Answer :

Answer:

To compare the graph of \( f(x) = (x + 7)^3 - 8 \) to the parent function \( g(x) = x^3 \), we need to analyze the transformations applied to \( g(x) = x^3 \).

1. **Horizontal Translation**:

  The term \((x + 7)\) inside the function \( f(x) \) indicates a horizontal translation. Specifically, replacing \( x \) with \( x + 7 \) translates the graph to the left by 7 units. Therefore, every point on the graph of \( g(x) = x^3 \) is shifted 7 units to the left.

2. **Vertical Translation**:

  The term \(- 8\) outside the function \( f(x) \) indicates a vertical translation. Subtracting 8 from the function value shifts the graph down by 8 units. Therefore, every point on the graph of \( g(x) = x^3 \) is shifted 8 units down.

To summarize, the graph of \( f(x) = (x + 7)^3 - 8 \) compared to the graph of \( g(x) = x^3 \) is:

- **Shifted left by 7 units**.

- **Shifted down by 8 units**.

**Visualization**:

- The vertex of the parent function \( g(x) = x^3 \) is at the origin (0,0).

- For the transformed function \( f(x) = (x + 7)^3 - 8 \), the vertex moves to \((-7, -8)\).

So, the point \((0, 0)\) on \( g(x) = x^3 \) will map to \((-7, -8)\) on \( f(x) = (x + 7)^3 - 8 \).

Here's a step-by-step summary of how the transformations affect the graph:

- Start with the parent function \( g(x) = x^3 \).

- Shift the entire graph 7 units to the left, resulting in \( (x + 7)^3 \).

- Then, shift the entire graph 8 units down, resulting in \( (x + 7)^3 - 8 \).

The transformations lead to the following changes in the graph:

\[

\begin{aligned}

&\text{Original function:} & g(x) = x^3 \\

&\text{Horizontal shift (left by 7):} & g(x + 7) = (x + 7)^3 \\

&\text{Vertical shift (down by 8):} & f(x) = (x + 7)^3 - 8 \\

\end{aligned}

\]