The graph of the function $f(x)=(x+2)(x+6)$ is shown below.

Which statement about the function is true?

A. The function is positive for all real values of $x$ where $x>-4$.
B. The function is negative for all real values of $x$ where $-6 < x < -2$.
C. The function is positive for all real values of $x$ where $x<-6$ or $x>-3$.
D. The function is negative for all real values of $x$ where $x<-2$.

Find the roots of the quadratic function f ( x ) = ( x + 2 ) ( x + 6 ) by setting f ( x ) = 0 , which gives x = − 2 and x = − 6 .
Determine the intervals where the function is positive or negative. Since the parabola opens upwards, 0"> f ( x ) > 0 when x < − 6 or -2"> x > − 2 , and f ( x ) < 0 when − 6 < x < − 2 .
Compare the intervals with the given statements to find the correct one.
The correct statement is: The function is negative for all real values of x where − 6 < x < − 2 , so the final answer is \boxed{\text{The function is negative for all real values of } x \text{ where } -6 0 when x < − 6 or -2"> x > − 2 , and f ( x ) < 0 when − 6 < x < − 2 .


Comparing with the Given Statements Now, let's compare these intervals with the given statements:

The function is positive for all real values of x where -4"> x > − 4 . This is false because the function is negative between − 6 and − 2 .

The function is negative for all real values of x where − 6 < x < − 2 . This is true.

The function is positive for all real values of x where x < − 6 or -3"> x > − 3 . This is false because the function is positive when x < − 6 or -2"> x > − 2 .

The function is negative for all real values of x where x < − 2 . This is false because the function is positive when -2"> x > − 2 .

Final Answer The correct statement is: The function is negative for all real values of x where − 6 < x < − 2 .


Examples
Understanding when a function is positive or negative is crucial in many real-world applications. For instance, in business, a function might represent the profit margin of a product. Knowing the intervals where the function is positive (profit) or negative (loss) helps in making informed decisions about pricing, production, and marketing strategies. Similarly, in physics, a function could describe the height of a projectile. Identifying when the function is positive helps determine the time intervals when the projectile is above the ground.

Answered by GinnyAnswer | 2025-07-04

The correct statement regarding the function f ( x ) = ( x + 2 ) ( x + 6 ) is that it is negative for all real values of x where − 6 < x < − 2 . Thus, the answer is option B.
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Answered by Anonymous | 2025-07-06