AP PreCalculus Unit 4: Functions Involving Parameters, Vectors, and Matrices Practice Exam

Assessment for Unit 4: Functions Involving Parameters, Vectors, and Matrices

Estimated Time: 65 minutes

Exam Details

Questions: 49 total (45 multiple choice, 4 short answer)
Time: 65 minutes (recommended)
Topics Covered: Unit 4: Functions Involving Parameters, Vectors, and Matrices
Format: Multiple Choice, Short Answer questions matching real exam difficulty
Scoring: Instant results with detailed explanations

A. Multiple-Choice Questions (45 Questions)

Select the one best answer for each question.

1. A curve is defined by the parametric equations $x(t) = t^2 - 2$ and $y(t) = t$ for the interval $-2 \le t \le 2$. Which of the following statements best describes the graph of the curve in the $xy$-plane?

(A)A parabola opening to the right, traced from $(2, -2)$ to $(2, 2)$.(B)A parabola opening to the right, traced from $(2, 2)$ to $(2, -2)$.(C)A parabola opening upward, traced from $(-2, 2)$ to $(2, 2)$.(D)A parabola opening upward, traced from $(2, -2)$ to $(2, 2)$.

2. The position of a particle at time $t$ is given by the parametric function $f(t) = (x(t), y(t))$, where $x(t) = 3\cos(t)$ and $y(t) = 3\sin(t)$ for $\pi \le t \le 2\pi$. Which of the following describes the path of the particle?

(A)The upper semicircle of radius 3 centered at the origin, moving from right to left.(B)The upper semicircle of radius 3 centered at the origin, moving from left to right.(C)The lower semicircle of radius 3 centered at the origin, moving from left to right.(D)The lower semicircle of radius 3 centered at the origin, moving from right to left.

3. [Skill: 2A | Topic: 4.1] A parametric function in the plane is defined by the table of values below, where points are generated in order of increasing parameter value $t$. $t$: $-2, -1, 0, 1, 2$ $x(t)$: $4, 1, 0, 1, 4$ $y(t)$: $-2, -1, 0, 1, 2$ Which of the following best describes the graph that is traced by the parametric function as $t$ increases from $-2$ to $2$?

(A)A right-opening parabola with vertex at $(0,0)$, traced upward from $(4,-2)$ through $(0,0)$ to $(4,2)$(B)A right-opening parabola with vertex at $(0,0)$, traced downward from $(4,2)$ through $(0,0)$ to $(4,-2)$(C)An upward-opening parabola with vertex at $(0,0)$, traced from $(-2,4)$ through $(0,0)$ to $(2,4)$(D)A circle centered at the origin, traced once counterclockwise

4. Consider the parametric function defined by $x(t) = \sqrt{t}$ and $y(t) = t - 1$ for $t \ge 0$. Which of the following best describes the graph of the function in the $xy$-plane?

(A)A line starting at $(0, -1)$ and increasing with a slope of 1.(B)A parabola opening upward with vertex at $(0, -1)$.(C)The right half of a parabola opening upward with vertex at $(0, -1)$.(D)The top half of a parabola opening to the right with vertex at $(-1, 0)$.

5. Let $A$ be a matrix of dimension $2 \times 4$ and let $B$ be a matrix of dimension $4 \times 3$. Which of the following correctly describes the existence and dimensions of the matrix products $AB$ and $BA$?

(A)The product $AB$ exists and has dimensions $2 \times 3$, but the product $BA$ does not exist.(B)The product $AB$ exists and has dimensions $2 \times 3$, and the product $BA$ exists and has dimensions $4 \times 4$.(C)The product $AB$ does not exist, but the product $BA$ exists and has dimensions $4 \times 2$.(D)Both products $AB$ and $BA$ exist, and they have the same dimensions.

6. Let $M = \begin{pmatrix} 2 & -3 \\ 4 & 1 \end{pmatrix}$ and $N = \begin{pmatrix} 5 & 0 \\ 2 & -1 \end{pmatrix}$. What is the product $MN$?

