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Functions Involving Parameters, Vectors, and Matrices

AP Precalculus · Topic 4

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Functions Involving Parameters, Vectors, and Matrices

Look at this roller coaster. At every moment the car has two facts: where it is, and which way it is going. One equation cannot hold both. This planet runs a…

English narration · English + 中文 subtitles burned in

4.1

Parametric Functions
Syllabus
Learning ObjectiveEssential Knowledge

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 $f$.

Source: College Board AP Course and Exam Description

A roller coaster: vectors and parametric motion describe path and velocity together
A roller coaster: vectors and parametric motion describe path and velocity together

Unit 4 is part of the course but is not assessed on the AP Precalculus Exam (which covers only Units 1–3); it builds tools used in later courses.

A parametric function 参数函数 describes a curve by giving both coordinates as functions of a third variable, the parameter 参数 $t$: $\big(x(t),\,y(t)\big)$. As $t$ runs through its domain, the point traces a path – and, unlike $y=f(x)$, the path may loop or cross itself.

Explore

What parametric form allows

In parametric form, x and y each depend on a parameter t. Sort each statement.

Vocabulary Train
English
parametric function/ˌpærəˈmetrɪk ˈfʌŋkʃn/
parameter/pəˈræmɪtə/
4.2

Parametric Functions and Planar Motion
Syllabus
Learning ObjectiveEssential Knowledge

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.

Source: College Board AP Course and Exam Description

Reading $t$ as time, a parametric function models motion in the plane. $x(t)$ and $y(t)$ give the horizontal and vertical position; the point's direction of travel is the order in which the curve is drawn as $t$ increases (mark it with arrows).

Worked example. A particle moves with $x(t)=2t-1$ and $y(t)=t^2$ for $-1\le t\le 2$. Describe its path.

Make a table first:

$t$ $-1$ $0$ $1$ $2$
$(x,y)$ $(-3,1)$ $(-1,0)$ $(1,1)$ $(3,4)$

To eliminate the parameter, solve $x=2t-1$ for $t=\tfrac{x+1}{2}$ and substitute: $y=\tfrac{(x+1)^2}{4}$. The path is the part of this parabola from $(-3,1)$ to $(3,4)$. Because $x$ increases with $t$, the particle moves left to right: down to the vertex $(-1,0)$ at $t=0$, then up.

The path x = 2t − 1, y = t² for t from −1 to 2: part of the parabola y = (x+1)²/4 from (−3, 1) down to the vertex (−1, 0) and up to (3, 4), with arrows showing the direction and the t-values marked
The same parabola could be traced in either direction; the arrows and $t$-values carry the information that $y=\tfrac{(x+1)^2}{4}$ alone loses.

Common mistake. Giving the whole parabola $y=\tfrac{(x+1)^2}{4}$ as the answer. The restriction $-1\le t\le 2$ becomes $-3\le x\le 3$, and the equation alone shows no direction.

Explore

A path traced by a parameter

As the parameter runs, the point moves and traces out a planar curve.

4.3

Parametric Functions and Rates of Change
Syllabus
Learning ObjectiveEssential Knowledge

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 $x(t) \neq 0$.

Source: College Board AP Course and Exam Description

Over an interval of $t$, the average rate of change of $x$ is $\dfrac{\Delta x}{\Delta t}$ and of $y$ is $\dfrac{\Delta y}{\Delta t}$. Their signs tell you which way the point moves (right/left, up/down); together they describe the motion's speed and direction along the path.

Worked example. Find the average rates of change for the same particle on $0\le t\le 2$ and on $-1\le t\le 0$.

$$0\le t\le 2:\quad \frac{\Delta x}{\Delta t}=\frac{3-(-1)}{2}=2,\quad \frac{\Delta y}{\Delta t}=\frac{4-0}{2}=2; \qquad -1\le t\le 0:\quad \frac{\Delta y}{\Delta t}=\frac{0-1}{1}=-1.$$

On the first interval it moves right and up on average; on the second it moves down while it moves right.

Explore

Add two vectors tip to tail

Drag the two vectors and watch their sum. A vector has both a direction and a magnitude, unlike a plain number.

4.4

Parametrically Defined Circles and Lines
Syllabus
Learning ObjectiveEssential Knowledge

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 $t$.

Source: College Board AP Course and Exam Description

A circle of radius $R$ centered at $(h,k)$ is $x=h+R\cos t,\ y=k+R\sin t$. A line through $(x_0,y_0)$ with direction $(a,b)$ is $x=x_0+at,\ y=y_0+bt$. Adjusting the coefficients changes the start point, speed, and direction of tracing.

