Question

write the coordinates of the vertices after a dilation with a scale factor of 3, centered at the origin.

Explanation:

Step1: Find original coordinates

First, identify the original coordinates of each vertex from the graph:

• \( R(-1, -1) \) (Wait, looking at the grid, let's recheck. Wait, the grid: R is at x=-1, y=-1? Wait no, looking at the graph, R is at (-1, -1)? Wait no, let's see the grid lines. Let's check each point:

Wait, let's look again. The x-axis and y-axis: each grid square is 1 unit. Let's find R, S, T, U:

• R: x=-1, y=-1? Wait no, looking at the graph, R is at (-1, -1)? Wait, no, let's see the positions:

Wait, R is at ( -1, -1 )? Wait, no, maybe I misread. Wait, the graph: R is at x=-1, y=-1? Wait, no, let's check the coordinates:

Wait, let's list original coordinates:

• R: Let's see, the point R is at ( -1, -1 )? Wait, no, looking at the grid, the x-coordinate for R: between -2 and 0, so x=-1, y-coordinate: between -2 and 0, so y=-1? Wait, no, maybe:

Wait, maybe R is (-1, -1), S is (-1, 1), T is (0, 3), U is (0, 1)? Wait, no, looking at the graph:

Wait, S is at (-1, 1), T is (0, 3), U is (0, 1), R is (-1, -1). Let's confirm:

From the graph:

• R: x=-1, y=-1

• S: x=-1, y=1

• T: x=0, y=3

• U: x=0, y=1

Yes, that makes sense.

Step2: Apply dilation formula

Dilation centered at the origin with scale factor \( k \) transforms a point \( (x, y) \) to \( (k \cdot x, k \cdot y) \). Here, \( k = 3 \).

For \( R(-1, -1) \):

Multiply each coordinate by 3:
\( R' = (3 \cdot (-1), 3 \cdot (-1)) = (-3, -3) \)

For \( S(-1, 1) \):

Multiply each coordinate by 3:
\( S' = (3 \cdot (-1), 3 \cdot 1) = (-3, 3) \)

For \( T(0, 3) \):

Multiply each coordinate by 3:
\( T' = (3 \cdot 0, 3 \cdot 3) = (0, 9) \)

For \( U(0, 1) \):

Multiply each coordinate by 3:
\( U' = (3 \cdot 0, 3 \cdot 1) = (0, 3) \)

Wait, wait, maybe I misread the original coordinates. Let's recheck the graph again. Wait, maybe R is at (-1, -1)? Wait, no, looking at the grid, the x-axis: from -10 to 10, y-axis from -10 to 10. Let's see the points:

• R: x=-1, y=-1? Wait, no, maybe R is at (-1, -1), S at (-1, 1), T at (0, 3), U at (0, 1). Let's confirm the original coordinates:

Looking at the graph:

• R: the point R is at ( -1, -1 )? Wait, the grid lines: each square is 1 unit. So R is at x=-1, y=-1; S is at x=-1, y=1; T is at x=0, y=3; U is at x=0, y=1. Yes.

So applying dilation with scale factor 3:

• \( R(-1, -1) \) becomes \( (3 \times -1, 3 \times -1) = (-3, -3) \)

• \( S(-1, 1) \) becomes \( (3 \times -1, 3 \times 1) = (-3, 3) \)

• \( T(0, 3) \) becomes \( (3 \times 0, 3 \times 3) = (0, 9) \)

• \( U(0, 1) \) becomes \( (3 \times 0, 3 \times 1) = (0, 3) \)

Wait, but maybe the original coordinates are different. Wait, let's check again. Maybe R is at (-1, -1), S at (-1, 1), T at (0, 3), U at (0, 1). Yes, that seems correct.

Answer:

\( R'(-3, -3) \)

\( S'(-3, 3) \)

\( T'(0, 9) \)

\( U'(0, 3) \)