(A)$\begin{pmatrix} $10 & 0 $\\ $8 & -1 $\end{pmatrix}$(B)$\begin{pmatrix} $4 & 3 $\\ $22 & -1 $\end{pmatrix}$(C)$\begin{pmatrix} $4 & 3 $\\ $22 & 1 $\end{pmatrix}$(D)$\begin{pmatrix} $16 & -3 $\\ $22 & -1 $\end{pmatrix}$

7. Matrices $A$ and $B$ are defined as follows: $$ A = \begin{pmatrix} 1 & 4 \\ -2 & 3 \\ 0 & 5 \end{pmatrix}, \quad B = \begin{pmatrix} 2 & -1 \\ 6 & 0 \end{pmatrix} $$ If $C = AB$, what is the value of the entry $c_{21}$ (the element in the second row and first column of $C$)?

(A)14(B)2(C)-4(D)12

8. Let $A = \begin{pmatrix} 3 & k \\ 1 & 2 \end{pmatrix}$ and $X = \begin{pmatrix} 2 \\ -1 \end{pmatrix}$. If the product $AX = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$, what is the value of the parameter $k$?

(A)-5(B)1(C)5(D)7

9. Let $A$ be the matrix defined by $A = \begin{pmatrix} 4 & 2 \\ -3 & -1 \end{pmatrix}$. Which of the following represents $A^{-1}$?

(A)$\begin{pmatrix} $-0.5 & -1 $\\ $1.5 & 2 $\end{pmatrix}$(B)$\begin{pmatrix} $-2 & -4 $\\ $6 & 8 $\end{pmatrix}$(C)$\begin{pmatrix} $-1 & -2 $\\ $3 & 4 $\end{pmatrix}$(D)$\begin{pmatrix} $2 & 1 $\\ $-1.5 & -0.5 $\end{pmatrix}$

10. For what value of constant $k$ does the matrix $M = \begin{pmatrix} 6 & k \\ -2 & 5 \end{pmatrix}$ not have an inverse?

(A)-15(B)-1.2(C)1.2(D)15

11. Matrices $A$ and $B$ are inverses of each other. If $A = \begin{pmatrix} 3 & 1 \\ 4 & 2 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & -0.5 \\ x & 1.5 \end{pmatrix}$, what is the value of $x$?

(A)-4(B)-2(C)2(D)4

12. Which of the following matrices represents a linear transformation that maps distinct vectors to the same vector (is not one-to-one), and why?

(A)$\begin{pmatrix} 5 & 0 \\ 0 & 5 \end{pmatrix}$, because the matrix entries are identical on the diagonal.(B)$\begin{pmatrix} 3 & -2 \\ 1 & 4 \end{pmatrix}$, because the determinant is positive.(C)$\begin{pmatrix} 2 & 4 \\ 3 & 6 \end{pmatrix}$, because the determinant is zero.(D)$\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$, because it is the identity matrix.

13. A linear transformation $L: \mathbb{R}^2 \to \mathbb{R}^2$ is defined by the matrix $A = \begin{pmatrix} 2 & -3 \\ 1 & 4 \end{pmatrix}$, such that $L(\vec{v}) = A\vec{v}$. Which of the following vectors is the image of the input vector $\vec{v} = \langle 5, 2 \rangle$ under this transformation?

(A)$\langle 4, 13 \rangle$(B)$\langle 16, 13 \rangle$(C)$\langle 7, 7 \rangle$(D)$\langle 4, 9 \rangle$

14. The linear transformation $T$ maps a vector $\vec{u} = \langle x, y \rangle$ to a vector $\vec{w}$ using the rule $T(\vec{u}) = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}$. If a triangle has vertices at points $(2, 0)$, $(4, 1)$, and $(2, 3)$, which of the following gives the coordinates of the vertices of the transformed triangle?

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

15. A linear transformation maps input vectors to output vectors according to the equations $x' = 3x + 2y$ and $y' = -x + 5y$, where $\langle x, y \rangle$ is the input vector and $\langle x', y' \rangle$ is the output vector. If the input vectors are represented by the matrix $B = \begin{pmatrix} 1 & 0 \\ -2 & 4 \end{pmatrix}$, where each column is a vector, which matrix represents the corresponding output vectors?

(A)$\begin{pmatrix} -1 & 8 \\ -11 & 20 \end{pmatrix}$(B)$\begin{pmatrix} 3 & 2 \\ -1 & 5 \end{pmatrix}$(C)$\begin{pmatrix} 7 & 12 \\ -9 & 20 \end{pmatrix}$(D)$\begin{pmatrix} -1 & -11 \\ 8 & 20 \end{pmatrix}$

16. The linear transformation $L: \mathbb{R}^2 \to \mathbb{R}^2$ rotates every vector counterclockwise by an angle of $\theta = \frac{2\pi}{3}$ radians about the origin. Which of the following matrices is associated with the transformation $L$?

