Question
Lucy draws a line with the equation y=3x+2. She translates the line by shifting it one unit to the right and two units up. Which is the equation of the new line?(1 point)Responsesy=4x+2y equals 4 x plus 2y=5x+3y equals 5 x plus 3y=3x−1y equals 3 x minus 1y=3x+1
Answers
GPT-4o mini
To find the equation of the new line after Lucy translates the line \( y = 3x + 2 \) one unit to the right and two units up, we need to apply the translations step by step.
1. **Shifting one unit to the right**: When we shift a line to the right, we replace \( x \) with \( x - 1 \) in the equation. So:
\[
y = 3(x - 1) + 2
\]
Simplifying this gives:
\[
y = 3x - 3 + 2 = 3x - 1
\]
2. **Shifting two units up**: Now, to shift the line up, we add 2 to the entire equation:
\[
y = (3x - 1) + 2
\]
Simplifying this gives:
\[
y = 3x + 1
\]
Thus, the equation of the new line after the translations is:
\[
y = 3x + 1
\]
The correct response is **y = 3x + 1**.
1. **Shifting one unit to the right**: When we shift a line to the right, we replace \( x \) with \( x - 1 \) in the equation. So:
\[
y = 3(x - 1) + 2
\]
Simplifying this gives:
\[
y = 3x - 3 + 2 = 3x - 1
\]
2. **Shifting two units up**: Now, to shift the line up, we add 2 to the entire equation:
\[
y = (3x - 1) + 2
\]
Simplifying this gives:
\[
y = 3x + 1
\]
Thus, the equation of the new line after the translations is:
\[
y = 3x + 1
\]
The correct response is **y = 3x + 1**.