The function given is \( f(x) = 4x^2 \), which is a vertical transformation of the parent function \( y = x^2 \).
Key Features of the Transformation:
-
Vertical Stretch: The presence of the coefficient \( 4 \) in front of \( x^2 \) indicates a vertical stretch by a factor of \( 4 \). This means that for every \( y \)-value of the parent function \( y = x^2 \), the \( y \)-value of the transformed function \( f(x) = 4x^2 \) is four times greater.
-
Coordinates Interpretation:
- The point \( (0, 0) \) remains the same since when \( x=0 \), \( f(0) = 4(0^2) = 0 \).
- The point \( (-1, 4) \) can be understood by substituting \( x = -1 \): \[ f(-1) = 4(-1)^2 = 4(1) = 4. \]
- The point \( (1, 4) \) similarly: \[ f(1) = 4(1)^2 = 4(1) = 4. \]
- The points \( (-2, 16) \) and \( (2, 16) \) are obtained as follows: \[ f(-2) = 4(-2)^2 = 4(4) = 16, \] \[ f(2) = 4(2)^2 = 4(4) = 16. \]
Conclusion on Transformations:
From the above interpretations, we confirm a vertical stretch occurred:
- This is a vertical stretch of the parent function \( y = x^2 \) due to the changes in the y-coordinates of each point.
This option correctly describes the impact of the factor \( 4 \) on the graph: the \( y \)-values have been stretched upward, while the \( x \)-coordinates remain unchanged in their distance from the \( y \)-axis. Therefore, the correct response to the transformation from the parent function \( y = x^2 \) to \( f(x) = 4x^2 \) is indeed the vertical stretch based on changes in the \( y \)-coordinates.
1 answer