AP Precalculus

A function 函数 is a rule that maps each input to exactly one output. The set of allowed inputs is the domain 定义域; the set of outputs is the range 值域. The input variable is the independent variable 自变量 and the output variable is the dependent variable 因变量. A function rule can be shown graphically, numerically, analytically (a formula), or verbally.

As the input changes, the output changes "in tandem" – together. Over an interval, a function is:

• increasing 递增 if, whenever $a, then $f(a) (bigger input, bigger output);
• decreasing 递减 if, whenever $a, then $f(a)>f(b)$ (bigger input, smaller output).

A graph of a function shows all its input–output pairs, so you can read this behavior straight off the picture.

The average rate of change 平均变化率 of a function over an interval is the change in output divided by the change in input – the single constant rate that would give the same total change. Over $[a,b]$:

The rate of change at a point measures how fast the output changes right at that input. You approximate it with average rates over small intervals around the point. Comparing two points, the one with the larger average rate (over small enough intervals) is changing faster. A positive rate means both quantities move the same way; a negative rate means they move in opposite ways.

Worked example. For $f(x)=x^2$, the average rate of change over $[1,4]$ is $\dfrac{f(4)-f(1)}{4-1}=\dfrac{16-1}{3}=5$. Over $[1,2]$ it is $\dfrac{4-1}{1}=3$ – the rate itself changes, which is exactly why $f$ is not linear.

The average rate of change over $[a,b]$ is the slope of the secant line 割线 from $(a,f(a))$ to $(b,f(b))$.

Completing the square gives the vertex of a parabola

• For a linear 线性 function, the average rate of change over any interval is constant – so the rate at which the rate changes is zero.
• For a quadratic 二次 function, the average rates of change over equal-length intervals themselves form a linear pattern – so those rates change at a constant rate.

This "rate of the rate" idea distinguishes function types: a constant second difference signals a quadratic.

A nonconstant polynomial 多项式 has the form

Key features:

• Where a polynomial switches between increasing and decreasing, it has a local (relative) extremum 局部极值. The greatest local maximum is the global (absolute) maximum 全局最大值; the least local minimum is the global minimum.
• Between any two distinct real zeros there is at least one local maximum or minimum.
• An even-degree polynomial has either a global maximum or a global minimum.
• A point of inflection 拐点 is where the graph changes concavity – from concave up 上凹 to concave down 下凹 or the reverse – i.e. where the rate of change switches between increasing and decreasing.

A zero 零点 (or root 根) of $p$ is a number $a$ with $p(a)=0$. For a real $a$, $(x-a)$ is a factor of $p$ exactly when $a$ is a zero. If the factor $(x-a)$ is repeated $n$ times, that zero has multiplicity 重数 $n$. Counting multiplicities, a degree-$n$ polynomial has exactly $n$ complex 复数 zeros.

A complex number on an Argand diagram: modulus is the distance, argument the angle

• A real zero $a$ gives an $x$-intercept at $(a,0)$; real zeros are the endpoints of the intervals where $p(x)\ge 0$ or $\le 0$.
• Non-real zeros come in conjugate 共轭 pairs: if $a+bi$ is a zero, so is $a-bi$.
• At a real zero of even multiplicity the graph is tangent to the $x$-axis (it touches but does not cross); at odd multiplicity it crosses.
• The degree equals the least $n$ for which the $n$th successive differences of equally-spaced outputs become constant.

End behavior 末端行为 describes where a function heads as the input grows without bound. For a nonconstant polynomial, as $x\to\pm\infty$ the output goes to $+\infty$ or $-\infty$, written e.g. $\lim_{x\to\infty}p(x)=\infty$. Which way depends entirely on the leading term $a_n x^n$, because for large $|x|$ it dominates all lower-degree terms: the sign of $a_n$ and whether $n$ is even or odd fix both ends.

Every polynomial's ends are decided by its degree's parity and the sign of its leading coefficient

A rational function 有理函数 is a quotient of two polynomials, $r(x)=\dfrac{\text{numerator}}{\text{denominator}}$. Its end behavior is governed by the quotient of the leading terms:

When the numerator has higher degree, the curve approaches a slanting asymptote

• Numerator degree > denominator degree: the quotient is a nonconstant polynomial, and $r$ follows that polynomial's end behavior. If that quotient is linear, the graph has a slant asymptote 斜渐近线.
• Equal degrees: the quotient is a constant, giving a horizontal asymptote 水平渐近线 $y=$ the ratio of leading coefficients.
• Numerator degree < denominator degree: the quotient tends to $0$, so the horizontal asymptote is $y=0$.

At a horizontal asymptote $y=b$, the outputs get and stay arbitrarily close to $b$: $\lim_{x\to\pm\infty}r(x)=b$.

The real zeros of a rational function are the real zeros of its numerator that are still in the domain. These zeros, together with the zeros of the denominator, split the number line into the intervals you test when solving inequalities $r(x)\ge 0$ or $r(x)\le 0$.

