Question

page 2 of 2
1 of 181 of 18 Items
Question
Given the parent function y=x2
write the equations with the following transformations in the form y=a(x−h)2+k
.

A. The graph has been shifted to the left 1 and up 5

B. The graph has been reflected over the x-axis and stretched vertically by a factor of 3.

C. The graph has been compressed vertically by a factor of 1/2 and shifted right 6

(6 points)

Answer 1

To rewrite the transformations of the parent function \( y = x^2 \) in the form \( y = a(x - h)^2 + k \), we will apply the necessary transformations for each case.

### A. The graph has been shifted to the left 1 and up 5

1. **Shift left 1**: This means we replace \( x \) with \( (x + 1) \).
2. **Shift up 5**: This means we add 5 to the entire function.

Putting it together:
\[
y = (x + 1)^2 + 5
\]

### B. The graph has been reflected over the x-axis and stretched vertically by a factor of 3

1. **Reflection over the x-axis**: This changes the sign of the function, giving us \( -x^2 \).
2. **Stretched vertically by a factor of 3**: We multiply by 3, getting \( -3x^2 \).

To write this in the desired form, we factor it:
\[
y = -3(x - 0)^2 + 0
\]

So:
\[
y = -3(x - 0)^2 + 0
\]

### C. The graph has been compressed vertically by a factor of \( \frac{1}{2} \) and shifted right 6

1. **Compressed vertically by a factor of \( \frac{1}{2} \)**: This means we multiply the function by \( \frac{1}{2} \), resulting in \( \frac{1}{2}x^2 \).
2. **Shifted right 6**: We replace \( x \) with \( (x - 6) \).

Putting it together:
\[
y = \frac{1}{2}(x - 6)^2
\]

### Summary of Transformations

A. \( y = (x + 1)^2 + 5 \)
B. \( y = -3(x - 0)^2 + 0 \)
C. \( y = \frac{1}{2}(x - 6)^2 \)

These are your transformed equations in the requested form.