Question

Given the parent function y=x^2
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 transform the parent function \( y = x^2 \) into the required forms, we will use the vertex form of a quadratic function, which is given by:

\[
y = a(x - h)^2 + k
\]

where:
- \( a \) controls the vertical stretch or compression and reflection,
- \( (h, k) \) represents the vertex of the parabola.

Now, let's apply the transformations for each case:

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

- Shifting left by 1 means \( h = -1 \) (since \( h \) is subtracted).
- Shifting up by 5 means \( k = 5 \).

Thus, the transformed equation is:

\[
y = 1(x + 1)^2 + 5 \quad \text{or simply}\quad y = (x + 1)^2 + 5
\]

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

- Reflecting over the x-axis means \( a = -3 \) (the negative sign indicates reflection and 3 indicates vertical stretch).
- The vertex remains at (0, 0) since there is no shifting.

Thus, the transformed equation is:

\[
y = -3(x - 0)^2 + 0 \quad \text{or simply}\quad y = -3x^2
\]

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

- Compressing vertically by a factor of \(\frac{1}{2}\) gives \( a = \frac{1}{2} \).
- Shifting right by 6 means \( h = 6 \).
- The vertex remains at \( k = 0 \) since there is no vertical shift.

Thus, the transformed equation is:

\[
y = \frac{1}{2}(x - 6)^2 + 0 \quad \text{or simply}\quad y = \frac{1}{2}(x - 6)^2
\]

### Summary of the Equations:

A. \( y = (x + 1)^2 + 5 \)

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

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