The Vertex of the Parabola - Examples, Exercises and Solutions

Question Types:
The Vertex of the Parabola: Finding a stationary pointThe Vertex of the Parabola: Linking function properties to its representation

Vertex of the Parabola

The vertex of the parabola indicates the highest or maximum point of a sad-faced parabola, and the lowest or minimum point of a happy-faced parabola.

The first method to find the vertex of the parabola: (with formula)

First step: We will find the XX of the vertex according to the formula x=(−b)2ax=\frac{(-b)}{2a}

Second step: We will place the XX of the vertex we have found into the original parabola equation to find the YY of the vertex.

Second method to find the vertex of the parabola: according to 2 points of intersection with the X-axis and use of symmetry

First step: Find two points of intersection of the parabola with the XX axis using the quadratic formula.

Second step: Find the XX of the vertex: the point that is exactly between two points of intersection. The calculation will be done through the average of two XXs of the intersection points.

Third step: Place the XX of the vertex we have found into the original parabola equation to solve for the YY of the vertex.

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

Practice The Vertex of the Parabola

Question 1

The following function has been graphed below:

\( f(x)=x^2-8x+16 \)

Calculate point C.

CCC

Question 2

The following function has been graphed below.

\( f(x)=-x^2+5x+6 \)

Calculate point C.

BBBAAACCC

Question 3

The following function has been graphed below:

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

Calculate point C.

CCCAAABBB

Question 4

The following function has been graphed below.

\( f(x)=x^2-6x+8 \)

Calculate point B.

BBB

Question 5

Find the vertex of the parabola

\( y=(x+1)^2 \)

Examples with solutions for The Vertex of the Parabola

Exercise #1

The following function has been graphed below:

f(x)=x2−8x+16 f(x)=x^2-8x+16

Calculate point C.

CCC

Video Solution

Step-by-Step Solution

To solve the exercise, first note that point C lies on the X-axis.

Therefore, to find it, we need to understand what is the X value when Y equals 0.

 

Let's set the equation equal to 0:

0=x²-8x+16

We'll use the preferred method (trinomial or quadratic formula) to find the X values, and we'll discover that

X=4

 

Answer

(4,0) (4,0)

Exercise #2

The following function has been graphed below.

f(x)=−x2+5x+6 f(x)=-x^2+5x+6

Calculate point C.

BBBAAACCC

Video Solution

Step-by-Step Solution

To solve the question, let's recall the formula for finding the vertex of a parabola:

Let's substitute the known data into the formula:

-5/2(-1)=-5/-2=2.5

In other words, the x-coordinate of the vertex of the parabola is found when the X value equals 2.5,

Now let's substitute this into the parabola equation and find the Y value

-(2.5)²+5*2.5+6= 12.25

Therefore, the coordinates of the vertex of the parabola are (2.5,12.25).

Answer

(212,1214) (2\frac{1}{2},12\frac{1}{4})

Exercise #3

The following function has been graphed below:

f(x)=x2−6x f(x)=x^2-6x

Calculate point C.

CCCAAABBB

Video Solution

Answer

(3,−9) (3,-9)

Exercise #4

The following function has been graphed below.

f(x)=x2−6x+8 f(x)=x^2-6x+8

Calculate point B.

BBB

Video Solution

Answer

(3,−1) (3,-1)

Exercise #5

Find the vertex of the parabola

y=(x+1)2 y=(x+1)^2

Video Solution

Answer

(−1,0) (-1,0)

Question 1

Find the vertex of the parabola

\( y=(x-1)^2-1 \)

Question 2

Find the vertex of the parabola

\( y=(x-3)^2-1 \)

Question 3

Find the vertex of the parabola

\( y=x^2+3 \)

Question 4

Find the vertex of the parabola

\( y=x^2 \)

Question 5

Find the vertex of the parabola

\( y=x^2-6 \)

Exercise #6

Find the vertex of the parabola

y=(x−1)2−1 y=(x-1)^2-1

Video Solution

Answer

(1,−1) (1,-1)

Exercise #7

Find the vertex of the parabola

y=(x−3)2−1 y=(x-3)^2-1

Video Solution

Answer

(3,−1) (3,-1)

Exercise #8

Find the vertex of the parabola

y=x2+3 y=x^2+3

Video Solution

Answer

(0,3) (0,3)

Exercise #9

Find the vertex of the parabola

y=x2 y=x^2

Video Solution

Answer

(0,0) (0,0)

Exercise #10

Find the vertex of the parabola

y=x2−6 y=x^2-6

Video Solution

Answer

(0,−6) (0,-6)

Question 1

Find the vertex of the parabola

\( y=(x+1)^2-1 \)

Question 2

Find the vertex of the parabola

\( y=(x-3)^2 \)

Question 3

Find the vertex of the parabola

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

Question 4

Find the vertex of the parabola

\( y=(x-7)-7 \)

Question 5

Find the vertex of the parabola

\( y=(x+8)^2-9 \)

Exercise #11

Find the vertex of the parabola

y=(x+1)2−1 y=(x+1)^2-1

Video Solution

Answer

(−1,−1) (-1,-1)

Exercise #12

Find the vertex of the parabola

y=(x−3)2 y=(x-3)^2

Video Solution

Answer

(3,0) (3,0)

Exercise #13

Find the vertex of the parabola

y=(x−3)2+6 y=(x-3)^2+6

Video Solution

Answer

(3,6) (3,6)

Exercise #14

Find the vertex of the parabola

y=(x−7)−7 y=(x-7)-7

Video Solution

Answer

(7,−7) (7,-7)

Exercise #15

Find the vertex of the parabola

y=(x+8)2−9 y=(x+8)^2-9

Video Solution

Answer

(−8,−9) (-8,-9)

Topics learned in later sections

  1. Symmetry in a parabola