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:
\[
\begin{aligned}
&\text{Original function:} & g(x) = x^3 \\
&\text{Horizontal shift (left by 7):} & g(x + 7) = (x + 7)^3 \\
&\text{Vertical shift (down by 8):} & f(x) = (x + 7)^3 - 8 \\
\end{aligned}
\]
