Unit 4

Functions Involving Parameters, Vectors, and Matrices: the one-page sheet

Beyond the exam. The AP® Precalculus exam does not test this unit. Many classes teach it, and it prepares you for calculus.

4.1 Parametric Functions

A parametric function f(t) = (x(t), y(t)) sends each value of the parameter t to a point. As t increases the points trace a curve with a direction. The curve is drawn in the xy -plane, and it need not be the graph of a function of x .

A parametric function $f(t) = (x(t), y(t))$ turns each input $t$ , the parameter, into a point of the plane.
To sketch one, make a table: values of $t$ , then $x(t)$ and $y(t)$ , then plot the points in order of increasing $t$ and add arrows.
The curve can go left, turn around, or cross itself. One $x$ with two $y$ -values means the curve is not the graph of a function of $x$ .
If $x(t)$ is easy to solve for $t$ , substitute into $y(t)$ to write $y$ as a function of $x$ .

x(t) and y(t) give one point (x(t), y(t)) the points trace a curve in the plane the curve gets a direction of travel, shown with arrows the curve is not the graph of a function of x , even though x(t) and y(t) are both functions of t

parameter: An extra input variable, often written t, that both coordinates of a curve depend on. Each value of the parameter picks out one point, and increasing it sets the order in which the points are traced.
parametric function: A function f(t) = (x(t), y(t)) that sends each value of a parameter t to the point (x(t), y(t)). As t runs through an interval, the points trace a curve with a direction of travel.

4.2 Parametric Functions Modeling Planar Motion

A particle moving in the plane has position (x(t), y(t)) at time t . Where x(t) increases it moves right, where y(t) increases it moves up, and the turning points of each coordinate give the farthest right, left, highest and lowest positions. Read each coordinate separately, then put the two together.

In planar motion the position at time $t$ is $(x(t), y(t))$ . Read the horizontal motion from $x(t)$ and the vertical motion from $y(t)$ , one at a time.
$x(t)$ increasing: moving right. $x(t)$ decreasing: moving left. $y(t)$ increasing: moving up. $y(t)$ decreasing: moving down.
Where $x(t)$ has a maximum the particle is farthest right; where $y(t)$ has a maximum it is highest.
A model is only good for the times the context allows, such as from launch until the ball lands.

the particle moves right during that time the particle moves down during that time the particle turns from moving right to moving left; that point is farthest right nearby the particle is still moving, so it is not at rest

planar motion: Motion of an object in a flat plane, described by a parametric function: at time t the object is at (x(t), y(t)). It moves right where x(t) increases and up where y(t) increases.

4.3 Parametric Functions and Rates of Change

For a parametric function on [t_1, t_2] , the average rates of change of x(t) and y(t) say how fast each coordinate changed per unit of t . Their quotient, the change in y over the change in x , is the slope of the segment joining the two points, an average steepness of the path.

The average rate of change of $x$ on $[t_1, t_2]$ is $\frac{x(t_2) - x(t_1)}{t_2 - t_1}$ ; the same for $y$ . Units: units of $x$ per unit of $t$ .
The slope of the segment joining the two points is $\frac{y(t_2) - y(t_1)}{x(t_2) - x(t_1)}$ : the average rate of change of $y$ with respect to $x$ .
That slope equals (rate of $y$ ) divided by (rate of $x$ ), as long as the change in $x$ is not $0$ .
Over equal intervals of $t$ , average rates that increase mean the rate of change is increasing.

find the two points (x(t_1), y(t_1)) and (x(t_2), y(t_2)) the average rate of change of x on [t_1, t_2] ; do the same for y the slope of the segment joining the two points, how steep the path is there on average the segment is vertical and its slope is undefined

parametric rate of change: For a parametric function (x(t), y(t)) on an interval [t1, t2]: the average rate of change of x, the average rate of change of y, and their quotient, the change in y over the change in x, which is the slope of the segment joining the two points.

4.4 Parametrically Defined Circles and Lines

(h + rcos t, k + rsin t) for 0 ≤ t ≤ 2π traces the circle with center (h, k) and radius r once counterclockwise, starting at (h + r, k) . A segment from P to Q is P + (Q − P)t for 0 ≤ t ≤ 1 , coordinate by coordinate. Both describe not only a set of points but a trip along it.

