Functions Involving Parameters, Vectors, and Matrices · ⁨الدوال التي تتضمن المعلمات والمتجهات والمصفوفات⁩

Learning Objective 4.1.A: Construct a graph or table of values for a parametric function represented analytically.

• 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.
• 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.
• 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.
• 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 $t$.
• 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 $f$.

Learning Objective 4.2.A: Identify key characteristics of a parametric planar motion function that are related to position.

• 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 $t$.
• 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.
• 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.

Learning Objective 4.3.A: Identify key characteristics of a parametric planar motion function that are related to direction and rate of change.

• 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.
• 4.3.A.2 At any given point in the plane, the direction of planar motion may be different for different values of $t$.
• 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.
• 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 $x(t) \neq 0$.

Learning Objective 4.4.A: Express motion around a circle or along a line segment parametrically.

• 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 $0 \leq t \leq 2\pi$.
• 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.
• 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 $t$.

Learning Objective 4.5.A: Construct a graph of an equation involving two variables.

• 4.5.A.1 An equation involving two variables can implicitly describe one or more functions.
• 4.5.A.2 An equation involving two variables can be graphed by finding solutions to the equation.
• 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.

Learning Objective 4.5.B: Determine how the two quantities related in an implicitly defined function vary together.

• 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.
• 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.

Learning Objective 4.6.A: Represent conic sections with horizontal or vertical symmetry analytically.

• 4.6.A.1 A parabola with vertex $(h,\, k)$ can, if $a \neq 0$, 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.
• 4.6.A.2 An ellipse centered at $(h,\, k)$ with horizontal radius $a$ and vertical radius $b$ can be represented analytically as $\dfrac{(x - h)^2}{a^2} + \dfrac{(y - k)^2}{b^2} = 1$. A circle is a special case of an ellipse where $a = b$.
• 4.6.A.3 A hyperbola centered at $(h,\, k)$ with vertical and horizonal lines of symmetry can be represented algebraically as $\dfrac{(x - h)^2}{a^2} - \dfrac{(y - k)^2}{b^2} = 1$ for a hyperbola opening left and right, or as $\dfrac{(y - k)^2}{b^2} - \dfrac{(x - h)^2}{a^2} = 1$ for a hyperbola opening up and down. The asymptotes are $y - k = \pm \dfrac{b}{a}(x - h)$.

Learning Objective 4.7.A: Represent a curve in the plane parametrically.

• 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.
• 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 $t$.

Learning Objective 4.7.B: Represent conic sections parametrically.

• 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 $t$.
• 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 $0 \leq t \leq 2\pi$.
• 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 $0 \leq t \leq 2\pi$. 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 $0 \leq t \leq 2\pi$.

Learning Objective 4.8.A: Identify characteristics of a vector.

• 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.
• 4.8.A.2 A vector $\overrightarrow{P_1 P_2}$ 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 $P_1 = P_2$.
• 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.
• 4.8.A.4 For a vector represented geometrically in the plane, the components of the vector can be found using trigonometry.

Learning Objective 4.8.B: Determine sums and products involving vectors.

• 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.
• 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.
• 4.8.B.3 The dot product of two vectors is the sum of the products of their corresponding components. That is, $\langle a_1,\, b_1 \rangle \cdot \langle a_2,\, b_2 \rangle = a_1 a_2 + b_1 b_2$.

Learning Objective 4.8.C: Determine a unit vector for a given vector.

• 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.
• 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 $\vec{j} = \langle 0,\, 1 \rangle$.

Learning Objective 4.8.D: Determine angle measures between vectors and magnitudes of vectors involved in vector addition.

• 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.
• 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.

Learning Objective 4.9.A: Represent planar motion in terms of vector-valued functions.

• 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.
• 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.

Learning Objective 4.10.A: Determine the product of two matrices.

• 4.10.A.1 An $n \times m$ matrix is an array consisting of $n$ rows and $m$ columns.
• 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.

Learning Objective 4.11.A: Determine the inverse of a $2 \times 2$ matrix.

• 4.11.A.1 The identity matrix, $I$, is a square matrix consisting of $1$s on the diagonal from the top left to bottom right and $0$s everywhere else.
• 4.11.A.2 Multiplying a square matrix by its corresponding identity matrix results in the original square matrix.
• 4.11.A.3 The product of a square matrix and its inverse, when it exists, is the identity matrix of the same size.
• 4.11.A.4 The inverse of a $2 \times 2$ matrix, when it exists, can be calculated with or without technology.

Learning Objective 4.11.B: Apply the value of the determinant to invertibility and vectors.

• 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 $\det(A)$.
• 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.
• 4.11.B.3 The square matrix $A$ has an inverse if and only if $\det(A) \neq 0$.

Learning Objective 4.12.A: Determine the output vectors of a linear transformation using a $2 \times 2$ matrix.

• 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.
• 4.12.A.2 A linear transformation will map the zero vector to the zero vector.
• 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.
• 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 $\mathbb{R}^2$.
• 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 $L(\vec{v}) = A\vec{v}$.

Learning Objective 4.13.A: Determine the association between a linear transformation and a matrix.

• 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 $\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$.
• 4.13.A.2 The mapping of the unit vectors in a linear transformation provides valuable information for determining the associated matrix.
• 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.
• 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.

Learning Objective 4.13.B: Determine the composition of two linear transformations.

• 4.13.B.1 The composition of two linear transformations is a linear transformation.
• 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.

Learning Objective 4.13.C: Determine the inverse of a linear transformation.

• 4.13.C.1 Two linear transformations are inverses if their composition maps any vector to itself.
• 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 $A$.

Learning Objective 4.14.A: Construct a model of a scenario involving transitions between two states using matrices.

• 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.

Learning Objective 4.14.B: Apply matrix models to predict future and past states for $n$ transition steps.

• 4.14.B.1 The product of a matrix that models transitions between states and a corresponding state vector can predict future states.
• 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.
• 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.