(A)$\begin{pmatrix} -\frac{1}{2} & -\frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & -\frac{1}{2} \end{pmatrix}$(B)$\begin{pmatrix} -\frac{1}{2} & \frac{\sqrt{3}}{2} \\ -\frac{\sqrt{3}}{2} & -\frac{1}{2} \end{pmatrix}$(C)$\begin{pmatrix} \frac{\sqrt{3}}{2} & -\frac{1}{2} \\ \frac{1}{2} & \frac{\sqrt{3}}{2} \end{pmatrix}$(D)$\begin{pmatrix} -\frac{\sqrt{3}}{2} & -\frac{1}{2} \\ \frac{1}{2} & -\frac{\sqrt{3}}{2} \end{pmatrix}$

17. Let $f$ and $g$ be linear transformations defined by $f(\vec{v}) = A\vec{v}$ and $g(\vec{v}) = B\vec{v}$, where $A = \begin{pmatrix} 2 & 1 \\ 0 & -1 \end{pmatrix}$ and $B = \begin{pmatrix} 3 & 0 \\ 1 & 2 \end{pmatrix}$. Which of the following matrices represents the composite transformation $h(\vec{v}) = f(g(\vec{v}))$?

(A)$\begin{pmatrix} 6 & 3 \\ 2 & -1 \end{pmatrix}$(B)$\begin{pmatrix} 7 & 2 \\ -1 & -2 \end{pmatrix}$(C)$\begin{pmatrix} 6 & 0 \\ 0 & -2 \end{pmatrix}$(D)$\begin{pmatrix} 5 & 1 \\ 1 & 1 \end{pmatrix}$

18. A linear transformation $T$ is represented by the matrix $M = \begin{pmatrix} -2 & 4 \\ 3 & 1 \end{pmatrix}$. If a region $R$ in the $xy$-plane has an area of 5 square units, what is the area of the image of $R$ under the transformation $T$?

(A)14(B)50(C)70(D)-70

19. A logistics company manages a fleet of delivery trucks that operate between two main distribution centers, Center A and Center B. Each morning, the trucks are dispatched from their current location. Historical data indicates the following transition probabilities for the trucks' locations by the end of the day: - Of the trucks starting at Center A, 70% return to Center A and 30% end up at Center B. - Of the trucks starting at Center B, 45% end up at Center A and 55% return to Center B. Let $V_t = \begin{bmatrix} a_t \\ b_t \end{bmatrix}$ represent the number of trucks at Center A and Center B, respectively, on day $t$. Which of the following matrix equations correctly models the distribution of trucks on day $t+1$ based on the distribution on day $t$?

(A)$V_{t+1} = \begin{bmatrix} 0.70 & 0.45 \\ 0.30 & 0.55 \end{bmatrix} V_t$(B)$V_{t+1} = \begin{bmatrix} 0.70 & 0.30 \\ 0.45 & 0.55 \end{bmatrix} V_t$(C)$V_{t+1} = \begin{bmatrix} 0.30 & 0.55 \\ 0.70 & 0.45 \end{bmatrix} V_t$(D)$V_{t+1} = \begin{bmatrix} 0.70 & 0.55 \\ 0.30 & 0.45 \end{bmatrix} V_t$

20. A simplified weather model for a specific region classifies each day as either Sunny (S) or Rainy (R). The transition between these states from one day to the next is modeled by the matrix equation $W_{n+1} = T W_n$, where $W_n = \begin{bmatrix} S_n \\ R_n \end{bmatrix}$ and $T = \begin{bmatrix} 0.8 & 0.4 \\ 0.2 & 0.6 \end{bmatrix}$. If the weather on Day 0 is represented by the initial state vector $W_0 = \begin{bmatrix} 100 \\ 0 \end{bmatrix}$ (indicating a 100% certainty of a Sunny day), what is the state vector $W_2$ representing the probabilities for the weather on Day 2?