A parametric circle: $x(t) = h + r\cos t$ , $y(t) = k + r\sin t$ , $0 \le t \le 2\pi$ , traces the circle with center $(h, k)$ and radius $r$ once, counterclockwise from $(h + r, k)$ .
Change the direction with $k - r\sin t$ ; change the speed with $\cos(bt)$ and $\sin(bt)$ ; trace an arc with a shorter interval of $t$ .
A segment from $(x_1, y_1)$ to $(x_2, y_2)$ : $x(t)$ $= x_1 + (x_2 - x_1)t$ , $y(t)$ $= y_1 + (y_2 - y_1)t$ , $0 \le t \le 1$ .
If $x(t) = x_0 + at$ and $y(t) = y_0 + bt$ with $a \ne 0$ , the line has slope $\frac{b}{a}$ .

every point is r units from (h, k) , so the curve is a circle with center (h, k) and radius r the point starts at (h + r, k) and goes once around counterclockwise the same circle from the same start, but clockwise the points lie on a line whose slope is the rate of y over the rate of x

parametric circle: A circle or line written with parametric equations. The circle with center (h, k) and radius r: x(t) = h + r cos t, y(t) = k + r sin t, which goes once around counterclockwise from (h + r, k) for 0 ≤ t ≤ 2π. The segment from (x1, y1) to (x2, y2): x(t) = x1 + (x2 − x1)t, y(t) = y1 + (y2 − y1)t for 0 ≤ t ≤ 1.

4.5 Implicitly Defined Functions

An implicitly defined function is an equation linking x and y ; its graph is every solution point. Substitute a value to find points, count the y -values for one x to decide whether the curve is a function of x , and solve for y to split it into functions.

An implicitly defined function is given by an equation in $x$ and $y$ , such as $x^2 + xy + y^2 = 7$ , instead of $y = f(x)$ . Its graph is the set of solutions.
To test a point, substitute both coordinates. To find points, substitute one coordinate and solve for the other.
If one $x$ gives two $y$ -values, the curve is not a function of $x$ . Solving for $y$ (sometimes with the quadratic formula) splits it into functions.
An implicit equation can define a function you cannot easily solve for, such as $y^3 + y = x$ .

its graph is every point (x, y) that makes the equation true you get an equation in y alone; each solution gives a point of the curve the curve is not the graph of a function of x , but it can be split into pieces that are the curve has no points above or below that x , which bounds how far left or right it reaches

implicitly defined function: A relationship between x and y given by an equation, such as x² + xy + y² = 7, rather than by a formula y = f(x). Its graph is every point that satisfies the equation; it may or may not be the graph of a function of x.

4.6 Conic Sections

Conic sections are the parabola, the ellipse and the hyperbola, all implicitly defined curves. In standard form you read off the center or vertex, the distances a and b from the denominators' square roots, and for a hyperbola the asymptotes y − k = ±b/a (x − h) . The signs of the squared terms tell which conic you have.

Parabola: $y - k = a(x - h)^2$ opens up or down; $x - h = a(y - k)^2$ opens right ( $a > 0$ ) or left.
Ellipse: $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2}$ $= 1$ , center $(h, k)$ ; vertices $a$ units across and $b$ units up and down. $a = b$ gives a circle.
Hyperbola: $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2}$ $= 1$ opens left and right, vertices $(h \pm a, k)$ , asymptotes $y - k = \pm\frac{b}{a}(x - h)$ . With $y$ first it opens up and down.
Both squares with the same sign: ellipse. Opposite signs: hyperbola. One square: parabola.

the curve is a parabola with vertex (h, k) an ellipse with center (h, k) , reaching a units across and b units up and down a hyperbola with center (h, k) , opening left and right, with asymptotes of slope ±b/a complete the square in each variable, then read the conic off the signs

conic section: A curve made by slicing a double cone with a plane: a parabola, an ellipse (a circle is a special one) or a hyperbola. Ellipse: (x − h)²/a² + (y − k)²/b² = 1. Hyperbola: (x − h)²/a² − (y − k)²/b² = 1, with asymptotes y − k = ±(b/a)(x − h).

