Certainly! Let's go through the detailed, step-by-step process of determining the transformation and resulting function.
1. Identify the Original Function:
- The original function is \( f(x) = x^2 \).
2. Transformation Applied:
- The given transformation involves the function \( f(x) \) being modified. However, without specific details on the transformation, we must consider common types of transformations that can be applied to \( f(x) = x^2 \). Common transformations for quadratic functions include:
- Vertical stretching or compressing
- Reflecting
- Translating (shifting up, down, left, or right)
3. Assume a Vertical Scaling:
- In this case, we will assume a vertical scaling transformation for simplicity. Vertical scaling involves multiplying the function \( f(x) \) by a constant factor.
4. Determine the Transformed Function \( g(x) \):
- Let the scaling factor be \( a \). When vertically scaling a quadratic function \( f(x) = x^2 \) by a factor of \( a \), we get the new function:
[tex]\[
g(x) = a \cdot f(x)
\][/tex]
- Substituting \( f(x) = x^2 \) into the equation, we get:
[tex]\[
g(x) = a \cdot x^2
\][/tex]
5. Specify the Transformation Type:
- The type of transformation applied to \( f(x) \) is vertical scaling.
6. Construct the Final Answer:
- The function \( g \) is a vertical scaling of the function \( f \) by a factor \( a \).
- The resulting function after the transformation is \( g(x) = a \cdot x^2 \).
- The scaling factor in this context is \( a \).
Here is the final answer, filling in the blanks appropriately:
The function \( f(x)=x^2 \) has been transformed, resulting in function \( g \).
Function [tex]\( g \)[/tex] is a vertical scaling of function [tex]\( f \)[/tex]. [tex]\( g(x) = a \cdot x^2 \)[/tex], where [tex]\( a \)[/tex] is the scaling factor.