Worked example. The circle with center $(3,-2)$ and radius $5$ is $x=3+5\cos t,\ y=-2+5\sin t$ for $0\le t<2\pi$. It starts at $(8,-2)$ and turns counterclockwise. Writing $y=-2-5\sin t$ instead traces the same circle clockwise.

Worked example. The segment from $(1,4)$ to $(7,1)$ has direction $(7-1,\,1-4)=(6,-3)$, so $x=1+6t,\ y=4-3t$ for $0\le t\le 1$. At $t=0$ the point is at $(1,4)$; at $t=1$ it reaches $(7,1)$.

4.5

Implicitly Defined Functions
Syllabus
Learning ObjectiveEssential Knowledge

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.

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.

Source: College Board AP Course and Exam Description

An implicit 隐式 equation relates $x$ and $y$ without solving for either, such as $x^2+y^2=25$. Its graph may fail the vertical-line test (not a function), so it is often split into pieces or described parametrically.

Explore

Explicit or implicit?

An implicit equation relates x and y without solving for y. Sort each.

Vocabulary Train
English
implicit/ɪmˈplɪsɪt/
4.6

Conic Sections
Syllabus
Learning ObjectiveEssential Knowledge

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 $y - k = \pm \dfrac{b}{a}(x - h)$.

Source: College Board AP Course and Exam Description

Conic sections: one cone, four curves

Conic sections 圆锥曲线 – circles, ellipses, parabolas, and hyperbolas – are the curves from slicing a cone, each given by a quadratic equation in $x$ and $y$. Their standard forms reveal centers, vertices, axes, and asymptotes.

Worked example. Identify $4x^2+9y^2-16x+18y-11=0$.

Complete the square in each variable:

$$4(x^2-4x)+9(y^2+2y)=11 \;\Rightarrow\; 4(x-2)^2-16+9(y+1)^2-9=11 \;\Rightarrow\; 4(x-2)^2+9(y+1)^2=36.$$

Divide by $36$: $\dfrac{(x-2)^2}{9}+\dfrac{(y+1)^2}{4}=1$. This is an ellipse 椭圆 with center $(2,-1)$, reaching $3$ units left and right (to $(-1,-1)$ and $(5,-1)$) and $2$ units up and down (to $(2,1)$ and $(2,-3)$).

Worked example. The hyperbola 双曲线 $\dfrac{x^2}{16}-\dfrac{y^2}{9}=1$ has vertices $(\pm4,0)$ and asymptotes $y=\pm\tfrac34x$. The minus sign makes it a hyperbola, and the positive $x^2$ term means it opens left and right.

Common mistake. Forgetting to subtract the constants that completing the square adds ($-16$ and $-9$ above), or dividing only one term by $36$.

Elliptical orbits: conic sections describe closed paths with two foci — a key parametric model
Elliptical orbits: conic sections describe closed paths with two foci — a key parametric model
Explore

Which conic section?

The squared terms tell you the conic. Sort each equation.

Vocabulary Train
English
Conic sections/ˈkɒnɪk ˈsekʃnz/
ellipse/ɪˈlɪps/
hyperbola/haɪˈpɜːbələ/
4.7

Parametrizing Implicit Curves
Syllabus
Learning ObjectiveEssential Knowledge

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 $t$.

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

Source: College Board AP Course and Exam Description

Many implicit curves can be parametrized – rewritten as $\big(x(t),y(t)\big)$ – which makes them easier to graph and to treat as motion. The circle above is the basic example; ellipses use $x=h+A\cos t,\ y=k+B\sin t$.

For the ellipse above, $x=2+3\cos t,\ y=-1+2\sin t$, $0\le t<2\pi$.

Explore

Turn a circle equation into motion

The implicit circle x squared plus y squared equals 4 becomes the parametric x = 2 cos t, y = 2 sin t as the angle sweeps.

4.8

Vectors
Syllabus
Learning ObjectiveEssential Knowledge

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.

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, $\langle a_1,\, b_1 \rangle \cdot \langle a_2,\, b_2 \rangle = a_1 a_2 + b_1 b_2$.

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

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.

Source: College Board AP Course and Exam Description

Resolving a vector into components

A vector 向量 has both magnitude 大小 (length) and direction. Write it by components $\langle a,b\rangle$. Add vectors component-by-component; scale by multiplying each component; the magnitude is $\sqrt{a^2+b^2}$. Vectors model displacements, velocities, and forces.

