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)

Question

Responses

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

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

y=5x+3
y equals 5 x plus 3

y=3x−1

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. Translation to the right: After translating a line one unit to the right, the x-coordinate increases by 1. This means we substitute \( x \) with \( x - 1 \) in the equation. \[ y = 3(x - 1) + 2 \]

2. Simplifying the equation: \[ y = 3x - 3 + 2 = 3x - 1 \]

3. Translation upwards: Now we translate the line upwards by 2 units, which means we add 2 to the entire equation: \[ y = (3x - 1) + 2 = 3x + 1 \]

The resulting equation after both translations is: \[ y = 3x + 1 \]

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

Answer 2

A point is translated on the coordinate plane from (4,−1) to (4,2) . Journey writes the function g(y)=y+k to describe the translation. What is the value of k?(1 point)
Responses

−3
negative 3

3
3

0
0

−5

Answer 3

To find the value of \( k \) in the function \( g(y) = y + k \) that describes the translation from the point \( (4, -1) \) to \( (4, 2) \), we need to determine how the \( y \)-coordinate changes.

The original point has a \( y \)-coordinate of \(-1\), and the new point has a \( y \)-coordinate of \( 2\). The change in the \( y \)-coordinate can be calculated as follows:

\[ \text{Change} = \text{new } y - \text{old } y = 2 - (-1) = 2 + 1 = 3 \]

Thus, the translation involves increasing the \( y \)-coordinate by \( 3 \).

This means that \( k \) must be \( 3 \), because in the function \( g(y) = y + k \):

\[ g(y) = y + 3 \]

Thus, the value of \( k \) is:

3.

Answer 4

A vertical line separates 2 triangles. The triangle to the left of the line is made up of points upper P, upper Q, and upper R, while the triangle on the right is made up of points upper P prime, upper Q prime, and upper R prime.

Triangle PQR is reflected over a vertical line of reflection to create triangle P′Q′R′ . What are the coordinates of point R′ ?

(1 point)
Responses

(−2,7)
left parenthesis negative 2 comma 7 right parenthesis

(11,1)
left parenthesis 11 comma 1 right parenthesis

(−3,1)
left parenthesis negative 3 comma 1 right parenthesis

(1,−3)

Answer 5

To find the coordinates of point \( R' \) after reflecting point \( R \) over a vertical line of reflection, we need to know the coordinates of point \( R \) and the equation of the vertical line of reflection.

Assuming that:

• The vertical line of reflection is \( x = d \), where \( d \) is the x-coordinate of the line.
• The coordinates of point \( R \) are \( (x_R, y_R) \).

The formula for reflecting a point \( (x, y) \) over a vertical line \( x = d \) is given by: \[ (x', y) = (2d - x, y) \] where \( (x', y) \) represents the coordinates of the reflected point.

However, we need the coordinates of point \( R \) (denoted as \( (x_R, y_R) \)) and the position of the vertical line \( x = d \) to explicitly calculate \( R' \).

Since specific coordinates for point \( R \) and the vertical line \( d \) are not provided in the question, I cannot provide an exact answer. Please provide the coordinates for point \( R \) and the equation of the vertical line of reflection to calculate the coordinates for point \( R' \).