Product Representation - Examples, Exercises and Solutions

Question Types:
Product Representation: Transition between different representations of a functionProduct Representation: Linking function properties to its representation

Factored form of the quadratic function

This form is called factored because it uses the factors of a multiplication.

With this form, we can easily identify the points of intersection of the function with the XX axis.
The factored form of the quadratic function looks like this:
y=(x−t)×(x−k)y=(x-t) \times (x-k)

Detailed explanation

Suggested Topics to Practice in Advance

  1. Standard Form of the Quadratic Function

Practice Product Representation

Question 1

Find the standard representation of the following function

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

Question 2

Determine the points of intersection of the function

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

With the X

Question 3

Find the standard representation of the following function

\( f(x)=3x(x+4) \)

Question 4

Find the standard representation of the following function

\( f(x)=(x+2)(x-4) \)

Question 5

Find the standard representation of the following function

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

Examples with solutions for Product Representation

Exercise #1

Find the standard representation of the following function

f(x)=(x−2)(x+5) f(x)=(x-2)(x+5)

Video Solution

Step-by-Step Solution

We will begin by using the distributive property in order to expand the following expression.

(a+1)⋆(b+2) = ab+2a+b+2

We will then proceed to insert the known values into the equation and solve as follows:

(x-2)(x+5) =

x²-2x+5x+-2*5=

x²+3x-10

And that's the solution!

Answer

f(x)+x2+3x−10 f(x)+x^2+3x-10

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

Find the standard representation of the following function

f(x)=3x(x+4) f(x)=3x(x+4)

Video Solution

Answer

f(x)=3x2+12x f(x)=3x^2+12x

Exercise #4

Find the standard representation of the following function

f(x)=(x+2)(x−4) f(x)=(x+2)(x-4)

Video Solution

Answer

f(x)=x2−2x−8 f(x)=x^2-2x-8

Exercise #5

Find the standard representation of the following function

f(x)=(x−6)(x−2) f(x)=(x-6)(x-2)

Video Solution

Answer

f(x)=x2−8x+12 f(x)=x^2-8x+12

Question 1

Find the standard representation of the following function

\( f(x)=-x(x-8) \)

Question 2

Consider the following function:

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

Determine the points of intersection with x.

Question 3

Determine the points of intersection of the function

\( y=(4x+8)(x+1) \)

With the X

Question 4

Determine the points of intersection of the function

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

With the X

Question 5

Determine the points of intersection of the function

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

With the X

Exercise #6

Find the standard representation of the following function

f(x)=−x(x−8) f(x)=-x(x-8)

Video Solution

Answer

f(x)=−x2+8x f(x)=-x^2+8x

Exercise #7

Consider the following function:

y=x(x−1) y=x(x-1)

Determine the points of intersection with x.

Video Solution

Answer

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

Exercise #8

Determine the points of intersection of the function

y=(4x+8)(x+1) y=(4x+8)(x+1)

With the X

Video Solution

Answer

(−1,0),(−2,0) (-1,0),(-2,0)

Exercise #9

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)

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

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

With the X

Question 2

Determine the points of intersection of the function

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

With the X

Question 3

Determine the points of intersection of the function

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

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

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

With the X

Exercise #11

Determine the points of intersection of the function

y=(x−1)(x−1) y=(x-1)(x-1)

With the X

Video Solution

Answer

(1,0) (1,0)

Exercise #12

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 #13

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

With the X

Video Solution

Answer

(−2,0),(−7,0) (-2,0),(-7,0)

Topics learned in later sections

  1. Ways to represent a quadratic function
  2. Vertex form of the quadratic equation