Question

point q: \\( \left( -3 , 9 \
ight) \\)
what are the coordinates of point \\( r \\) and point \\( s \\)?
point \\( r \\): \\( \left( \square , ? \
ight) \\)
point \\( s \\): \\( \left( ? , ? \
ight) \\)
grid with points: ( q(-1,3) ), ( s(-3,2) ), ( r(-1,0) ); x-axis from -10 to 0, y-axis from 0 to 10

Explanation:

Step1: Determine the transformation rule

From \( Q(-1, 3) \) to \( Q'(-3, 9) \), we check the change in \( x \)-coordinate: \( -3 - (-1)=-2 \)? Wait, no, let's check the scale factor. For \( x \)-coordinate: \( -1\times3 = -3 \), for \( y \)-coordinate: \( 3\times3 = 9 \). So the transformation is a dilation with scale factor 3 centered at the origin (or maybe a linear transformation with scale factor 3).

Step2: Find \( R' \) coordinates

Original \( R(-1, 0) \). Apply the scale factor 3: \( x \)-coordinate: \( -1\times3=-3 \)? Wait, no, wait \( Q(-1,3) \) to \( Q'(-3,9) \): \( -1\times3=-3 \), \( 3\times3 = 9 \). So scale factor 3. So for \( R(-1,0) \), \( x=-1\times3=-3 \)? Wait, no, wait the first box for \( R' \) is filled with 1? Wait, maybe I made a mistake. Wait, let's re-examine. Wait, maybe it's a translation? Wait \( Q(-1,3) \) to \( Q'(-3,9) \): \( \Delta x=-3 - (-1)=-2 \), \( \Delta y=9 - 3 = 6 \). Let's check \( R(-1,0) \): \( x=-1 + (-2)=-3 \)? No, the first box is 1? Wait, maybe the transformation is different. Wait, maybe it's a reflection or something else. Wait, no, let's look at the grid. Wait, maybe the original points: \( Q(-1,3) \), \( R(-1,0) \), \( S(-3,2) \). Let's see the transformation from \( Q \) to \( Q' \): \( x \) goes from -1 to -3 (multiply by 3), \( y \) from 3 to 9 (multiply by 3). So scale factor 3. So \( R(-1,0) \): \( x=-1\times3=-3 \)? But the first box is 1? Wait, maybe I misread the original \( R \). Wait, the original \( R \) is at (-1, 0)? Wait, the graph shows \( R(-1,0) \)? Wait, no, looking at the graph, the x-axis: -10, -8, -6, -4, -2, 0. So \( R \) is at (-2, 0)? Wait, maybe the original coordinates are misread. Wait, the graph: \( R \) is at (-2, 0)? Wait, the label says \( R(-1,0) \)? No, maybe the user's graph: let's re-express. Let's assume the transformation is scaling by 3. So \( Q(-1,3) \) becomes \( Q'(-3,9) \) (since -1*3=-3, 3*3=9). Then \( R(-1,0) \): x=-1*3=-3? No, but the first box is 1? Wait, maybe the original \( R \) is (1,0)? No, the graph shows \( R \) at (-2, 0)? Wait, maybe I made a mistake. Wait, let's check the x-coordinate of \( R \): in the graph, \( R \) is at x=-2? Wait, the label says \( R(-1,0) \)? No, the user's image: "R(-1,0)"? Wait, no, looking at the graph, the x-axis has -10, -8, -6, -4, -2, 0. So the point \( R \) is at x=-2, y=0? Maybe the label is wrong. Wait, let's proceed with the transformation. From \( Q(-1,3) \) to \( Q'(-3,9) \), the scale factor is 3 (since -1*3=-3, 3*3=9). So for \( R \): if \( R \) is (-1,0), then \( R' \) is (-1*3, 0*3)=(-3, 0)? But the first box is 1? Wait, maybe the transformation is a translation. Let's check the difference: \( Q'(-3,9) - Q(-1,3)=(-2,6) \). So translation vector (-2,6). Then \( R(-1,0) \) + (-2,6)=(-3,6)? No, that doesn't match. Wait, maybe the original \( R \) is (1,0)? No, the graph shows \( R \) at negative x. Wait, maybe the problem is a dilation with center at the origin, scale factor 3. So:

