Question

which of the functions shown is the graph of $y = -(x + 1)^3$?

Explanation:

Step1: Analyze the function \( y = -(x + 1)^3 \)

The parent function is \( y = x^3 \), which has a point - symmetric graph about the origin. For the function \( y=-(x + 1)^3\), the negative sign reflects the graph of \( y = x^3 \) over the \( x \) - axis, and the \( x+1 \) inside the cube function shifts the graph of \( y=-x^3 \) 1 unit to the left.
The general shape of \( y = x^3 \) is an increasing curve. After reflection over the \( x \) - axis (\( y=-x^3 \)), it becomes a decreasing curve. After shifting 1 unit to the left (\( y =-(x + 1)^3 \)), the point of inflection (the point where the concavity changes) is at \( x=- 1,y = 0 \) (when \( x=-1 \), \( y=-(-1 + 1)^3=0 \)).

Step2: Analyze the graphs

• For the graph of \( B(x) \): It looks like a parabola - shaped graph (maybe a quadratic function), so it is not a cubic function.
• For the graph of \( C(x) \): It is an increasing function for \( x>-2 \) (from the graph), which does not match the behavior of \( y =-(x + 1)^3 \) (which should be decreasing for \( x>-1 \) and increasing for \( x < - 1\) but with a reflection).
• For the graph of \( A(x) \): The graph has a point of inflection around \( x=-1 \), and its behavior (decreasing when \( x>-1 \) and increasing when \( x < - 1\) after reflection) matches the function \( y=-(x + 1)^3 \). When we substitute \( x = 0\) into \( y=-(x + 1)^3\), we get \( y=-(0 + 1)^3=-1\). Looking at the graph of \( A(x) \), when \( x = 0\), the \( y\) - value is negative, which also matches.

Answer:

\( A(x) \) (the graph labeled \( A(x) \))