the expression of the quadratic function

The symmetrical axis of the expression must be found.

f(x)=−2x2+16 f(x)=-2x^2+16

Video Solution

Answer

x=0 x=0

Exercise #14

Calculate the axis of symmetry of the quadratic function below:

f(x)=3x2+6x−6 f(x)=3x^2+6x-6

Video Solution

Answer

x=−1 x=-1

Exercise #15

A quadratic function is graphed below.

What is the axis of symmetry for the graph f(x)=x2+4x f(x)=x^2+4x ?

Video Solution

Answer

x=−2 x=-2