Question

write the coordinates of the vertices after a dilation with a scale factor of 4, centered at the origin.
graph with grid, axes, and points s, p, q, r
p((□,□))
q((□,□))
r((□,□))
s((□,□))

Explanation:

Step1: Find original coordinates

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

• \( P \): From the grid, \( P \) is at \( (1, -1) \) (wait, no, looking at the graph, let's recheck. Wait, the x-axis and y-axis: let's see the points. Wait, S is at (-2, 0), P: let's see, the x-coordinate is 1? Wait no, the grid lines: each square is 1 unit. Let's look again. The point P: x is 1? Wait, no, the x-axis: from origin (0,0), moving right 1? Wait, no, the point P: let's see, the coordinates: S is (-2, 0), P: let's check the x and y. Wait, the y-coordinate of P is -1, x-coordinate: let's see, between 0 and 2? Wait, maybe I misread. Wait, the graph: S is at (-2, 0), P: let's see, the x is 1? No, wait, the point P: x=1, y=-1? Wait, no, maybe the original coordinates:

Wait, let's list the original coordinates:

• \( S \): (-2, 0)
• \( P \): (1, -1)? Wait, no, looking at the graph, the point P is at (1, -1)? Wait, no, maybe the x is 1, y is -1? Wait, no, let's check the grid. Each square is 1 unit. So:

• \( P \): x=1, y=-1? Wait, no, the point P is at (1, -1)? Wait, no, maybe I made a mistake. Wait, the point Q: x=2, y=-1? Wait, R is at (2, 0), Q is at (2, -1), P is at (1, -1), S is at (-2, 0). Yes, that makes sense. So original coordinates:

• \( P(1, -1) \)
• \( Q(2, -1) \)
• \( R(2, 0) \)
• \( S(-2, 0) \)

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, \( k = 4 \).

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

Multiply x and y by 4:
\( P' = (4 \cdot 1, 4 \cdot (-1)) = (4, -4) \)

For \( Q(2, -1) \):

Multiply x and y by 4:
\( Q' = (4 \cdot 2, 4 \cdot (-1)) = (8, -4) \)

For \( R(2, 0) \):

Multiply x and y by 4:
\( R' = (4 \cdot 2, 4 \cdot 0) = (8, 0) \)

For \( S(-2, 0) \):

Multiply x and y by 4:
\( S' = (4 \cdot (-2), 4 \cdot 0) = (-8, 0) \)

Wait, wait, maybe I misread the original coordinates. Let's recheck the graph. Let's look at the points:

• S: on the x-axis, at x=-2, y=0: so S(-2, 0)
• P: let's see, the x-coordinate: between 0 and 2? Wait, the grid lines: each square is 1 unit. So from origin (0,0), moving right 1 unit (x=1), down 1 unit (y=-1): so P(1, -1)
• Q: x=2, y=-1: Q(2, -1)
• R: x=2, y=0: R(2, 0)

Yes, that's correct. So dilation with scale factor 4:

\( (x, y) \to (4x, 4y) \)

So:

• P(1, -1) → (41, 4(-1)) = (4, -4)
• Q(2, -1) → (42, 4(-1)) = (8, -4)
• R(2, 0) → (42, 40) = (8, 0)
• S(-2, 0) → (4(-2), 40) = (-8, 0)

Wait, but maybe the original coordinates are different. Wait, maybe I made a mistake in P's x-coordinate. Let's check the graph again. The point P: is it at (1, -1) or (1, -1)? Wait, the graph shows P between x=0 and x=2, y=-1. So x=1, y=-1. Yes.

Alternatively, maybe the original coordinates are:

Wait, maybe the x-coordinate of P is 1, y is -1. Let's confirm with the grid. Each square is 1 unit, so from (0,0), moving right 1, down 1: (1, -1). Correct.

So applying dilation:

P' = (41, 4(-1)) = (4, -4)

Q' = (42, 4(-1)) = (8, -4)

R' = (42, 40) = (8, 0)

S' = (4(-2), 40) = (-8, 0)

Answer:

\( P'(4, -4) \), \( Q'(8, -4) \), \( R'(8, 0) \), \( S'(-8, 0) \)