(A)$\begin{bmatrix} 80 \\ 20 \end{bmatrix}$(B)$\begin{bmatrix} 72 \\ 28 \end{bmatrix}$(C)$\begin{bmatrix} 64 \\ 36 \end{bmatrix}$(D)$\begin{bmatrix} 100 \\ 0 \end{bmatrix}$

21. A biology student uses a matrix model to track the movement of a beetle population between two distinct zones in a terrarium. The population distribution at hour $k$ is given by the vector $P_k$, and the distribution at the next hour is given by $P_{k+1} = M P_k$, where $M$ is an invertible matrix. If the student observes the population distribution at hour 10 ($P_{10}$), which of the following expressions correctly represents the population distribution at hour 8 ($P_8$)?

(A)$M^2 P_{10}$(B)$M^{-1} P_{10}$(C)$(M^{-1})^2 P_{10}$(D)$P_{10} M^{-2}$

22. A particle moves in the $xy$-plane such that its position $(x(t), y(t))$ for time $t > -5$ is given by the parametric equations $x(t) = t^2 - 2t - 8$ and $y(t) = \ln(t + 5)$. At which points does the path of the particle intersect the $y$-axis?

(A)$(0, \ln 3)$ and $(0, \ln 9)$(B)$(\ln 3, 0)$ and $(\ln 9, 0)$(C)$(-8, \ln 5)$(D)$(16, 0)$

23. The position of an object moving in a plane is defined by the parametric functions $x(t) = 5 - 3\sin(2t)$ and $y(t) = 4 + \cos(2t)$. What is the domain of the curve traced by the object (the set of all possible $x$-coordinates)?

(A)$[-3, 3]$(B)$[2, 8]$(C)$[3, 5]$(D)$[4, 6]$

24. A parametric function is defined by $x(t) = 2t + 4$ and $y(t) = t^2 - 1$ for all real numbers $t$. Which of the following Cartesian equations represents the graph of this function?

(A)$y = \frac{1}{4}(x-4)^2 - 1$(B)$y = \frac{1}{2}(x-4)^2 - 1$(C)$y = 4(x-4)^2 - 1$(D)$y = (2x+4)^2 - 1$

25. [Skill: 2.B | Topic: 4.3] The table below gives values for the parametric functions $x(t)$ and $y(t)$ at selected values of time $t$. | $t$ | $x(t)$ | $y(t)$ | |:---:|:---:|:---:| | 0 | 1 | 2 | | 1 | 3 | 6 | | 2 | 6 | 9 | | 3 | 10 | 11 | Based on the values in the table, which of the following best describes the motion of the particle for $0 \le t \le 3$?

(A)The particle moves to the right and up, with the horizontal speed increasing and the vertical speed decreasing.(B)The particle moves to the right and up, with the horizontal speed decreasing and the vertical speed increasing.(C)The particle moves to the right and up, with both horizontal and vertical speeds increasing.(D)The particle moves to the right and up, with both horizontal and vertical speeds decreasing.

26. [Skill: 1.C | Topic: 4.3] A particle moves in the $xy$-plane such that its position for $t \ge 0$ is given by the parametric equations $x(t) = t^2 + 1$ and $y(t) = 3t - 4$. What is the slope of the line segment connecting the point on the curve at $t=1$ to the point on the curve at $t=3$?

(A)3/4(B)3(C)4/3(D)4

27. [Skill: 3.B | Topic: 4.3] The position of a particle moving in the $xy$-plane is defined by the parametric functions $x(t) = \sqrt{t}$ and $y(t) = 2t - 3$ for $t \ge 0$. Which of the following Cartesian equations describes the path of the particle in the $xy$-plane?

(A)$y = 2x^2 - 3$ for all real $x$(B)$y = 2x^2 - 3$ for $x \ge 0$(C)$y = \sqrt{\frac{x+3}{2}}$ for $x \ge -3$(D)$x = \sqrt{\frac{y+3}{2}}$ for all real $y$

28. A particle moves in the $xy$-plane along a linear path from the point $P(-2, 5)$ to the point $Q(4, -3)$ over the time interval $0 \le t \le 2$. Which of the following systems of parametric equations represents the position of the particle $(x(t), y(t))$ ?

(A)$x(t) = -2 + 3t$ and $y(t) = 5 - 4t$(B)$x(t) = -2 + 6t$ and $y(t) = 5 - 8t$(C)$x(t) = 4 - 3t$ and $y(t) = -3 + 4t$(D)$x(t) = -2 + 2t$ and $y(t) = 5 - 3t$

29. A circle in the $xy$-plane has a radius of 3 and is centered at the point $(2, -1)$. A particle moves around this circle such that at time $t=0$, the particle is located at $(2, 2)$. The particle moves in a clockwise direction with a period of $2\pi$. Which of the following parametric systems could describe the motion of the particle?

