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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

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To translate the line given by the equation \( y = 3x + 2 \) one unit to the right and two units up, we will perform the following steps:

  1. Shift One Unit to the Right:

    • To shift a graph one unit to the right, we replace \( x \) with \( x - 1 \).
    • The new equation becomes: \[ y = 3(x - 1) + 2 \]
    • This simplifies to: \[ y = 3x - 3 + 2 \] \[ y = 3x - 1 \]

  2. Shift Two Units Up:

    • To shift the graph two units up, we add 2 to the entire equation.
    • Therefore, we take the modified equation \( y = 3x - 1 \) and add 2: \[ y = 3x - 1 + 2 \]
    • This results in: \[ y = 3x + 1 \]

Thus, the equation of the translated line is: \[ y = 3x + 1 \]