Question

Interpret the key features of the graph what type of transformation occurred from the parent function
f(x)= 4x^2
Points (0,0), (-1,4),(1,4),(-2,16) and (2,16)
This is a vertical stretch of the parent function y =x^2 due to the changes in the x-intercept
This is a vertical stretch of the parent function y =x^2 due to the changes in the y-coordinates of each point
A vertical stretch y =x^2 due to changes in the x-coordinates of each point
Vertical stretch y =x^2 due to changes in y-intercept

Answer 1

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:
1. **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.

2. **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.