4.7 Parametrization of Implicitly Defined Functions

To parametrize a curve given by an equation, choose x(t) and y(t) that make the equation true for every t : (t, f(t)) for a graph, (h + acos t, k + bsin t) for an ellipse, (h + asec t, k + btan t) for a hyperbola. To go back, solve for the trig functions and use a Pythagorean identity.

A parametrization of a curve is a parametric function that traces it. A graph $y = f(x)$ : $(t, f(t))$ . A curve $x = g(y)$ : $(g(t), t)$ .
Ellipse $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2}$ $= 1$ : $(h + a\cos t, k + b\sin t)$ , $0 \le t \le 2\pi$ .
Hyperbola $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2}$ $= 1$ : $(h + a\sec t, k + b\tan t)$ ; $-\frac{\pi}{2} < t < \frac{\pi}{2}$ gives the right branch.
A curve has many parametrizations; they can differ in start, direction and speed.

x(t) = t , y(t) = f(t) traces it left to right cos² t + sin² t = 1 makes every point satisfy the ellipse equation sec² t − tan² t = 1 makes every point satisfy the hyperbola equation you get back the equation in x and y , with the parameter removed

parametrization: A parametric function (x(t), y(t)) that traces a given curve. For example, (t, f(t)) parametrizes y = f(x), and (h + a cos t, k + b sin t) parametrizes an ellipse. A curve has many parametrizations, differing in start, direction and speed.

4.8 Vectors

A vector ⟨a, b⟩ is a size and a direction: a across, b up, length √(a² + b²) . Vectors add tip to tail (component by component), scale by numbers, and give a unit vector when divided by their length. The dot product ac + bd measures the angle between two vectors and is 0 for perpendicular ones.

A vector has a magnitude and a direction. $\mathbf{v} = \left\langle a, b \right\rangle = a\mathbf{i} + b\mathbf{j}$ moves $a$ across and $b$ up; its magnitude is $\left\| \mathbf{v} \right\| = \sqrt{a^2 + b^2}$ .
A vector of magnitude $m$ at angle $\theta$ from the positive $x$ -axis is $\left\langle m\cos\theta, m\sin\theta \right\rangle$ .
Add and subtract component by component. A scalar multiple $k\mathbf{v}$ stretches by $|k|$ and reverses if $k < 0$ . A unit vector $\frac{\mathbf{v}}{\left\| \mathbf{v} \right\|}$ has length $1$ .
Dot product: $\left\langle a, b \right\rangle \cdot \left\langle c, d \right\rangle$ $= ac$ $+ bd$ $= \left\| \mathbf{u} \right\| \left\| \mathbf{v} \right\| \cos\theta$ , $\theta$ the angle between them.

its components are ⟨x_2 − x_1, y_2 − y_1⟩ , wherever it is drawn the magnitude ‖v‖ , the vector's length the sum runs from the first tail to the second tip; in components, add matching entries the dot product; it is 0 exactly when the vectors are perpendicular

vector: A quantity with a magnitude and a direction, written ⟨a, b⟩ or ai + bj: it moves a units across and b units up. Its magnitude is √(a² + b²). Vectors add component by component, a scalar multiple kv scales the length by |k|, a unit vector has length 1, and the dot product ⟨a, b⟩ · ⟨c, d⟩ = ac + bd is 0 for perpendicular vectors.

4.9 Vector-Valued Functions

A vector-valued function p(t) = ⟨x(t), y(t)⟩ is a parametric function written as position vectors. The displacement p(t_2) − p(t_1) says how far and which way the object moved, and dividing by t_2 − t_1 gives the average rate of change vector, made of the average rates of x and y .

