AP®︎ Precalculus: Unit 4 Practice Test

Essential Knowledge: 4.1.A.1

A parametric function in $\mathbb{R}^{2}$, the set of all ordered pairs of two real numbers, consists of a set of two parametric equations in which two dependent variables, $x$ and $y$, are dependent on a single independent variable, $t$, called the parameter.

Essential Knowledge: 4.1.A.2

Because variables $x$ and $y$ are dependent on the independent variable, $t$, the coordinates $(x_i, y_i)$ at time $t_i$ can be written as functions of $t$ and can be expressed as the single parametric function $f(t) = (x(t), y(t))$, where in this case $x$ and $y$ are names of two functions.

Essential Knowledge: 4.1.A.3

A numerical table of values can be generated for the parametric function $f(t) = (x(t), y(t))$ by evaluating $x_i$ and $y_i$ at several values of $t_i$ within the domain.

Essential Knowledge: 4.1.A.4

A graph of a parametric function can be sketched by connecting several points from the numerical table of values in order of increasing value of $t$.

Essential Knowledge: 4.1.A.5

The domain of the parametric function $f$ is often restricted, which results in start and end points on the graph of $f$.

Essential Knowledge: 4.2.A.1

A parametric function given by $f(t) = (x(t), y(t))$ can be used to model particle motion in the plane. The graph of this function indicates the position of a particle at time $t$.

Essential Knowledge: 4.2.A.2

The horizontal and vertical extrema of a particle’s motion can be determined by identifying the maximum and minimum values of the functions $x(t)$ and $y(t)$, respectively.

Essential Knowledge: 4.2.A.3

The real zeros of the function $x(t)$ correspond to $y$-intercepts, and the real zeros of $y(t)$ correspond to $x$-intercepts.

Essential Knowledge: 4.3.A.1

As the parameter increases, the direction of planar motion of a particle can be analyzed in terms of $x$ and $y$ independently. If $x(t)$ is increasing or decreasing, the direction of motion is to the right or left, respectively. If $y(t)$ is increasing or decreasing, the direction of motion is up or down, respectively.

Essential Knowledge: 4.3.A.2

At any given point in the plane, the direction of planar motion may be different for different values of $t$.

Essential Knowledge: 4.3.A.3

The same curve in the plane can be parametrized in different ways and can be traversed in different directions with different parametric functions.

Essential Knowledge: 4.3.A.4

Over a given interval $[t_1, t_2]$ within the domain, the average rate of change can be computed for $x(t)$ and $y(t)$ independently. The ratio of the average rate of change of $y$ to the average rate of change of $x$ gives the slope of the graph between the points on the curve corresponding to $t_1$ and $t_2$, so long as the average rate of change of $x(t) \ne 0$.

Essential Knowledge: 4.4.A.1

A complete counterclockwise revolution around the unit circle that starts and ends at $(1, 0)$ and is centered at the origin can be modeled by $(x(t), y(t)) = (\cos t, \sin t)$ with domain $0 \leq t \leq 2\pi$.

Essential Knowledge: 4.4.A.2

Transformations of the parametric function $(x(t), y(t)) = (\cos t, \sin t)$ can model any circular path traversed in the plane.

Essential Knowledge: 4.4.A.3

A linear path along the line segment from the point $(x_1, y_1)$ to the point $(x_2, y_2)$ can be parametrized many ways, including using an initial position $(x_1, y_1)$ and rates of change for $x$ with respect to $t$ and $y$ with respect to $t$.

Essential Knowledge: 4.5.A.1

An equation involving two variables can implicitly describe one or more functions.

Essential Knowledge: 4.5.A.2

An equation involving two variables can be graphed by finding solutions to the equation.

Essential Knowledge: 4.5.A.3

Solving for one of the variables in an equation involving two variables can define a function whose graph is part or all of the graph of the equation.

Essential Knowledge: 4.5.B.1

For ordered pairs on the graph of an implicitly defined function that are close together, if the ratio of the change in the two variables is positive, then the two variables simultaneously increase or both decrease; conversely, if the ratio is negative, then as one variable increases, the other decreases.

Essential Knowledge: 4.5.B.2

The rate of change of $x$ with respect to $y$ or of $y$ with respect to $x$ can be zero, indicating vertical or horizontal intervals, respectively.

Essential Knowledge: 4.6.A.1

A parabola with vertex $(h, k)$ can, if $a \ne 0$, be represented analytically as $x - h = a(y - k)^2$ if it opens left or right, or as $y - k = a(x - h)^2$ if it opens up or down.

Essential Knowledge: 4.6.A.2

An ellipse centered at $(h, k)$ with horizontal radius $a$ and vertical radius $b$ can be represented analytically as $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$. A circle is a special case of an ellipse where $a = b$.

Essential Knowledge: 4.6.A.3

A hyperbola centered at $(h, k)$ with vertical and horizonal lines of symmetry can be represented algebraically as $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$ for a hyperbola opening left and right, or as $\frac{(y - k)^2}{b^2} - \frac{(x - h)^2}{a^2} = 1$ for a hyperbola opening up and down. The asymptotes are $y - k = \pm \frac{b}{a}(x - h)$.