(A)$x(t) = 2 + 3\cos(t)$ and $y(t) = -1 + 3\sin(t)$(B)$x(t) = 2 + 3\cos(t)$ and $y(t) = -1 - 3\sin(t)$(C)$x(t) = 2 + 3\sin(t)$ and $y(t) = -1 + 3\cos(t)$(D)$x(t) = 2 - 3\sin(t)$ and $y(t) = -1 + 3\cos(t)$

30. A particle moves along the circle defined by $x^2 + y^2 = 16$. The particle starts at the point $(4, 0)$ at time $t=0$ and moves counterclockwise, completing one full revolution every 5 seconds. Which of the following parametric equations describes the $x$-coordinate of the particle's position as a function of $t$ ?

(A)$x(t) = 4\cos\left(\frac{\pi}{5}t\right)$(B)$x(t) = 4\cos\left(\frac{2\pi}{5}t\right)$(C)$x(t) = 4\cos(5t)$(D)$x(t) = 4\cos(10\pi t)$

31. A relation between $x$ and $y$ is defined by the equation $(y+2)^2 = x - 3$. Which of the following equations represents a function $y = h(x)$ defined implicitly by this relation such that the range of $h$ is $(-\infty, -2]$?

(A)$h(x) = \sqrt{x-3} - 2$(B)$h(x) = -\sqrt{x-3} - 2$(C)$h(x) = 2 - \sqrt{x-3}$(D)$h(x) = -2 + \sqrt{x-3}$

32. The equation $3x^2 + y^2 = 12$ defines a relation between $x$ and $y$. If $y = g(x)$ is a function defined implicitly by this relation such that $g(0) > 0$, what is the domain of $g$?

(A)$[-4, 4]$(B)$[-2, 2]$(C)$[-\sqrt{12}, \sqrt{12}]$(D)$[0, \infty)$

33. A conic section is defined by the equation $x = \frac{1}{8}(y-3)^2 - 2$. Which of the following correctly describes the direction in which the conic section opens and the coordinates of its vertex?

(A)Opens to the right; vertex $(-2, 3)$(B)Opens to the right; vertex $(2, -3)$(C)Opens upward; vertex $(-2, 3)$(D)Opens upward; vertex $(2, -3)$

34. An ellipse is defined by the equation $\frac{(x-1)^2}{16} + \frac{(y+4)^2}{9} = 1$. Which of the following lists the coordinates of the vertices of the ellipse?

(A)$(1, -7)$ and $(1, -1)$(B)$(-3, -4)$ and $(5, -4)$(C)$(-5, 4)$ and $(3, 4)$(D)$(-4, 1)$ and $(6, 1)$

35. A hyperbola has its center at $(2, -1)$ and a vertical transverse axis. Which of the following could be an equation for this hyperbola?

(A)$\frac{(x-2)^2}{16} - \frac{(y+1)^2}{9} = 1$(B)$\frac{(y-1)^2}{9} - \frac{(x+2)^2}{16} = 1$(C)$\frac{(y+1)^2}{9} - \frac{(x-2)^2}{16} = 1$(D)$\frac{(x+2)^2}{16} + \frac{(y-1)^2}{9} = 1$

36. A curve in the plane is defined by the equation $y = \frac{3}{x-1}$. Which of the following sets of parametric equations represents this curve?

(A)$x(t) = t$ and $y(t) = \frac{3}{t} - 1$(B)$x(t) = t + 1$ and $y(t) = \frac{3}{t}$(C)$x(t) = \frac{3}{t}$ and $y(t) = t + 1$(D)$x(t) = t - 1$ and $y(t) = \frac{3}{t}$

37. An ellipse is defined by the equation $\frac{(x+3)^2}{4} + \frac{(y-2)^2}{25} = 1$. Which of the following pairs of parametric equations could describe the graph of the ellipse?