A vector-valued function $\mathbf{p}(t)$ $= \left\langle x(t), y(t) \right\rangle$ $= x(t)\mathbf{i}$ $+ y(t)\mathbf{j}$ gives a vector for each input $t$ .
Drawn from the origin, $\mathbf{p}(t)$ is a position vector; its tip traces the curve $(x(t), y(t))$ .
Displacement on $[t_1, t_2]$ : $\mathbf{p}(t_2) - \mathbf{p}(t_1)$ . Distance between the two positions: its magnitude.
Average rate of change of $\mathbf{p}$ on $[t_1, t_2]$ : $\frac{\mathbf{p}(t_2) - \mathbf{p}(t_1)}{t_2 - t_1}$ , whose components are the average rates of change of $x$ and $y$ .

each t gives a position vector from the origin to the point the tip of the position vector traces the same path as the parametric function the displacement vector, from the first position to the second; its magnitude is the distance between them the average rate of change of p : the average rates of x and y , as one vector

vector-valued function: A function that gives a vector for each input, p(t) = ⟨x(t), y(t)⟩ = x(t)i + y(t)j. Drawn from the origin, p(t) is the position vector of the point (x(t), y(t)), and its tip traces the parametric curve. p(t2) − p(t1) is the displacement, and dividing by t2 − t1 gives the average rate of change vector.

4.10 Matrices

A matrix is a rectangle of numbers with dimensions rows by columns. Matrices of the same size add entry by entry. A product AB is defined when A has as many columns as B has rows; each entry is a row of A times a column of B . The order of the factors matters.

A matrix is a rectangular array of numbers. An $m \times n$ matrix has $m$ rows and $n$ columns; $a_{ij}$ is the entry in row $i$ , column $j$ .
Add matrices of the same dimensions entry by entry; a number times a matrix multiplies every entry.
Matrix multiplication: entry $(i, j)$ of $AB$ is row $i$ of $A$ times column $j$ of $B$ , multiplied entry by entry and added. It needs columns of $A$ = rows of $B$ .
Order matters: usually $AB \ne BA$ . The identity $I = \left[ \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right]$ gives $AI = IA = A$ .

its dimensions are m×n , rows first AB is defined; an m×n times an n×p gives an m×p multiply matching entries and add: that is the entry in row i , column j of AB BA usually differs from AB , or is undefined at all

matrix: A rectangular array of numbers with m rows and n columns (dimensions m × n). Matrices of the same dimensions add entry by entry. The product AB is defined when A has as many columns as B has rows; the entry in row i, column j is row i of A times column j of B, multiplied entry by entry and added.

4.11 The Inverse and Determinant of a Matrix

For a 2×2 matrix the determinant ad − bc is the signed area of the parallelogram made by its columns. If it is not 0 , the inverse is 1/(ad − bc) [d, −b; −c, a] , which undoes the matrix and solves AX = B as X = A⁻¹B . If it is 0 , there is no inverse.

The determinant of $\left[ \begin{matrix} a & b \\ c & d \end{matrix} \right]$ is $ad - bc$ . Its absolute value is the area of the parallelogram made by the two columns.
The inverse matrix $A^{-1}$ satisfies $AA^{-1} = A^{-1}A = I$ . For $\det A \ne 0$ : $A^{-1}$ $= \frac{1}{ad - bc}\left[ \begin{matrix} d & -b \\ -c & a \end{matrix} \right]$ .
$\det A = 0$ means no inverse: two different inputs give the same product, so nothing can undo $A$ .
If $AX = B$ and $A$ has an inverse, then $X = A^{-1}B$ .

the determinant det A they make a parallelogram whose area is |det A| A has an inverse: swap a and d , negate b and c , divide by det A the columns lie on one line, the parallelogram is flat, and A has no inverse

determinant: For a 2 × 2 matrix with rows (a, b) and (c, d), the number ad − bc. Its absolute value is the area of the parallelogram made by the columns. The inverse matrix A⁻¹, with AA⁻¹ = A⁻¹A = I, exists exactly when the determinant is not 0, and then it is (1/(ad − bc)) times the matrix with rows (d, −b) and (−c, a).

4.12 Linear Transformations and Matrices

A linear transformation sends ⟨x, y⟩ to ⟨ax + by, cx + dy⟩ , which is the matrix [a, b; c, d] times the input. Its columns are the images of ⟨1, 0⟩ and ⟨0, 1⟩ . Rotations, reflections, dilations and shears are all linear, and every area is multiplied by the absolute value of the determinant.