Essential Knowledge: 4.7.A.1

A parametrization $(x(t), y(t))$ for an implicitly defined function will, when $x(t)$ and $y(t)$ are substituted for $x$ and $y$, respectively, satisfy the corresponding equation for every value of $t$ in the domain.

Essential Knowledge: 4.7.A.2

If $f$ is a function of $x$, then $y = f(x)$ can be parametrized as $(x(t), y(t)) = (t, f(t))$. If $f$ is invertible, its inverse can be parametrized as $(x(t), y(t)) = (f(t), t)$ for an appropriate interval of $t$.

Essential Knowledge: 4.7.B.1

A parabola can be parametrized in the same way that any equation that can be solved for $x$ or $y$ can be parametrized. Equations that can be solved for $x$ can be parametrized as $(x(t), y(t)) = (f(t), t)$ by solving for $x$ and replacing $y$ with $t$. Equations that can be solved for $y$ can be parametrized as $(x(t), y(t)) = (t, f(t))$ by solving for $y$ and replacing $x$ with $t$.

Essential Knowledge: 4.7.B.2

An ellipse can be parametrized using the trigonometric functions $x(t) = h + a\cos t$ and $y(t) = k + b\sin t$ for $0 \le t \le 2\pi$.

Essential Knowledge: 4.7.B.3

A hyperbola can be parametrized using trigonometric functions. For a hyperbola that opens left and right, the functions are $x(t) = h + a\sec t$ and $y(t) = k + b\tan t$ for $0 \le t \le 2\pi$. For a hyperbola that opens up and down, the functions are $x(t) = h + a\tan t$ and $y(t) = k + b\sec t$ for $0 \le t \le 2\pi$.

Essential Knowledge: 4.8.A.1

A vector is a directed line segment. When a vector is placed in the plane, the point at the beginning of the line segment is called the tail, and the point at the end of the line segment is called the head. The length of the line segment is the magnitude of the vector.

Essential Knowledge: 4.8.A.2

A vector $\overrightarrow{P_1P_2}$ with two components can be plotted in the $xy$-plane from $P_1=(x_1,y_1)$ to $P_2=(x_2,y_2)$. The vector is identified by $a$ and $b$, where $a=x_2-x_1$ and $b=y_2-y_1$. The vector can be expressed as $\langle a,b\rangle$. A zero vector $\langle 0,0\rangle$ is the trivial case when $P_1=P_2$.

Essential Knowledge: 4.8.A.3

The direction of the vector is parallel to the line segment from the origin to the point with coordinates $(a,b)$. The magnitude of the vector is the square root of the sum of the squares of the components.

Essential Knowledge: 4.8.A.4

For a vector represented geometrically in the plane, the components of the vector can be found using trigonometry.

Essential Knowledge: 4.8.B.1

The multiplication of a constant and a vector results in a new vector whose components are found by multiplying the constant by each of the components of the original vector. The new vector is parallel to the original vector.

Essential Knowledge: 4.8.B.2

The sum of two vectors in $\mathbb{R}^2$ is a new vector whose components are found by adding the corresponding components of the original vectors. The new vector can be represented graphically as a vector whose tail corresponds to the tail of the first vector and whose head corresponds to the head of the second vector when the second vector’s tail is located at the first vector’s head.

Essential Knowledge: 4.8.B.3

The dot product of two vectors is the sum of the products of their corresponding components. That is, $\langle a_1,b_1\rangle\cdot\langle a_2,b_2\rangle=a_1a_2+b_1b_2$.

Essential Knowledge: 4.8.C.1

A unit vector is a vector of magnitude 1. A unit vector in the same direction as a given nonzero vector can be found by scalar multiplying the vector by the reciprocal of its magnitude.

Essential Knowledge: 4.8.C.2

The vector $\langle a,b\rangle$ can be expressed as $a\vec{i}+b\vec{j}$ in $\mathbb{R}^2$, where $\vec{i}$ and $\vec{j}$ are unit vectors in the $x$ and $y$ directions, respectively. That is, $\vec{i}=\langle 1,0\rangle$ and $\vec{j}=\langle 0,1\rangle$.

Essential Knowledge: 4.8.D.1

The dot product is geometrically equivalent to the product of the magnitudes of the two vectors and the cosine of the angle between them. Therefore, if the dot product of two nonzero vectors is zero, then the vectors are perpendicular.

Essential Knowledge: 4.8.D.2

The Law of Sines and Law of Cosines can be used to determine side lengths and angle measures of triangles formed by vector addition.

Essential Knowledge: 4.9.A.1

The position of a particle moving in a plane that is given by the parametric function $f(t) = (x(t), y(t))$ may be expressed as a vector-valued function, $p(t) = x(t)\vec{i} + y(t)\vec{j}$ or $p(t) = \langle x(t), y(t)\rangle$. The magnitude of the position vector at time $t$ gives the distance of the particle from the origin.

Essential Knowledge: 4.9.A.2

The vector-valued function $v(t) = \langle x(t), y(t)\rangle$ can be used to express the velocity of a particle moving in a plane at different times, $t$. At time $t$, the sign of $x(t)$ indicates if the particle is moving right or left, and the sign of $y(t)$ indicates if the particle is moving up or down. The magnitude of the velocity vector at time $t$ gives the speed of the particle.