(A)$x(t) = -3 + 4 \cos t$ and $y(t) = 2 + 25 \sin t$(B)$x(t) = 3 + 2 \cos t$ and $y(t) = -2 + 5 \sin t$(C)$x(t) = -3 + 2 \cos t$ and $y(t) = 2 + 5 \sin t$(D)$x(t) = -3 + 5 \cos t$ and $y(t) = 2 + 2 \sin t$

38. A particle moves along a curve defined by the parametric equations $x(t) = 4 \sec t + 1$ and $y(t) = 3 \tan t - 2$ for $t$ in the domain $[0, 2\pi)$ where $\cos t \neq 0$. Which of the following Cartesian equations represents the curve traced by the particle?

(A)$\frac{(x-1)^2}{16} - \frac{(y+2)^2}{9} = 1$(B)$\frac{(y+2)^2}{9} - \frac{(x-1)^2}{16} = 1$(C)$\frac{(x-1)^2}{16} + \frac{(y+2)^2}{9} = 1$(D)$\frac{(x+1)^2}{4} - \frac{(y-2)^2}{3} = 1$

39. Let $\vec{u} = \langle 3, -2 \rangle$ and $\vec{v} = \langle -1, 5 \rangle$. What is the magnitude of the vector $\vec{w} = 2\vec{u} - \vec{v}$?

(A)$\sqrt{130}$(B)$\sqrt{85}$(C)9(D)5

40. Which of the following represents the unit vector in the same direction as the vector $\vec{v} = \langle -5, 12 \rangle$?

(A)$\langle $-$\frac{5}{13}$, $\frac{12}{13} \rangle$(B)$\langle $-$\frac{5}{17}$, $\frac{12}{17} \rangle$(C)$\langle $-$\frac{5}{\sqrt{17}}$, $\frac{12}{\sqrt{17}} \rangle$(D)$\langle $-5, 12 $\rangle$

41. Two vectors, $\vec{a}$ and $\vec{b}$, have magnitudes $|\vec{a}| = 6$ and $|\vec{b}| = 4$. If the angle between the two vectors is $\frac{2\pi}{3}$, what is the value of the dot product $\vec{a} \cdot \vec{b}$?

(A)-12(B)-12$\sqrt{3}$(C)12(D)24

42. A particle moves in the $xy$-plane. For time $t \ge 0$, the horizontal position of the particle is given by $x(t) = 2t + 3$ and the vertical position is given by $y(t) = t^2$. Which of the following vector-valued functions $\vec{p}(t)$ represents the position of the particle?

(A)$\vec{p}(t) = \langle 2, 2t \rangle$(B)$\vec{p}(t) = \langle 2t + 3, t^2 \rangle$(C)$\vec{p}(t) = (2t + 3) + t^2$(D)$\vec{p}(t) = \langle t^2, 2t + 3 \rangle$

43. The position of a robotic vehicle moving on a flat surface is defined by the vector-valued function $\vec{r}(t) = \langle 5t - 2, 12t + 1 \rangle$, where $t$ is measured in seconds and distance is measured in meters. What is the speed of the vehicle, in meters per second, at time $t = 3$?

(A)13(B)17(C)39(D)$\sqrt{13^2 + 37^2}$

44. A vector-valued function is given by $\vec{p}(t) = \langle t - 2, (t - 2)^2 + 1 \rangle$ for all real numbers $t$. Which of the following Cartesian equations describes the path traced by the particle in the $xy$-plane?

(A)$y = x^2 + 1$(B)$y = (x - 2)^2 + 1$(C)$x = y^2 + 1$(D)$y = x + 1$

45. The relationship between two variables, $x$ and $y$, is defined by the equation $x^2 - y^2 = 9$. The graph of this relation passes through the point $(5, 4)$. Which of the following statements correctly describes the relationship between the changes in $x$ and $y$ for points on the graph that are very close to $(5, 4)$?

(A)The ratio of the change in $y$ to the change in $x$ is negative, which implies that as $x$ increases, $y$ decreases.(B)The ratio of the change in $y$ to the change in $x$ is negative, which implies that $x$ and $y$ simultaneously increase or simultaneously decrease.(C)The ratio of the change in $y$ to the change in $x$ is positive, which implies that as $x$ increases, $y$ decreases.(D)The ratio of the change in $y$ to the change in $x$ is positive, which implies that $x$ and $y$ simultaneously increase or simultaneously decrease.

B. Short Answer Questions (4 Questions)

Answer all parts of each question. Answers must be in essay form. Outlines or lists alone are not acceptable.

Question 46:

Question 47:

Question 48:

Question 49:

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