Positive and Negative intervals of a Quadratic Function

Positive and Negative Domains - Examples, Exercises and Solutions
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The Quadratic Function
Positive and Negative Domains
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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

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Test yourself on positive and negative domains!

einstein

The graph of the function below does not intersect the \( x \)-axis.

The parabola's vertex is marked A.

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

AAAX

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Positivity and Negativity Sets of the Parabola

Pay attention!
Do not confuse the intervals of increase and decrease with the sets of positivity and negativity.
The intervals of increase and decrease describe when the function is increasing or decreasing, regardless of its position, above or below the XX axis.
On the other hand, the sets of positivity and negativity describe when the function is positive - above the XX axis or negative - below the XX axis, regardless of whether the function is increasing or decreasing.

Examples and exercises with solutions of Positivity and Negativity Sets of the Quadratic Function

Exercise #1

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 #2

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?

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Video Solution

Answer

2 < x , x < 4

Exercise #3

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

Exercise #4

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 #5

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)

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Test your knowledge
Question 1

The graph of the function below does not intersect the \( x \)-axis.

The parabola's vertex is marked A.

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

AAAX

Question 2

The graph of the function below does not intersect the \( x \)-axis.

The parabola's vertex is marked A.

Find all values of \( x \) where

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

AAAX

Question 3

The graph of the function below intersects the \( x \)-axis at point A (the vertex of the parabola).

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

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