Question

write the coordinates of the vertices after a dilation with a scale factor of 4, centered at the origin.
q. ( , )
r. ( , )
s. ( , )
t. ( , )

Explanation:

Step1: Find original coordinates

First, identify the original coordinates of each vertex. From the graph:

• \( T \): Let's assume the original coordinates (before dilation) of \( T \) is \( (0, 1) \)? Wait, no, looking at the grid, let's re - examine. Wait, the axes: x - axis and y - axis. Wait, the original figure: Let's find \( Q \), \( T \), \( S \), \( R \). Wait, from the graph, let's get the original coordinates:
• \( Q \): Let's see, x - coordinate (horizontal) and y - coordinate (vertical). For point \( Q \), moving left (negative x) and up/down? Wait, the grid: each square is 1 unit? Let's check the original points:
• \( Q \): Let's say original \( Q = (-2, 1) \)? Wait, no, maybe I misread. Wait, the original figure: Let's look at the coordinates. Wait, the original \( T \) is at \( (0, 1) \)? No, wait, the original points: Let's take \( Q \), \( T \), \( S \), \( R \). Let's find their original coordinates:
• \( Q \): \( (-2, 1) \)? Wait, no, maybe the original coordinates are:
• \( Q(-2, 1) \), \( T(0, 1) \), \( S(0, - 1) \), \( R(-2, - 1) \)? Wait, no, maybe the original coordinates are:

Wait, perhaps the original coordinates are:

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

Step2: Apply dilation with scale factor 4

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

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

Multiply each coordinate by 4: \( x'=-2\times4=-8 \), \( y' = 1\times4 = 4 \). So \( Q'(-8,4) \).

For point \( T(0,1) \):

Multiply by 4: \( x'=0\times4 = 0 \), \( y'=1\times4 = 4 \). So \( T'(0,4) \). Wait, but the problem has \( Q \), \( R \), \( S \), \( r \)? Wait, the problem has \( Q \), \( R \), \( S \), \( r \)? Wait, the question is about \( Q \), \( R \), \( S \), \( r \)? Wait, maybe the original points are \( Q(-2,1) \), \( R(-2,-1) \), \( S(0,-1) \), \( T(0,1) \). Let's correct:

Original coordinates:

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

Applying dilation with scale factor 4 (centered at origin):

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

New \( Q' \): \( x=-2\times4=-8 \), \( y = 1\times4 = 4 \). So \( Q'(-8,4) \).

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

New \( T' \): \( x = 0\times4=0 \), \( y=1\times4 = 4 \). \( T'(0,4) \).

For \( S(0,-1) \):

New \( S' \): \( x = 0\times4 = 0 \), \( y=-1\times4=-4 \). \( S'(0,-4) \).

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

New \( R' \): \( x=-2\times4=-8 \), \( y=-1\times4=-4 \). \( R'(-8,-4) \).

Wait, but the problem has \( Q \), \( R \), \( S \), \( r \)? Maybe a typo, but assuming the points are \( Q \), \( R \), \( S \), \( T \) (maybe \( r \) is \( T \)?). But let's proceed with the given labels:

If we assume the original points:

• \( Q(-2,1) \)
• \( R(-2,-1) \)
• \( S(0,-1) \)
• \( T(0,1) \) (maybe \( r \) is \( T \))

After dilation with scale factor 4:

• \( Q'(-2\times4,1\times4)=(-8,4) \)
• \( R'(-2\times4,-1\times4)=(-8,-4) \)
• \( S'(0\times4,-1\times4)=(0,-4) \)
• \( T'(0\times4,1\times4)=(0,4) \) (if \( r \) is \( T \))

Answer:

• \( Q'(-8,4) \)
• \( R'(-8,-4) \)
• \( S'(0,-4) \)
• \( T'(0,4) \) (assuming \( r \) is \( T \))

(If the original coordinates were misread, but the key is to multiply each coordinate by 4. For example, if original \( Q = (-2,1) \), then \( Q'=(-8,4) \); if original \( R = (-2,-1) \), \( R'=(-8,-4) \); if original \( S=(0,-1) \), \( S'=(0,-4) \); if original \( T=(0,1) \), \( T'=(0,4) \))