A vertical asymptote 垂直渐近线 occurs at $x=a$ when $a$ is a zero of the denominator but not cancelled by the numerator – more precisely, when its multiplicity in the denominator exceeds its multiplicity in the numerator. Near it, the denominator is nearly zero, so $r$ shoots to $\pm\infty$: $\lim_{x\to a^{\pm}}r(x)=\pm\infty$. Check each side, since the two sides can go opposite ways.

Worked example. Describe $r(x)=\dfrac{2x^2+3}{x^2-1}$. The numerator and denominator have equal degree, so the horizontal asymptote is $y=\dfrac{2}{1}=2$. The denominator $x^2-1$ is zero at $x=\pm1$ and neither cancels, so there are vertical asymptotes at $x=1$ and $x=-1$.

A rational function approaching a vertical asymptote and a horizontal asymptote

A hole 空洞 (removable point) occurs at $x=c$ when a factor cancels – the multiplicity of the zero $c$ in the numerator is at least its multiplicity in the denominator. The graph is missing a single point. Its height is the limit of the simplified function: if inputs near $c$ give outputs near $L$, the hole is at $(c,L)$, and $\lim_{x\to c}r(x)=L$.

The same expression, written differently, reveals different features:

• Factored form shows real zeros, and hence $x$-intercepts, holes, vertical asymptotes, domain, and range.
• Standard form (expanded) shows the degree and leading term, and hence end behavior.

Polynomial long division 多项式长除法 rewrites $f(x)=g(x)\,q(x)+r(x)$, where $q$ is the quotient 商 and $r$ the remainder 余数 (of smaller degree than $g$). The quotient gives the equation of a slant asymptote when the numerator's degree is one more than the denominator's.

A transformation 变换 builds a new function from an old one $f$:

Adding to the output or input shifts the curve; a multiplier stretches it

• Translations 平移 (shifts): $f(x)+k$ moves up/down; $f(x-h)$ moves right/left.
• Dilations 伸缩 (stretches): $a\,f(x)$ stretches vertically; $f(bx)$ stretches horizontally.
• Reflections 反射: $-f(x)$ flips over the $x$-axis; $f(-x)$ flips over the $y$-axis.

Combine additive shifts and multiplicative stretches to model shifted, scaled versions of a known shape.

Transforming the parent parabola: shift, stretch, and reflect

Choosing a model 模型 means matching a function type to how a quantity changes. A linear model fits a constant rate of change; a quadratic fits a constant second difference; a polynomial fits data with several turns. State the assumptions 假设 your model relies on (for example, that the pattern continues), because a model is only valid where those assumptions hold.

To construct a model, use given features – points, intercepts, zeros with multiplicities, end behavior – to write the function, then use it to answer questions in context. Always check the answer against the domain that makes sense for the situation, and interpret outputs with their real-world units.

• Read a function through its rate of change: average rate = secant slope; a constant rate means linear, a linear-changing rate means quadratic.
• Factor a polynomial to find zeros (x-intercepts) and their multiplicity; even multiplicity touches, odd crosses the axis.
• End behaviour is set by the leading term; a rational function's asymptotes come from the degrees of top and bottom.
• Build models from key features (points, zeros, end behaviour) and state your assumptions.
• Describe transformations precisely: shift $f(x-h)+k$, stretch $af(bx)$, reflect $-f(x)$ / $f(-x)$.

A sequence 数列 is a function from the whole numbers to the real numbers, so its graph is discrete points, not a curve.

An arithmetic sequence climbs in equal steps; a geometric one multiplies by a ratio

• An arithmetic sequence 等差数列 has a common difference 公差 $d$ (a constant rate of change): $a_n = a_0 + dn$, or from a known term, $a_n = a_k + d(n-k)$.
• A geometric sequence 等比数列 has a common ratio 公比 $r$ (a constant proportional change): $g_n = g_0\,r^{\,n}$, or $g_n = g_k\,r^{\,(n-k)}$.

An increasing arithmetic sequence grows by the same amount each step; an increasing geometric sequence grows by a larger amount each step.

Worked example. An arithmetic sequence with $a_0=3$ and $d=5$ has $a_4=3+5(4)=23$. A geometric sequence with $g_0=2$ and $r=3$ has $g_4=2\cdot3^4=162$ – addition versus multiplication makes the geometric one pull far ahead.

Sequences have continuous cousins:

• Linear functions $f(x)=b+mx$ mirror arithmetic sequences – an initial value plus repeated addition of the slope $m$. Point form: $f(x)=y_i+m(x-x_i)$.
• Exponential functions $f(x)=ab^x$ mirror geometric sequences – an initial value times repeated multiplication by the base $b$. Point form: $f(x)=y_i\,r^{\,(x-x_i)}$.

The difference: linear = repeated addition, exponential = repeated multiplication. (A sequence and its function may have different domains.)

The general exponential function 指数函数 is $f(x)=ab^x$ with initial value 初始值 $a\neq 0$ and base 底数 $b>0,\ b\neq 1$. Domain: all real numbers.