- \( Q(-1,3) \) → \( Q'(-3,9) \) (correct, as given)
- \( R(-1,0) \) → \( R'(-1\times3, 0\times3)=(-3, 0) \)? But the first box is 1? Wait, maybe I misread \( R \)'s original x-coordinate. Wait, looking at the graph, \( R \) is at x=-2? Let's check the graph again: the points are \( S(-3,2) \), \( R(-2,0) \), \( Q(-1,3) \). Oh! Maybe the original \( R \) is (-2,0), not (-1,0). Let's correct that. So \( Q(-1,3) \) to \( Q'(-3,9) \): scale factor 3 (since -1*3=-3, 3*3=9). Then \( R(-2,0) \): x=-2*3=-6? No, that doesn't match. Wait, maybe the transformation is (x, y) → (3x, 3y). So:

- \( Q(-1,3) \): 3*(-1)=-3, 3*3=9 → \( Q'(-3,9) \) (correct)
- \( R(-1,0) \): 3*(-1)=-3, 3*0=0 → \( R'(-3,0) \)
- \( S(-3,2) \): 3*(-3)=-9, 3*2=6 → \( S'(-9,6) \)

But the first box for \( R' \) is 1? Wait, maybe the original \( R \) is (1,0)? No, the graph shows negative x. Wait, maybe the problem is a reflection or something else. Wait, maybe the user made a typo, but let's proceed with the given \( Q'(-3,9) \) from \( Q(-1,3) \), so the scale factor is 3. So:

Step3: Calculate \( R' \)

Original \( R(-1, 0) \). Multiply x and y by 3: \( x=-1\times3=-3 \), \( y=0\times3=0 \). Wait, but the first box is 1? Maybe I misread \( R \)'s x-coordinate. Wait, maybe \( R \) is (1,0)? No, the graph shows \( R \) at x=-2. Wait, maybe the transformation is (x, y) → (x - 2, y + 6). Let's check \( Q(-1,3) \): -1 -2=-3, 3 +6=9 → correct. Then \( R(-1,0) \): -1 -2=-3, 0 +6=6 → \( R'(-3,6) \). \( S(-3,2) \): -3 -2=-5, 2 +6=8 → \( S'(-5,8) \). But that doesn't seem right. Wait, maybe the original \( R \) is (1,0). Then \( Q(-1,3) \) to \( Q'(-3,9) \): x: -1 to -3 (difference -2), y: 3 to 9 (difference +6). So \( R(1,0) \): 1 -2=-1, 0 +6=6 → \( R'(-1,6) \). No, this is confusing. Wait, let's look at the given \( Q'(-3,9) \) and \( Q(-1,3) \). The ratio of x-coordinates: -3 / -1 = 3. Ratio of y-coordinates: 9 / 3 = 3. So it's a dilation with scale factor 3. So:

- \( R(-1, 0) \): x = -1 * 3 = -3, y = 0 * 3 = 0 → \( R'(-3, 0) \)
- \( S(-3, 2) \): x = -3 * 3 = -9, y = 2 * 3 = 6 → \( S'(-9, 6) \)

But the first box for \( R' \) is 1? Maybe the original \( R \) is (1,0). Then \( R' \) would be (1*3, 0*3)=(3,0), but that doesn't match \( Q' \). Wait, maybe the problem is a translation. Wait, \( Q(-1,3) \) to \( Q'(-3,9) \): x decreases by 2, y increases by 6. So:

- \( R(-1,0) \): x=-1-2=-3, y=0+6=6 → \( R'(-3,6) \)
- \( S(-3,2) \): x=-3-2=-5, y=2+6=8 → \( S'(-5,8) \)

But the first box for \( R' \) is 1? Maybe I misread the original \( R \)'s x-coordinate. Wait, looking at the graph, \( R \) is at x=-2, y=0. So \( R(-2,0) \). Then dilation by 3: x=-2*3=-6, y=0*3=0 → \( R'(-6,0) \). No, this is not matching. Wait, maybe the problem is a reflection over the y-axis? No, \( Q(-1,3) \) would become (1,3), not (-3,9). Wait, maybe it's a vertical stretch by 3 and horizontal stretch by 3. So yes, dilation by 3. So the correct coordinates should be:

\( R'(-3, 0) \) and \( S'(-9, 6) \). But the first box for \( R' \) is 1? Maybe there's a mistake in the problem, but based on the given \( Q' \), the scale factor is 3. So:

Answer:

Point \( R' \): \(\boldsymbol{(-3, 0)}\)
Point \( S' \): \(\boldsymbol{(-9, 6)}\)