A linear transformation of the plane has the form $L(\left\langle x, y \right\rangle)$ $= \left\langle ax + by, cx + dy \right\rangle$ : multiplication by $\left[ \begin{matrix} a & b \\ c & d \end{matrix} \right]$ .
The columns of the matrix are $L(\left\langle 1, 0 \right\rangle)$ and $L(\left\langle 0, 1 \right\rangle)$ , so a picture of where the unit square goes gives the matrix.
Rotation by $\theta$ counterclockwise: $\left[ \begin{matrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{matrix} \right]$ . Reflection across the $x$ -axis: $\left[ \begin{matrix} 1 & 0 \\ 0 & -1 \end{matrix} \right]$ . Dilation by $k$ : $\left[ \begin{matrix} k & 0 \\ 0 & k \end{matrix} \right]$ .
Areas are multiplied by $\left| \det \right|$ ; a negative determinant reverses orientation.

L is multiplication by the matrix [a, b; c, d] , with the input written as a column you get the first and second columns of the matrix the origin stays put, lines stay lines and the grid becomes a grid of parallelograms its absolute value is the factor by which every area is multiplied; a negative sign means the picture is flipped

linear transformation: A rule that sends each vector ⟨x, y⟩ to ⟨ax + by, cx + dy⟩: multiplication by the matrix with rows (a, b) and (c, d). Its columns are the images of ⟨1, 0⟩ and ⟨0, 1⟩. It keeps the origin fixed and lines straight, and multiplies every area by the absolute value of its determinant.

4.13 Matrices as Functions

A matrix is a function on vectors. Doing B and then A is the single matrix AB , read right to left, and repeating A gives A^n . When det A ≠ 0 , A⁻¹ undoes A and recovers an input from its output; when det A = 0 , two inputs share an output and nothing can undo it.

A $2 \times 2$ matrix $A$ is a function from vectors to vectors: input $\mathbf{v}$ , output $A\mathbf{v}$ .
Composition of linear transformations: "first $B$ , then $A$ " has matrix $AB$ . The first move is on the right, next to $\mathbf{v}$ .
The inverse transformation has matrix $A^{-1}$ , and exists exactly when $\det A \ne 0$ . To find the input with output $\mathbf{w}$ , compute $A^{-1}\mathbf{w}$ .
Repeating $A$ $n$ times has matrix $A^n$ , and $\det(AB) = \det A \cdot \det B$ .

it returns the output Av , so A acts as a function on vectors the output is A(Bv) = (AB)v : the matrix of the composition is AB , written right to left A⁻¹ is the matrix of the inverse function: it sends each output back to its input two inputs share an output, so the transformation has no inverse

matrix as a function: A matrix used as a function on vectors: input v, output Av. Doing B and then A is the composition of linear transformations, with matrix AB (the first move on the right). The inverse function has matrix A⁻¹, which exists exactly when det A is not 0.

4.14 Matrices Modeling Contexts

A transition matrix holds the chances of moving between states in one step, with "from" columns that add up to 1. Multiplying a state vector by T moves it one step forward, T^n moves it n steps, T⁻¹ moves it one step back, and a steady state is a vector that T leaves unchanged.

A transition matrix $T$ : the entry in row $i$ , column $j$ is the chance of moving from state $j$ to state $i$ in one step. Each column adds up to $1$ .
A state vector lists how many (or what share) are in each state. One step later: $\mathbf{x}_{n+1} = T\mathbf{x}_n$ ; after $n$ steps: $\mathbf{x}_n = T^n\mathbf{x}_0$ .
One step earlier: $\mathbf{x}_{n-1} = T^{-1}\mathbf{x}_n$ .
A steady state satisfies $T\mathbf{x} = \mathbf{x}$ . Over many steps the state vector usually settles toward it.

the transition matrix T ; each column adds up to 1 the expected state one step later: x_(n + 1) = Tx_n x_n = T^nx_0 , the prediction n steps ahead the state one step earlier, as long as T has an inverse

transition matrix: A matrix whose entry in row i, column j is the chance of moving from state j to state i in one step, so each column adds up to 1. A state vector lists how many (or what share) are in each state; with T the transition matrix, one step later it is T times the state vector, n steps later Tⁿ times it, and one step earlier T⁻¹ times it.