Positive and Negative Domains - Examples, Exercises and Solutions

Question Types:
Positive and Negative Domains: Graphical representation with dataPositive and Negative Domains: Comparisons to zero or between parabolas

Positive and Negative intervals of a Quadratic Function

To find out when the parabola is positive and when it is negative, we must plot its graph.
Then we will look at
When the graph of the parabola is above the XX axis, with a positive YY value, the set is positive
When the graph of the parabola is below the XX axis, with a negative YY value, the set is negative
Let's see it in an illustration:

Representation of the Positive and Negative domains of a Quadratic Function

We will ask ourselves:
When is the graph of the parabola above the XX axis? 
When X>−1 X>-1 or X<−6X<-6
Therefore, the sets of positivity of the function are: X>−1 X>-1,X<−6X<-6
Now we will ask When is the graph of the parabola below the XX axis? 
When 6<X<−16<X<-1
Therefore, the set of negativity of the function is: −6<X<−1-6<X<-1

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

Practice Positive and Negative Domains

Question 1

The graph of the function below intersects the\( x \)-axis at points A and B.

The vertex of the parabola is marked at point C.

Find all values of \( x \) where\( f\left(x\right) < 0 \).

AAABBBCCCX

Question 2

Look at the function graphed below.

Find all values of \( x \)

where \( f\left(x\right) < 0 \).

000-3-3-3333XY

Question 3

Find all values of x

where \( f\left(x\right) > 0 \).

XXXYYY-6-6-6-10-10-10-2-2-2

Question 4

Find all values of \( x \)

where \( f\left(x\right) > 0 \).

XXXYYY555

Question 5

Find all values of \( x \)

where\( f\left(x\right) < 0 \) .

XXXYYY888

Examples with solutions for Positive and Negative Domains

Exercise #1

The graph of the function below intersects thex x -axis at points A and B.

The vertex of the parabola is marked at point C.

Find all values of x x where f\left(x\right) < 0 .

AAABBBCCCX

Video Solution

Answer

A < x < B

Exercise #2

Look at the function graphed below.

Find all values of x x

where f\left(x\right) < 0 .

000-3-3-3333XY

Video Solution

Answer

x > 3 or x < -3

Exercise #3

Find all values of x

where f\left(x\right) > 0 .

XXXYYY-6-6-6-10-10-10-2-2-2

Video Solution

Answer

x > -2 or x > -10

Exercise #4

Find all values of x x

where f\left(x\right) > 0 .

XXXYYY555

Video Solution

Answer

x < 5 or x > 5

Exercise #5

Find all values of x x

where f\left(x\right) < 0 .

XXXYYY888

Video Solution

Answer

x < 8 or 8 < x

Question 1

Based on the data in the sketch, find for which X values the graph of the function \( f\left(x\right) > 0 \)

XXXYYY222888555

Question 2

The following function is graphed below:

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

For which values of x is
\( f(x)<0 \) true?

Question 3

The following function is graphed below:

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

For which values of x is

\( f(x)>0 \) true?

BBBAAAEEEOOO

Question 4

The following function is graphed below:

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

For which values of x is

\( f(x)>0 \) true?

BBBAAACCCOOO

Question 5

The following function is graphed below:

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

For which values of x is

\( f(x)<0 \) true?

BBBAAACCCOOO

Exercise #6

Based on the data in the sketch, find for which X values the graph of the function f\left(x\right) > 0

XXXYYY222888555

Video Solution

Answer

2 < x < 8

Exercise #7

The following function is graphed below:

f(x)=−2x2+4x−6 f(x)=-2x^2+4x-6

For which values of x is
f(x)<0 true?

Video Solution

Answer

For all values of x

Exercise #8

The following function is graphed below:

y=x2−6x+8 y=x^2-6x+8

For which values of x is

f(x)>0 true?

BBBAAAEEEOOO

Video Solution

Answer

2 < x , x < 4

Exercise #9

The following function is graphed below:

y=−x2+5x+6 y=-x^2+5x+6

For which values of x is

f(x)>0 true?

BBBAAACCCOOO

Video Solution

Answer

-1 < x < 6

Exercise #10

The following function is graphed below:

y=−x2+5x+6 y=-x^2+5x+6

For which values of x is

f(x)<0 true?

BBBAAACCCOOO

Video Solution

Answer

Answers (a) and (b)

Question 1

The following function is graphed below:

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

For which values of x is

\( f(x)<0 \) true?

BBBAAAEEEOOO

Question 2

The following function is graphed below:

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

For which values of x is

\( f(x)>0 \) true?

Exercise #11

The following function is graphed below:

y=x2−6x+8 y=x^2-6x+8

For which values of x is

f(x)<0 true?

BBBAAAEEEOOO

Video Solution

Answer

2 < x < 4

Exercise #12

The following function is graphed below:

f(x)=−2x2+4x−6 f(x)=-2x^2+4x-6

For which values of x is

f(x)>0 true?

Video Solution

Answer

No answer

Topics learned in later sections

  1. Increasing and Decreasing Intervals of a Parabola
  2. Vertex of a parabola
  3. Symmetry in a parabola