Multiple Choice

1. A point $P(3, -5)$ is transformed by a reflection in the x-axis. What are the coordinates of the image point $P'$?

   $P'(3, -5)$

   $P'(-3, -5)$

   $P'(-3, 5)$

   $P'(3, 5)$

2. Which matrix represents a reflection in the y-axis?

   $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$

   $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$

   $\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$

   $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$

3. A point $Q(2, 4)$ undergoes a dilation from the origin with a scale factor of $3$. What are the coordinates of the image point $Q'$?

   $Q'(6, 4)$

   $Q'(2/3, 4/3)$

   $Q'(6, 12)$

   $Q'(5, 7)$

4. A point $R(-2, 7)$ is transformed by a horizontal stretch with a scale factor of $2$. What are the coordinates of the image point $R'$?

   $R'(-4, 14)$

   $R'(-1, 7)$

   $R'(-4, 7)$

   $R'(-2, 14)$

5. Given the transformation matrix $M = \begin{pmatrix} 1 & 0 \\ 3 & 1 \end{pmatrix}$, what are the coordinates of the image of the point $S(4, 1)$ after this transformation?

   $S'(4, 1)$

   $S'(1, 4)$

   $S'(4, 13)$

   $S'(7, 1)$

6. A point $T(6, -2)$ is first reflected in the line $y=x$, and then dilated from the origin by a scale factor of $0.5$. What are the final coordinates of the image point $T''$?

   $T''(3, -1)$

   $T''(-3, 1)$

   $T''(1, -3)$

   $T''(-1, 3)$

7. The matrix $\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$ represents which geometric transformation?

   Reflection in the y-axis

   Dilation from the origin with scale factor -1

   Reflection in the x-axis

   Reflection in the line $y = -x$

8. A triangle has vertices at $A(1, 0)$, $B(0, 2)$, and $C(3, 1)$. What are the coordinates of the vertices of the image triangle $A'B'C'$ after a vertical stretch with a scale factor of $2$?

   $A'(2, 0)$, $B'(0, 4)$, $C'(6, 2)$

   $A'(1, 0)$, $B'(0, 1)$, $C'(3, 0.5)$

   $A'(1, 0)$, $B'(0, 4)$, $C'(3, 1)$

   $A'(1, 0)$, $B'(0, 4)$, $C'(3, 2)$

9. A point $P$ is transformed by the matrix $\begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix}$ to its image $P'(8, -3)$. What were the original coordinates of point $P$?

   $P(4, 3)$

   $P(4, -3)$

   $P(8, -6)$

   $P(16, -3)$

10. Which matrix maps a point $(x, y)$ to $(y, x)$?

    $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$

    $\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$

    $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$

    $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$

11. A square with vertices at $(0,0)$, $(1,0)$, $(1,1)$, and $(0,1)$ is transformed by the matrix $\begin{pmatrix} 3 & 0 \\ 0 & 2 \end{pmatrix}$. What is the area of the transformed square?

    $1$ square unit

    $6$ square units

    $2$ square units

    $3$ square units

12. A point $(x, y)$ is first transformed by a horizontal stretch with a scale factor of $2$, and then by a reflection in the y-axis. What is the single matrix that represents this combined transformation?

    $\begin{pmatrix} 1 & 0 \\ 0 & -2 \end{pmatrix}$

    $\begin{pmatrix} 2 & 0 \\ 0 & -1 \end{pmatrix}$

    $\begin{pmatrix} -2 & 0 \\ 0 & 1 \end{pmatrix}$

    $\begin{pmatrix} -1 & 0 \\ 0 & 2 \end{pmatrix}$

13. A point $K(1, 5)$ is translated by the vector $\begin{pmatrix} 3 \\ -2 \end{pmatrix}$. What are the coordinates of the image point $K'$?

    $K'(-2, 7)$

    $K'(4, 7)$

    $K'(1, 5)$

    $K'(4, 3)$

14. A point $L(2, -3)$ undergoes a horizontal shear with a shear factor of $4$. What are the coordinates of the image point $L'$?

    $L'(-10, -3)$

    $L'(8, -3)$

    $L'(6, -3)$

    $L'(2, -15)$

15. Which matrix represents a vertical stretch with a scale factor of $5$?

    $\begin{pmatrix} 1 & 5 \\ 0 & 1 \end{pmatrix}$

    $\begin{pmatrix} 5 & 0 \\ 0 & 1 \end{pmatrix}$

    $\begin{pmatrix} 5 & 0 \\ 0 & 5 \end{pmatrix}$

    $\begin{pmatrix} 1 & 0 \\ 0 & 5 \end{pmatrix}$

16. A point $M(4, 2)$ is first dilated from the origin by a scale factor of $2$, and then reflected in the x-axis. What are the final coordinates of the image point $M''$?

    $M''(-8, -4)$

    $M''(8, 4)$

    $M''(8, -4)$

    $M''(4, -8)$

17. The image of a point $N$ after a reflection in the line $y=-x$ is $N'(5, -1)$. What were the original coordinates of point $N$?

