Zeros of a Fuction - Examples, Exercises and Solutions

Question Types:
Zeros of a Fuction: Calculating geometric shapesZeros of a Fuction: Finding a stationary pointZeros of a Fuction: Linking function properties to its representation

Finding the zeros of a quadratic function of the form \(y=ax^2+bx+c\)

Zero points of a function are its intersection points with the XX-axis.
To find them, we set Y=0 Y=0 ,
we get an equation that can sometimes be solved using a trinomial or the quadratic formula.

When trying to find the zero point, you can encounter three possible results:

  1. Two results -
    In this case, the function intersects the XX-axis at two different points.
  2. One result -
    In this case, the function intersects the XX-axis at only one point, meaning the vertex of the parabola is exactly on the XX-axis.
  3. No results -
    In this case, the function does not intersect the XX-axis at all, meaning it hovers above or below it.
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

Practice Zeros of a Fuction

Question 1

The following function has been graphed below:

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

Calculate point C.

CCC

Question 2

Determine the points of intersection of the function

\( y=(x-5)(x+5) \)

With the X

Question 3

The following function has been graphed below:

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

Calculate points A and B.

BBBAAACCC

Question 4

The following function has been graphed below:

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

Calculate points A and B.

\( \)AAABBB

Question 5

The following function has been graphed below:

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

Calculate point A.

AAACCC

Examples with solutions for Zeros of a Fuction

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

Determine the points of intersection of the function

y=(x−5)(x+5) y=(x-5)(x+5)

With the X

Video Solution

Step-by-Step Solution

In order to find the point of the intersection with the X-axis, we first need to establish that Y=0.

 

0 = (x-5)(x+5)

When we have an equation of this type, we know that one of these parentheses must be equal to 0, so we begin by checking the possible options.

x-5 = 0
x = 5

 

x+5 = 0
x = -5

That is, we have two points of intersection with the x-axis, when we discover their x points, and the y point is already known to us (0, as we placed it):

(5,0)(-5,0)

This is the solution!

Answer

(5,0),(−5,0) (5,0),(-5,0)

Exercise #3

The following function has been graphed below:

f(x)=−x2+5x+6 f(x)=-x^2+5x+6

Calculate points A and B.

BBBAAACCC

Video Solution

Answer

(−1,0),(6,0) (-1,0),(6,0)

Exercise #4

The following function has been graphed below:

f(x)=x2−6x+5 f(x)=x^2-6x+5

Calculate points A and B.

AAABBB

Video Solution

Answer

(1,0),(5,0) (1,0),(5,0)

Exercise #5

The following function has been graphed below:

f(x)=x2−8x+16 f(x)=x^2-8x+16

Calculate point A.

AAACCC

Video Solution

Answer

(0,16) (0,16)

Question 1

The following function has been graphed below:

\( f(x)=x^2-3x-4 \)

Calculate points A and B.

CCCAAABBB

Question 2

The following function has been graphed below:

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

Calculate points A and B.

AAABBB

Question 3

The following function has been graphed below:

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

Calculate points A and B.

AAABBB

Question 4

Determine the points of intersection of the function

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

With the X

Question 5

Determine the points of intersection of the function

\( y=(x-1)(x+10) \)

With the X

Exercise #6

The following function has been graphed below:

f(x)=x2−3x−4 f(x)=x^2-3x-4

Calculate points A and B.

CCCAAABBB

Video Solution

Answer

A(−1,0),B(4,0) A(-1,0),B(4,0)

Exercise #7

The following function has been graphed below:

f(x)=x2−6x f(x)=x^2-6x

Calculate points A and B.

AAABBB

Video Solution

Answer

(6,0),(0,0) (6,0),\lparen0,0)

Exercise #8

The following function has been graphed below:

f(x)=x2−6x+8 f(x)=x^2-6x+8

Calculate points A and B.

AAABBB

Video Solution

Answer

(2,0),(4,0) (2,0),(4,0)

Exercise #9

Determine the points of intersection of the function

y=(x−2)(x+3) y=(x-2)(x+3)

With the X

Video Solution

Answer

(−3,0),(2,0) (-3,0),(2,0)

Exercise #10

Determine the points of intersection of the function

y=(x−1)(x+10) y=(x-1)(x+10)

With the X

Video Solution

Answer

(1,0),(−10,0) (1,0),(-10,0)

Question 1

Determine the points of intersection of the function

\( y=(x-3)(x+3) \)

With the X

Question 2

Determine the points of intersection of the function

\( y=x(x+5) \)

With the X

Question 3

Determine the points of intersection of the function

\( y=(x+7)(x+2) \)

With the X

Question 4

Determine the points of intersection of the function

\( y=(x+3)(x-3) \)

With the X

Question 5

Determine the points of intersection of the function

\( y=(x-11)(x+1) \)

With the X

Exercise #11

Determine the points of intersection of the function

y=(x−3)(x+3) y=(x-3)(x+3)

With the X

Video Solution

Answer

(−3,0),(3,0) (-3,0),(3,0)

Exercise #12

Determine the points of intersection of the function

y=x(x+5) y=x(x+5)

With the X

Video Solution

Answer

(−5,0),(0,0) (-5,0),(0,0)

Exercise #13

Determine the points of intersection of the function

y=(x+7)(x+2) y=(x+7)(x+2)

With the X

Video Solution

Answer

(−2,0),(−7,0) (-2,0),(-7,0)

Exercise #14

Determine the points of intersection of the function

y=(x+3)(x−3) y=(x+3)(x-3)

With the X

Video Solution

Answer

(3,0),(−3,0) (3,0),(-3,0)

Exercise #15

Determine the points of intersection of the function

y=(x−11)(x+1) y=(x-11)(x+1)

With the X

Video Solution

Answer

(−1,0),(11,0) (-1,0),(11,0)

Topics learned in later sections

  1. Positive and Negative intervals of a Quadratic Function
  2. Vertex of a parabola
  3. Symmetry in a parabola