Question

Given the parent function y=x2y=x2 , write the equations with the following transformations in the form y=a(x−h)2+ky=a(x−h)2+k . A. The parent graph has been shifted to the left 1 and up 5. B. The parent graph has been reflected over the x-axis and stretched vertically by a factor of 3. C. The parent graph has been compressed vertically by a factor of 1/2 and shifted right 6. (6 points)

Answer 1

To write the equations in the form \( y = a(x - h)^2 + k \) based on the given transformations of the parent function \( y = x^2 \), we will handle each transformation step-by-step.

### A. Shifted to the left by 1 and up by 5.
- Shifting left by 1 means \( h = -1 \).
- Shifting up by 5 means \( k = 5 \).
- Thus, the equation becomes:
\[
y = 1(x - (-1))^2 + 5 = (x + 1)^2 + 5
\]

### B. Reflected over the x-axis and stretched vertically by a factor of 3.
- A reflection over the x-axis involves changing the sign of \( a \), so \( a = -3 \) (because we are stretching by a factor of 3 in the negative direction).
- There are no shifts involved, so \( h = 0 \) and \( k = 0 \).
- Thus, the equation becomes:
\[
y = -3(x - 0)^2 + 0 = -3x^2
\]

### C. Compressed vertically by a factor of 1/2 and shifted right by 6.
- A vertical compression by a factor of 1/2 means \( a = \frac{1}{2} \).
- Shifting right by 6 means \( h = 6 \).
- There is no vertical shift, so \( k = 0 \).
- Thus, the equation becomes:
\[
y = \frac{1}{2}(x - 6)^2 + 0 = \frac{1}{2}(x - 6)^2
\]

### Summary of the Final Equations:
A. \( y = (x + 1)^2 + 5 \)

B. \( y = -3x^2 \)

C. \( y = \frac{1}{2}(x - 6)^2 \)