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. What is the equation of the new line?

Answer 1

To translate the line defined by the equation \( y = 3x + 2 \) one unit to the right and two units up, we need to adjust the equation accordingly.

1. **Translate one unit to the right:**
- Shifting a line to the right involves replacing \( x \) with \( x - 1 \). So, we replace \( x \) in the original equation:
\[
y = 3(x - 1) + 2
\]
Simplifying this:
\[
y = 3x - 3 + 2
\]
\[
y = 3x - 1
\]

2. **Translate two units up:**
- To shift the line up, we add 2 to the \( y \) value:
\[
y = 3x - 1 + 2
\]
Simplifying this:
\[
y = 3x + 1
\]

Thus, the equation of the new line after the translation is:

\[
\boxed{y = 3x + 1}
\]