Question
Given the parent function y = z write the equations with the following transformations in the form y = a(zh)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)
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)
Answers
GPT-4o mini
Let's break down each of the transformations step by step, starting from the parent function \( y = z \).
### A. The graph has been shifted to the left 1 and up 5.
1. Shifting to the left by 1: This means replacing \( z \) with \( z + 1 \).
2. Shifting up by 5: This involves adding 5 to the function.
So, the transformed equation is:
\[
y = (z + 1)^2 + 5
\]
This is the form of \( y = a(z - h)^2 + k \) with \( a = 1 \), \( h = -1 \), and \( k = 5 \).
### B. The graph has been reflected over the x-axis and stretched vertically by a factor of 3.
1. Reflecting over the x-axis means we take the negative of the function: This changes the sign of \( y \).
2. Stretching vertically by a factor of 3 means multiplying the function by 3.
Thus, the transformed equation is:
\[
y = -3(z)^2
\]
This is the form of \( y = a(z - h)^2 + k \) where \( a = -3 \), \( h = 0 \), and \( k = 0 \).
### C. The graph has been compressed vertically by a factor of \( \frac{1}{2} \) and shifted right 6.
1. Compressing vertically by a factor of \( \frac{1}{2} \) means multiplying the function by \( \frac{1}{2} \).
2. Shifting right by 6 means replacing \( z \) with \( z - 6 \).
Putting these transformations together gives:
\[
y = \frac{1}{2}(z - 6)^2
\]
This is the form of \( y = a(z - h)^2 + k \) with \( a = \frac{1}{2} \), \( h = 6 \), and \( k = 0 \).
### Summary of the Transformed Equations
- A: \( y = (z + 1)^2 + 5 \)
- B: \( y = -3(z)^2 \)
- C: \( y = \frac{1}{2}(z - 6)^2 \)
### A. The graph has been shifted to the left 1 and up 5.
1. Shifting to the left by 1: This means replacing \( z \) with \( z + 1 \).
2. Shifting up by 5: This involves adding 5 to the function.
So, the transformed equation is:
\[
y = (z + 1)^2 + 5
\]
This is the form of \( y = a(z - h)^2 + k \) with \( a = 1 \), \( h = -1 \), and \( k = 5 \).
### B. The graph has been reflected over the x-axis and stretched vertically by a factor of 3.
1. Reflecting over the x-axis means we take the negative of the function: This changes the sign of \( y \).
2. Stretching vertically by a factor of 3 means multiplying the function by 3.
Thus, the transformed equation is:
\[
y = -3(z)^2
\]
This is the form of \( y = a(z - h)^2 + k \) where \( a = -3 \), \( h = 0 \), and \( k = 0 \).
### C. The graph has been compressed vertically by a factor of \( \frac{1}{2} \) and shifted right 6.
1. Compressing vertically by a factor of \( \frac{1}{2} \) means multiplying the function by \( \frac{1}{2} \).
2. Shifting right by 6 means replacing \( z \) with \( z - 6 \).
Putting these transformations together gives:
\[
y = \frac{1}{2}(z - 6)^2
\]
This is the form of \( y = a(z - h)^2 + k \) with \( a = \frac{1}{2} \), \( h = 6 \), and \( k = 0 \).
### Summary of the Transformed Equations
- A: \( y = (z + 1)^2 + 5 \)
- B: \( y = -3(z)^2 \)
- C: \( y = \frac{1}{2}(z - 6)^2 \)