Worked example. Let $\vec u=\langle 3,-4\rangle$ and $\vec v=\langle -1,2\rangle$. Then $\vec u+2\vec v=\langle 3-2,\,-4+4\rangle=\langle 1,0\rangle$, and $\lvert\vec u\rvert=\sqrt{9+16}=5$. The unit vector in the direction of $\vec u$ divides by its magnitude: $\tfrac15\vec u=\left\langle \tfrac35,-\tfrac45\right\rangle$.

Adding vectors by the triangle law
Adding vectors by the triangle law
Explore

Add two vectors tip to tail

A vector has magnitude and direction. Adding two vectors places them tip to tail; the resultant runs from the first tail to the last tip. Drag each to see the sum.

Vocabulary Train
English
vector/ˈvektə/
magnitude/ˈmæɡnɪtjuːd/
4.9

Vector-Valued Functions
Syllabus
Learning ObjectiveEssential Knowledge

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.

Source: College Board AP Course and Exam Description

A vector-valued function 向量值函数 outputs a vector for each input, e.g. $\vec{r}(t)=\langle x(t),y(t)\rangle$ – the same information as a parametric function, packaged as a moving position vector.

A vector-valued line: start at a, then slide by t lots of the direction b
A vector-valued line: start at a, then slide by t lots of the direction b
Explore

Vector-valued functions

A vector-valued function outputs a vector; add the component vectors to get the resultant.

Vocabulary Train
English
vector-valued function/ˈvektə ˈvæljuːd ˈfʌŋkʃn/
4.10

Matrices
Syllabus
Learning ObjectiveEssential Knowledge

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.

Source: College Board AP Course and Exam Description

A matrix 矩阵 is a rectangular array of numbers. Add matrices of the same size entry-by-entry; multiply a matrix by a compatible one by combining rows with columns. Matrices store and transform data compactly.

Worked example.

$$\begin{bmatrix}1 & 2\\ 0 & 1\end{bmatrix}\begin{bmatrix}3 & 0\\ 1 & 2\end{bmatrix}=\begin{bmatrix}1\cdot3+2\cdot1 & 1\cdot0+2\cdot2\\ 0\cdot3+1\cdot1 & 0\cdot0+1\cdot2\end{bmatrix}=\begin{bmatrix}5 & 4\\ 1 & 2\end{bmatrix}.$$

In the other order the product is $\begin{bmatrix}3 & 6\\ 1 & 4\end{bmatrix}$, which is different. Matrix multiplication is not commutative: order matters.

Explore

A matrix transforms the plane

A 2x2 matrix moves every point of the plane; this one shears it sideways.

Vocabulary Train
English
matrix/ˈmeɪtrɪks/
4.11

The Inverse and Determinant of a Matrix
Syllabus
Learning ObjectiveEssential Knowledge

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.

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 $\det(A) \neq 0$.

Source: College Board AP Course and Exam Description

The determinant 行列式 of a $2\times 2$ matrix $\begin{bmatrix} a & b \\ c & d\end{bmatrix}$ is $ad-bc$. A matrix has an inverse 逆矩阵 exactly when its determinant is nonzero; the inverse undoes the matrix, and it solves matrix equations (like a reciprocal for numbers).

Worked example. For $A=\begin{bmatrix}3 & 1\\ 2 & 4\end{bmatrix}$, the determinant is $ad-bc=(3)(4)-(1)(2)=10$. Since it is nonzero, $A$ is invertible, and $A^{-1}=\dfrac{1}{10}\begin{bmatrix}4 & -1\\ -2 & 3\end{bmatrix}$ (swap the diagonal, negate the off-diagonal, divide by the determinant).

Worked example. Solve $3x+y=7,\ 2x+4y=8$ with $A^{-1}$. In matrix form $A\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}7\\8\end{bmatrix}$, so

$$\begin{bmatrix}x\\y\end{bmatrix}=A^{-1}\begin{bmatrix}7\\8\end{bmatrix}=\frac{1}{10}\begin{bmatrix}4\cdot7-1\cdot8\\ -2\cdot7+3\cdot8\end{bmatrix}=\frac{1}{10}\begin{bmatrix}20\\10\end{bmatrix}=\begin{bmatrix}2\\1\end{bmatrix}.$$

Check: $3(2)+1=7$ and $2(2)+4(1)=8$.

