Answer :
To solve the given problem, we will analyze the transformations applied to the original function \( f(x) = x^3 \) to correctly determine the graph of the function \( y = f(x+3) - 9 \).
### Step-by-Step Solution:
1. Understanding the function transformation:
We begin with the original function \( f(x) = x^3 \).
2. Horizontal Translation:
The expression \( f(x+3) \) applies a horizontal shift to the function \( f(x) \). Specifically, adding a constant inside the function argument (i.e., replacing \( x \) with \( x + 3 \)) results in a shift to the left.
- The graph of \( f(x+3) \) is the graph of \( f(x) \) shifted 3 units to the left.
3. Vertical Translation:
Next, we consider the expression \( -9 \) which is subtracted from the function. Subtracting a constant outside the function argument (i.e., \( f(x+3) - 9 \)) results in a vertical shift.
- The graph of \( f(x+3) - 9 \) is the graph of \( f(x+3) \) shifted 9 units downwards.
4. Combining Transformations:
The resulting function \( y = f(x+3) - 9 \) involves:
- A translation 3 units to the left due to the \( (x+3) \) term.
- A translation 9 units down due to the \( -9 \) term.
Thus, given these transformations, the correct description of the graph transformation is:
C. It is the graph of [tex]\( f \)[/tex] translated 9 units down and 3 units to the left.
### Step-by-Step Solution:
1. Understanding the function transformation:
We begin with the original function \( f(x) = x^3 \).
2. Horizontal Translation:
The expression \( f(x+3) \) applies a horizontal shift to the function \( f(x) \). Specifically, adding a constant inside the function argument (i.e., replacing \( x \) with \( x + 3 \)) results in a shift to the left.
- The graph of \( f(x+3) \) is the graph of \( f(x) \) shifted 3 units to the left.
3. Vertical Translation:
Next, we consider the expression \( -9 \) which is subtracted from the function. Subtracting a constant outside the function argument (i.e., \( f(x+3) - 9 \)) results in a vertical shift.
- The graph of \( f(x+3) - 9 \) is the graph of \( f(x+3) \) shifted 9 units downwards.
4. Combining Transformations:
The resulting function \( y = f(x+3) - 9 \) involves:
- A translation 3 units to the left due to the \( (x+3) \) term.
- A translation 9 units down due to the \( -9 \) term.
Thus, given these transformations, the correct description of the graph transformation is:
C. It is the graph of [tex]\( f \)[/tex] translated 9 units down and 3 units to the left.
