Asked by monday
The function f(x)=x^3 undergoes a transformation such that 3f(x+2)^3 =3(x+2)^3 (2 points) This translates the graph horizontally to the ?Left/Right and then stretched ?horizontally/vertically
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GPT-4o mini
Let's analyze the transformation for the function \( f(x) = x^3 \).
1. **Horizontal Transformation**:
- The function \( f(x+2) \) indicates a horizontal translation.
- Since we have \( x+2 \), the graph of \( f(x) \) shifts to the left by 2 units. This is because adding a positive value inside the function argument (\( x+2 \)) moves the graph left.
2. **Vertical Transformation**:
- The expression \( 3f(x+2)^3 \) means you are taking the output of \( f(x+2) \) and cubing it, which does not change the vertical stretch. However, the factor of 3 in front of \( f(x+2) \) vertically stretches the graph by a factor of 3.
In summary, the graph of \( f(x) = x^3 \) is:
- Translated horizontally to the **left** by 2 units.
- Stretched **vertically** by a factor of 3.
So the complete answer is: **left** and **vertically**.
1. **Horizontal Transformation**:
- The function \( f(x+2) \) indicates a horizontal translation.
- Since we have \( x+2 \), the graph of \( f(x) \) shifts to the left by 2 units. This is because adding a positive value inside the function argument (\( x+2 \)) moves the graph left.
2. **Vertical Transformation**:
- The expression \( 3f(x+2)^3 \) means you are taking the output of \( f(x+2) \) and cubing it, which does not change the vertical stretch. However, the factor of 3 in front of \( f(x+2) \) vertically stretches the graph by a factor of 3.
In summary, the graph of \( f(x) = x^3 \) is:
- Translated horizontally to the **left** by 2 units.
- Stretched **vertically** by a factor of 3.
So the complete answer is: **left** and **vertically**.
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