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) Responses
y=3x+1 y=3x-1 y=5x+3 y=4x+2
y=3x+1 y=3x-1 y=5x+3 y=4x+2
Answers
GPT-4o mini
To shift the line given by the equation \( y = 3x + 2 \) one unit to the right and two units up, we can perform the following transformations:
1. **Shift one unit to the right**: To shift the graph of a function \( f(x) \) to the right, we replace \( x \) with \( (x - 1) \). For the equation \( y = 3x + 2 \), this gives us:
\[
y = 3(x - 1) + 2 = 3x - 3 + 2 = 3x - 1
\]
2. **Shift two units up**: To shift the graph of a function up by \( c \) units, we add \( c \) to the entire function. In this case, we will add 2 to the equation we got from the first step:
\[
y = (3x - 1) + 2 = 3x + 1
\]
Thus, the equation of the new line after the transformations is:
\[
y = 3x + 1
\]
So the correct response is:
- **y=3x+1**
1. **Shift one unit to the right**: To shift the graph of a function \( f(x) \) to the right, we replace \( x \) with \( (x - 1) \). For the equation \( y = 3x + 2 \), this gives us:
\[
y = 3(x - 1) + 2 = 3x - 3 + 2 = 3x - 1
\]
2. **Shift two units up**: To shift the graph of a function up by \( c \) units, we add \( c \) to the entire function. In this case, we will add 2 to the equation we got from the first step:
\[
y = (3x - 1) + 2 = 3x + 1
\]
Thus, the equation of the new line after the transformations is:
\[
y = 3x + 1
\]
So the correct response is:
- **y=3x+1**