Question

the graph of the function $f(x) = -(x + 6)(x + 2)$ is shown below.
which statement about the function is true?
\\(\bigcirc\\) the function is increasing for all real values of $x$ where $x < -4$.
\\(\bigcirc\\) the function is increasing for all real values of $x$ where $-6 < x < -2$.
\\(\bigcirc\\) the function is decreasing for all real values of $x$ where $x < -6$ and where $x > -2$.
\\(\bigcirc\\) the function is decreasing for all real values of $x$ where $x < -4$.

Explanation:

Step1: Analyze the function's form

The function is \( f(x) = -(x + 6)(x + 2) \), which is a quadratic function. First, find the vertex. The roots are at \( x=-6 \) and \( x = -2 \), so the axis of symmetry (x-coordinate of vertex) is \( \frac{-6 + (-2)}{2}=\frac{-8}{2}=-4 \). The coefficient of \( x^2 \) (when expanded) is -1 (since \( -(x + 6)(x + 2)=-(x^2 + 8x + 12)=-x^2 - 8x - 12 \)), so the parabola opens downward.

Step2: Determine increasing/decreasing intervals

For a downward - opening parabola (\( a<0 \) in \( y = ax^2+bx + c \)), the function is increasing to the left of the vertex (\( x < - 4 \)) and decreasing to the right of the vertex (\( x>-4 \)). Let's analyze each option:

• Option 1: "The function is increasing for all real values of \( x \) where \( x < - 4 \)." Since the parabola opens downward and the vertex is at \( x=-4 \), to the left of \( x = - 4 \) (i.e., \( x < - 4 \)), as \( x \) increases towards \( - 4 \), \( f(x) \) increases. But let's check other options to be sure.
• Option 2: "The function is increasing for all real values of \( x \) where \( -6 < x < -2 \)." The interval \( -6 < x < -2 \) includes the vertex at \( x=-4 \). To the left of \( x=-4 \) (in \( -6 < x < -4 \)) the function is increasing, and to the right of \( x=-4 \) (in \( -4 < x < -2 \)) the function is decreasing. So the function is not increasing throughout \( -6 < x < -2 \).
• Option 3: "The function is decreasing for all real values of \( x \) where \( x < - 6 \) and where \( x > - 2 \)." For \( x < - 6 \), as \( x \) increases towards \( - 6 \), \( f(x) \) is increasing (since \( x < - 4 \)), so this is false.
• Option 4: "The function is decreasing for all real values of \( x \) where \( x < - 4 \)." But for \( x < - 4 \), the function is increasing (as \( x \) increases towards \( - 4 \)), so this is false.

Answer:

The function is increasing for all real values of \( x \) where \( x < - 4 \) (the first option: "The function is increasing for all real values of \( x \) where \( x < - 4 \)")