Question

write the coordinates of the vertices after a dilation with a scale factor of 2, centered at the origin.
q ( , )
r ( , )
s ( , )

Explanation:

Step1: Find original coordinates

First, we identify the original coordinates of points \( Q \), \( R \), and \( S \) from the graph.
- For point \( Q \): Looking at the grid, the \( x \)-coordinate is \(-4\) and the \( y \)-coordinate is \( 3 \) (wait, no, let's check again. Wait, the axes: the \( y \)-axis is vertical? Wait, no, in the graph, the \( x \)-axis is horizontal (right is positive \( x \)) and \( y \)-axis is vertical (up is positive \( y \))? Wait, no, the labels: the \( y \)-axis has 10 at the top, so maybe the axes are labeled with \( y \) on the horizontal? Wait, no, the standard coordinate system: \( x \)-axis horizontal, \( y \)-axis vertical. But in the graph, the \( y \)-axis is labeled with 10 on the right, so maybe the \( x \)-axis is vertical? Wait, no, the grid lines: let's look at the points. Point \( S \) is at \( (0, 5) \)? Wait, no, the original point \( S \): looking at the graph, \( S \) is at \( (0, 5) \)? Wait, no, the coordinates: let's see, the \( x \)-axis (horizontal) has labels from -10 to 10, and \( y \)-axis (vertical) has labels from -10 to 10. Wait, point \( Q \): \( x = -4 \), \( y = 3 \)? Wait, no, the green points: \( Q \) is at \( (-4, 3) \)? Wait, \( R \) is at \( (-4, 1) \)? Wait, \( S \) is at \( (0, 5) \)? Wait, no, let's re-examine. Wait, the problem says "Write the coordinates of the vertices after a dilation with a scale factor of 2, centered at the origin." So first, find original coordinates.

Wait, looking at the graph:
- Point \( Q \): \( x = -4 \), \( y = 3 \)? Wait, no, maybe \( Q \) is at \( (-4, 3) \), \( R \) at \( (-4, 1) \), \( S \) at \( (0, 5) \)? Wait, no, let's check the grid. Each grid square is 1 unit. So:

- Point \( Q \): \( x = -4 \), \( y = 3 \) (since it's 4 units left on \( x \)-axis, 3 units up on \( y \)-axis)
- Point \( R \): \( x = -4 \), \( y = 1 \) (4 units left, 1 unit up)
- Point \( S \): \( x = 0 \), \( y = 5 \) (on the \( y \)-axis, 5 units up)

Wait, no, maybe the \( x \)-axis is vertical? Wait, the labels: the \( x \)-axis has 10 at the bottom, so \( x \) increases downward? No, standard coordinate system: \( x \)-axis horizontal (right positive), \( y \)-axis vertical (up positive). So in the graph, the horizontal axis is \( x \)-axis (left to right: -10 to 10), vertical axis is \( y \)-axis (bottom to top: -10 to 10). Wait, point \( S \) is at \( (0, 5) \) (on the \( y \)-axis, 5 units up). Point \( R \) is at \( (-4, 1) \) (4 units left on \( x \), 1 unit up on \( y \)). Point \( Q \) is at \( (-4, 3) \) (4 units left, 3 units up).

Step2: Apply dilation with scale factor 2

Dilation centered at the origin with scale factor \( k \) transforms a point \( (x, y) \) to \( (k \cdot x, k \cdot y) \). So scale factor \( k = 2 \).

- For point \( Q \): Original coordinates \( (x, y) = (-4, 3) \). After dilation: \( (2 \cdot (-4), 2 \cdot 3) = (-8, 6) \)
- For point \( R \): Original coordinates \( (x, y) = (-4, 1) \). After dilation: \( (2 \cdot (-4), 2 \cdot 1) = (-8, 2) \)
- For point \( S \): Original coordinates \( (x, y) = (0, 5) \). After dilation: \( (2 \cdot 0, 2 \cdot 5) = (0, 10) \)

Wait, but maybe I misread the original coordinates. Let's check again. Wait, maybe the original coordinates are:

- \( Q \): \( (-4, 3) \)? Wait, no, looking at the graph, \( Q \) is at \( (-4, 3) \), \( R \) at \( (-4, 1) \), \( S \) at \( (0, 5) \). Wait, but let's confirm the axes. The \( x \)-axis (horizontal) has negative on the left, positive on the right. The \( y \)-axis (vertical) has negative at the bottom, positive at the top. So:

- \( Q \): \( x = -4 \), \( y = 3 \)
- \( R \): \( x = -4 \), \( y = 1 \)
- \( S \): \( x = 0 \), \( y = 5 \)

Then dilation with scale factor 2 (centered at origin) means multiply each coordinate by 2.

So:

- \( Q' = (2 \times -4, 2 \times 3) = (-8, 6) \)
- \( R' = (2 \times -4, 2 \times 1) = (-8, 2) \)
- \( S' = (2 \times 0, 2 \times 5) = (0, 10) \)

Wait, but maybe the original coordinates are different. Let's check the graph again. Wait, point \( S \) is at \( (0, 5) \)? Wait, the original \( S \) is at \( (0, 5) \), \( R \) at \( (-4, 1) \), \( Q \) at \( (-4, 3) \). Yes, that makes sense.

Step2: Apply dilation formula

The formula for dilation centered at the origin with scale factor \( k \) is \( (x', y') = (k \cdot x, k \cdot y) \), where \( (x, y) \) is the original coordinate and \( (x', y') \) is the dilated coordinate.

So:

- For \( Q(-4, 3) \): \( Q' = (2 \times -4, 2 \times 3) = (-8, 6) \)
- For \( R(-4, 1) \): \( R' = (2 \times -4, 2 \times 1) = (-8, 2) \)
- For \( S(0, 5) \): \( S' = (2 \times 0, 2 \times 5) = (0, 10) \)

Wait, but maybe I made a mistake in original coordinates. Let's check again. Wait, the graph: \( S \) is at \( (0, 5) \), \( R \) at \( (-4, 1) \), \( Q \) at \( (-4, 3) \). Yes. So after dilation with scale factor 2:

- \( Q'(-8, 6) \)
- \( R'(-8, 2) \)
- \( S'(0, 10) \)

Answer:

\( Q'(-8, 6) \), \( R'(-8, 2) \), \( S'(0, 10) \)