Question

the graph of the function $f(x)=-(x + 3)(x - 1)$ is shown below.
which statement about the function is true?
○ the function is positive for all real values of $x$ where $x < -1$.
○ the function is negative for all real values of $x$ where $x < -3$ and where $x > 1$.
○ the function is positive for all real values of $x$ where $x > 0$.
○ the function is negative for all real values of $x$ where $x < -3$ or $x > -1$.

Explanation:

Step1: Find roots of the function

Set $f(x)=0$, so $-(x+3)(x-1)=0$. The roots are $x=-3$ and $x=1$.

Step2: Analyze parabola direction

The coefficient of $x^2$ is $-1$ (expand $f(x)=-(x^2+2x-3)=-x^2-2x+3$), so the parabola opens downward.

Step3: Determine sign intervals

For a downward-opening parabola:

• $f(x)>0$ between the roots: $-3
• $f(x)<0$ outside the roots: $x<-3$ or $x>1$

Step4: Match with options

Compare intervals to the given statements.

Answer:

B. The function is negative for all real values of x where $x < -3$ and where $x > 1$.