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 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.
Write it like a human would.
.
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.
Write it like a human would.
Answers
GPT-4o mini
To transform the parent function \( y = x^2 \) into the desired forms, we can apply the transformations as specified for each case.
### A. Shifted Left 1 and Up 5
To shift the parent graph left by 1, we replace \( x \) with \( x + 1 \) (because shifting left is like adding a positive number). To shift up by 5, we add 5 to the whole function.
Putting this together, the equation becomes:
\[
y = (x + 1)^2 + 5
\]
### B. Reflected Over the X-axis and Stretched Vertically by a Factor of 3
Reflecting the graph over the x-axis involves multiplying the function by -1. Stretching vertically by a factor of 3 means we multiply the entire function by 3. Thus, we have:
\[
y = -3(x^2)
\]
In the vertex form, this can be rewritten as:
\[
y = -3(x - 0)^2 + 0
\]
### C. Compressed Vertically by a Factor of 1/2 and Shifted Right 6
To compress vertically by a factor of \( \frac{1}{2} \), we multiply the function by \( \frac{1}{2} \). Shifting right by 6 means we replace \( x \) with \( x - 6\).
Putting both transformations together, we get:
\[
y = \frac{1}{2}(x - 6)^2
\]
So, the equations for each transformation in the requested form are:
A. \( y = (x + 1)^2 + 5 \)
B. \( y = -3(x - 0)^2 + 0 \)
C. \( y = \frac{1}{2}(x - 6)^2 \)
### A. Shifted Left 1 and Up 5
To shift the parent graph left by 1, we replace \( x \) with \( x + 1 \) (because shifting left is like adding a positive number). To shift up by 5, we add 5 to the whole function.
Putting this together, the equation becomes:
\[
y = (x + 1)^2 + 5
\]
### B. Reflected Over the X-axis and Stretched Vertically by a Factor of 3
Reflecting the graph over the x-axis involves multiplying the function by -1. Stretching vertically by a factor of 3 means we multiply the entire function by 3. Thus, we have:
\[
y = -3(x^2)
\]
In the vertex form, this can be rewritten as:
\[
y = -3(x - 0)^2 + 0
\]
### C. Compressed Vertically by a Factor of 1/2 and Shifted Right 6
To compress vertically by a factor of \( \frac{1}{2} \), we multiply the function by \( \frac{1}{2} \). Shifting right by 6 means we replace \( x \) with \( x - 6\).
Putting both transformations together, we get:
\[
y = \frac{1}{2}(x - 6)^2
\]
So, the equations for each transformation in the requested form are:
A. \( y = (x + 1)^2 + 5 \)
B. \( y = -3(x - 0)^2 + 0 \)
C. \( y = \frac{1}{2}(x - 6)^2 \)