Explore

The determinant is an area factor

The determinant tells you how much a matrix scales area; here it scales area by four.

Vocabulary Train
English
determinant/dɪˈtɜːmɪnənt/
inverse/ɪnˈvɜːs/
4.12

Linear Transformations and Matrices
Syllabus
Learning ObjectiveEssential Knowledge

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 $L(\vec{v}) = A\vec{v}$.

Source: College Board AP Course and Exam Description

A matrix transforms the plane

A $2\times 2$ matrix acts as a linear transformation 线性变换 of the plane – rotating, reflecting, stretching, or shearing points – by multiplying each position vector. The determinant measures how the transformation scales area (and its sign tells whether orientation flips).

Worked example. The matrix $\begin{bmatrix}2 & 0\\ 0 & 3\end{bmatrix}$ stretches the plane by $2$ horizontally and $3$ vertically, sending the vector $\begin{bmatrix}1\\1\end{bmatrix}$ to $\begin{bmatrix}2\\3\end{bmatrix}$. Its determinant $2\times3=6$ means every area is multiplied by $6$, so the unit square becomes a $2\times3$ rectangle.

Worked example. The rotation by angle $\theta$ counterclockwise about the origin is $\begin{bmatrix}\cos\theta & -\sin\theta\\ \sin\theta & \cos\theta\end{bmatrix}$. For $\theta=90^\circ$ it is $\begin{bmatrix}0 & -1\\ 1 & 0\end{bmatrix}$, which sends $\begin{bmatrix}3\\1\end{bmatrix}$ to $\begin{bmatrix}-1\\3\end{bmatrix}$. Its determinant is $0\cdot0-(-1)(1)=1$: a rotation keeps every area the same.

A 2x2 matrix maps the unit square to a parallelogram; the determinant is the area scale
A 2x2 matrix maps the unit square to a parallelogram; the determinant is the area scale
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Transform the plane with a matrix

A $2\times2$ matrix maps every point to a new one, so it stretches, rotates or shears the whole grid. Change the entries and watch the unit square transform.

Vocabulary Train
English
linear transformation/ˈlɪnɪə trænsfɔːˈmeɪʃn/
4.13

Matrices as Functions
Syllabus
Learning ObjectiveEssential Knowledge

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.

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.

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 $A$.

Source: College Board AP Course and Exam Description

Because a matrix maps input vectors to output vectors, it is a function on the plane. Composing transformations corresponds to multiplying their matrices, and the inverse matrix reverses the mapping.

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A matrix as a function

A matrix takes an input vector and returns an output vector - a function on vectors.

4.14

Matrices Modeling Contexts
Syllabus
Learning ObjectiveEssential Knowledge

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.

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.

Source: College Board AP Course and Exam Description

Matrices model systems that step forward in stages – for example, populations moving between states each year. Repeated multiplication by a transition matrix advances the model one step at a time, so matrix powers predict long-run behavior.

Worked example. Each year $90\%$ of a town's residents who use bus A keep using it and $10\%$ switch to bus B; $20\%$ of bus B users switch to A and $80\%$ stay. Write the model with each column holding where one group goes:

$$T=\begin{bmatrix}0.9 & 0.2\\ 0.1 & 0.8\end{bmatrix}, \qquad T\begin{bmatrix}600\\400\end{bmatrix}=\begin{bmatrix}0.9(600)+0.2(400)\\ 0.1(600)+0.8(400)\end{bmatrix}=\begin{bmatrix}620\\380\end{bmatrix}.$$

After one year, $620$ use A and $380$ use B. The numbers stop changing when the flows balance, $0.1A=0.2B$, so $A=2B$. Out of $1000$ residents, that is about $667$ on A and $333$ on B in the long run.

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A good use of a matrix?

Matrices model transformations, systems and transitions. Sort each task.

4.14

Exam tips
  • A parametric function gives $x(t)$ and $y(t)$ separately — track the direction of motion as $t$ increases.
  • A vector has magnitude and direction; add component-by-component and find magnitude with $\sqrt{a^2+b^2}$.
  • The determinant of a $2\times2$ matrix is $ad-bc$ (mind the minus sign); it scales area under the transformation.
  • A $2\times2$ matrix transforms the plane (rotate, reflect, stretch); an inverse exists only when the determinant is non-zero.
  • Note this unit is not tested on the AP Precalculus Exam (Units 1–3 only), but it underpins later courses.

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