Lucy draws a line with the equation y=3x+2

Question

𝑦
=
3
𝑥
+
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)
Responses

y=4x+2
𝑦
=
4
𝑥
+
2
y equals 4 x plus 2

y=3x+1
𝑦
=
3
𝑥
+
1
y equals 3 x plus 1

y=3x−1
𝑦
=
3
𝑥
−
1
y equals 3 x minus 1

y=5x+3

Answer 1

To find the equation of the new line after translating the original line \( y = 3x + 2 \) one unit to the right and two units up, we can follow these steps:

1. Translating the line one unit to the right:

   • To shift a line \( y = mx + b \) to the right by \( h \) units, we replace \( x \) with \( x - h \). Here, \( h = 1 \), so the new equation becomes:
     \[ y = 3(x - 1) + 2 \] \[ y = 3x - 3 + 2 \] \[ y = 3x - 1 \]

2. Translating the line two units up:

   • To shift the line \( y = mx + b \) up by \( k \) units, we add \( k \) to the entire equation. Here, \( k = 2 \), so we add 2 to the last equation: \[ y = 3x - 1 + 2 \] \[ y = 3x + 1 \]

Thus, the equation of the new line after the translation is \( y = 3x + 1 \).

Therefore, the correct response is: y = 3x + 1