Essential Knowledge: 4.10.A.1

An $n \times m$ matrix is an array consisting of $n$ rows and $m$ columns.

Essential Knowledge: 4.10.A.2

Two matrices can be multiplied if the number of columns in the first matrix equals the number of rows in the second matrix. The product of the matrices is a new matrix in which the component in the $i$th row and $j$th column is the dot product of the $i$th row of the first matrix and the $j$th column of the second matrix.

Essential Knowledge: 4.11.A.1

The identity matrix, $I$, is a square matrix consisting of 1s on the diagonal from the top left to bottom right and 0s everywhere else.

Essential Knowledge: 4.11.A.2

Multiplying a square matrix by its corresponding identity matrix results in the original square matrix.

Essential Knowledge: 4.11.A.3

The product of a square matrix and its inverse, when it exists, is the identity matrix of the same size.

Essential Knowledge: 4.11.A.4

The inverse of a $2 \times 2$ matrix, when it exists, can be calculated with or without technology.

Essential Knowledge: 4.11.B.1

The determinant of the matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is $ad - bc$. The determinant can be calculated with or without technology and is denoted $\det(A)$.

Essential Knowledge: 4.11.B.2

If a $2 \times 2$ matrix consists of two column or row vectors from $\mathbb{R}^2$, then the nonzero absolute value of the determinant of the matrix is the area of the parallelogram spanned by the vectors represented in the columns or rows of the matrix. If the determinant equals 0, then the vectors are parallel.

Essential Knowledge: 4.11.B.3

The square matrix $A$ has an inverse if and only if $\det(A) \ne 0$.

Essential Knowledge: 4.12.A.1

A linear transformation is a function that maps an input vector to an output vector such that each component of the output vector is the sum of constant multiples of the input vector components.

Essential Knowledge: 4.12.A.2

A linear transformation will map the zero vector to the zero vector.

Essential Knowledge: 4.12.A.3

A single vector in $\mathbb{R}^{2}$ can be expressed as a $2 \times 1$ matrix. A set of $n$ vectors in $\mathbb{R}^{2}$ can be expressed as a $2 \times n$ matrix.

Essential Knowledge: 4.12.A.4

For a linear transformation, $L$, from $\mathbb{R}^{2}$ to $\mathbb{R}^{2}$, there is a unique $2 \times 2$ matrix, $A$, such that $L(\vec{v}) = A\vec{v}$ for vectors in $\mathbb{R}^{2}$. Conversely, for a given $2 \times 2$ matrix, $A$, the function $L(\vec{v}) = A\vec{v}$ is a linear transformation from $\mathbb{R}^{2}$ to $\mathbb{R}^{2}$.

Essential Knowledge: 4.12.A.5

Multiplication of a $2 \times 2$ transformation matrix, $A$, and a $2 \times n$ matrix of $n$ input vectors gives a $2 \times n$ matrix of the $n$ output vectors for the linear transformation $L(\vec{v}) = A\vec{v}$.

Essential Knowledge: 4.13.A.1

The linear transformation mapping $\langle x, y\rangle$ to $\langle a_{11}x + a_{12}y, a_{21}x + a_{22}y\rangle$ is associated with the matrix $\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$.

Essential Knowledge: 4.13.A.2

The mapping of the unit vectors in a linear transformation provides valuable information for determining the associated matrix.

Essential Knowledge: 4.13.A.3

The matrix $\begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$ is associated with a linear transformation of vectors that rotates every vector an angle $\theta$ counterclockwise about the origin.

Essential Knowledge: 4.13.A.4

The absolute value of the determinant of a $2 \times 2$ transformation matrix gives the magnitude of the dilation of regions in $\mathbb{R}^2$ under the transformation.

Essential Knowledge: 4.13.B.1

The composition of two linear transformations is a linear transformation.

Essential Knowledge: 4.13.B.2

The matrix associated with the composition of two linear transformations is the product of the matrices associated with each linear transformation.

Essential Knowledge: 4.13.C.1

Two linear transformations are inverses if their composition maps any vector to itself.

Essential Knowledge: 4.13.C.2

If a linear transformation, $L$, is given by $L(\vec{v}) = A\vec{v}$, then its inverse transformation is given by $L^{-1}(\vec{v}) = A^{-1}\vec{v}$, where $A^{-1}$ is the inverse of the matrix $A$.

Essential Knowledge: 4.14.A.1

A contextual scenario can indicate the rate of transitions between states as percent changes. A matrix can be constructed based on these rates to model how states change over discrete intervals.

Essential Knowledge: 4.14.B.1

The product of a matrix that models transitions between states and a corresponding state vector can predict future states.

Essential Knowledge: 4.14.B.2

Repeated multiplication of a matrix that models the transitions between states and corresponding resultant state vectors can predict the steady state, a distribution between states that does not change from one step to the next.

Essential Knowledge: 4.14.B.3

The product of the inverse of a matrix that models transitions between states and a corresponding state vector can predict past states.