    $N(-1, 5)$

    $N(5, -1)$

    $N(1, -5)$

    $N(-5, 1)$

18. A point $P(2, 3)$ is transformed to $P'(-2, 3)$. Which transformation matrix could produce this result?

    $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$

    $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$

    $\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$

    $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$

19. A point $F(1, 0)$ is first transformed by a vertical shear with a shear factor of $2$, and then reflected in the x-axis. What are the final coordinates of the image point $F''$?

    $F''(-1, -2)$

    $F''(-1, 0)$

    $F''(1, 2)$

    $F''(1, -2)$

20. A point $G(4, -2)$ is dilated from the origin by a scale factor of $0.5$, and then translated by the vector $\begin{pmatrix} 1 \\ 3 \end{pmatrix}$. What are the final coordinates of the image point $G''$?

    $G''(3, 2)$

    $G''(5, -1)$

    $G''(2, 1)$

    $G''(2, -1)$

21. A point $(x,y)$ is transformed by a dilation in the x-direction (from the y-axis) by a factor of $2$, followed by a reflection in the x-axis. Which matrix represents this combined transformation?

    $\begin{pmatrix} -2 & 0 \\ 0 & 1 \end{pmatrix}$

    $\begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix}$

    $\begin{pmatrix} 1 & 0 \\ 0 & -2 \end{pmatrix}$

    $\begin{pmatrix} 2 & 0 \\ 0 & -1 \end{pmatrix}$

22. Find the single transformation matrix that represents a reflection in the y-axis, followed by a dilation in the y-direction (from the x-axis) by a factor of $3$.

    $\begin{pmatrix} 1 & 0 \\ 0 & -3 \end{pmatrix}$

    $\begin{pmatrix} 1 & 0 \\ 0 & 3 \end{pmatrix}$

    $\begin{pmatrix} -1 & 0 \\ 0 & 3 \end{pmatrix}$

    $\begin{pmatrix} -3 & 0 \\ 0 & 1 \end{pmatrix}$

23. A shape undergoes a dilation in the x-direction (from the y-axis) by a factor of $0.5$, and then a dilation in the y-direction (from the x-axis) by a factor of $4$. What is the transformation matrix for this sequence of transformations?

    $\begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix}$

    $\begin{pmatrix} 0.5 & 0 \\ 0 & 4 \end{pmatrix}$

    $\begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix}$

    $\begin{pmatrix} 4 & 0 \\ 0 & 0.5 \end{pmatrix}$

24. Determine the transformation matrix that maps points after a reflection in the line $y=x$, followed by a reflection in the x-axis.

    $\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$

    $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$

    $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$

    $\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$

25. A transformation consists of a dilation by a factor of $3$ from the origin (uniform scaling), followed by a reflection in the line $y=-x$. What is the overall transformation matrix?

    $\begin{pmatrix} -3 & 0 \\ 0 & -3 \end{pmatrix}$

    $\begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix}$

    $\begin{pmatrix} 0 & 3 \\ 3 & 0 \end{pmatrix}$

    $\begin{pmatrix} 0 & -3 \\ -3 & 0 \end{pmatrix}$

26. Find the single matrix that performs the following sequence of transformations: a dilation in the x-direction (from the y-axis) by a factor of $2$, followed by a reflection in the y-axis, and finally a dilation in the y-direction (from the x-axis) by a factor of $0.5$.

    $\begin{pmatrix} 2 & 0 \\ 0 & -0.5 \end{pmatrix}$

    $\begin{pmatrix} -0.5 & 0 \\ 0 & 2 \end{pmatrix}$

    $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$

    $\begin{pmatrix} -2 & 0 \\ 0 & 0.5 \end{pmatrix}$

27. A point is first reflected in the x-axis. Then, it undergoes a dilation in the x-direction (from the y-axis) by a factor of $4$. Finally, it is reflected in the y-axis. What is the combined transformation matrix?

    $\begin{pmatrix} 4 & 0 \\ 0 & 1 \end{pmatrix}$

    $\begin{pmatrix} -4 & 0 \\ 0 & 1 \end{pmatrix}$

    $\begin{pmatrix} -4 & 0 \\ 0 & -1 \end{pmatrix}$

    $\begin{pmatrix} 1 & 0 \\ 0 & 4 \end{pmatrix}$

28. Determine the transformation matrix for a dilation in the x-direction (from the y-axis) by a factor of $-2$, followed by a reflection in the line $y=x$.

    $\begin{pmatrix} 0 & 1 \\ -2 & 0 \end{pmatrix}$

    $\begin{pmatrix} -2 & 0 \\ 0 & -1 \end{pmatrix}$

    $\begin{pmatrix} 0 & -2 \\ 1 & 0 \end{pmatrix}$

    $\begin{pmatrix} 1 & 0 \\ 0 & -2 \end{pmatrix}$