Exponential decay: a fixed percentage is lost each period

• $a>0,\ b>1$ gives exponential growth 指数增长; $a>0,\ 0 gives exponential decay 指数衰减.
• Outputs are proportional over equal-length input intervals. So an exponential is always increasing or always decreasing, and always concave up or always concave down – it has no extrema (except on a closed interval) and no points of inflection.

The exponent rules reshape exponential expressions and connect to graph transformations:

• Product: $b^m b^n = b^{m+n}$. A horizontal shift $b^{x+k}$ equals a vertical stretch $ab^x$ with $a=b^k$.
• Power: $(b^m)^n = b^{mn}$. A horizontal stretch $b^{cx}$ equals a base change $(b^c)^x$.
• Negative exponent: $b^{-n}=\dfrac{1}{b^n}$.
• Unit-fraction exponent: $b^{1/k}=\sqrt[k]{b}$ (the $k$th root 方根).

Exponentials model quantities that grow by a constant proportion over equal intervals (repeated multiplication).

• Build a model from a ratio and an initial value, or from two points (solve the system for $a$ and $b$).
• Sometimes a constant must be added to the data to reveal the proportional pattern.
• Use exponential regression 指数回归 on technology to fit a data set.
• The natural base 自然底数 $e\approx 2.718$ is the standard base for real-world models.

When a rate of change shifts only slightly, linear, quadratic, and exponential models may all seem to fit. Choose using context and fit quality:

• A model is appropriate if its residual plot 残差图 shows no pattern (a residual 残差 is actual minus predicted).
• The error is the gap between predicted and actual; context decides whether an over- or under-estimate is safer.

The composite function 复合函数 $(f\circ g)(x)=f(g(x))$ feeds $g$'s output into $f$. Its domain is the inputs of $g$ whose outputs lie in $f$'s domain.

A function as a machine; its inverse runs the machine backwards

• Composition is not commutative: $f(g(x))$ and $g(f(x))$ usually differ.
• To build $f(g(x))$ analytically, substitute $g(x)$ for every $x$ in $f$.
• The identity function 恒等函数 $f(x)=x$ leaves any function unchanged under composition.
• A function can also be decomposed into simpler pieces – useful for seeing an additive shift as composing with $x+k$, or a dilation as composing with $kx$.

A function is invertible 可逆 on a domain where each output comes from a unique input (you may restrict the domain to force this). The inverse function 反函数 $f^{-1}$ reverses the mapping: if $f(a)=b$ then $f^{-1}(b)=a$.

• $f\big(f^{-1}(x)\big)=f^{-1}\big(f(x)\big)=x$ (composition gives the identity).
• Domain and range swap: reverse each $(a,b)$ to $(b,a)$ in a table; reflect the graph over the line $y=x$.
• To find a formula: swap $x$ and $y$ in $y=f(x)$, then solve for $y$. Context may further limit where the inverse applies.

The logarithm 对数 answers "what exponent?": $\log_b c = a$ means exactly $b^a = c$ (with $b>0,\ b\neq 1$). An unwritten base means the common logarithm 常用对数 (base $10$). On a logarithmic scale, each unit is a multiplicative step of the base (…, $10^0,10^1,10^2,$ …).

The logarithm is the inverse of the exponential: $y=b^x$ and $y=\log_b x$ undo each other, so their graphs are reflections over $y=x$. Hence $\log_b(b^x)=x$ and $b^{\log_b x}=x$. The natural logarithm 自然对数 $\ln x = \log_e x$ is the inverse of $e^x$.

e^x and ln x are reflections of each other in the line y = x

The logarithmic function 对数函数 $f(x)=\log_b x$ has domain $x>0$ and range all reals. It is always increasing (for $b>1$) or always decreasing (for $0), always concave one way, with a vertical asymptote at $x=0$ – the mirror image of the exponential's horizontal asymptote. It grows very slowly for large $x$.

The log properties reverse the exponent rules:

To solve, use the fact that exponentials and logs are inverses:

• Isolate the exponential, then take a log of both sides (bring the exponent down with the power property).
• Isolate the log, then exponentiate both sides.
• Always check for extraneous solutions – the argument of a log must stay positive.

Worked example. Solve $2\cdot3^x=54$. Divide by $2$: $3^x=27=3^3$, so $x=3$. When the sides are not tidy powers, take logs instead: $5^x=20$ gives $x=\dfrac{\ln 20}{\ln 5}\approx1.86$.

Logarithms model quantities that change over huge multiplicative ranges (sound, acid strength, earthquakes). Build a log model from data, and use logarithmic regression for a data set. Because a log compresses large values, it turns proportional growth into a straight-line pattern – the idea behind semi-log plots.

A semi-log plot 半对数图 puts the output on a logarithmic axis and the input on a normal axis. On these axes, an exponential function $y=ab^x$ becomes a straight line, because $\log y = \log a + (\log b)\,x$ is linear in $x$. So: if data look linear on a semi-log plot, an exponential model fits; the line's slope gives $\log b$ and its intercept gives $\log a$.

• Distinguish arithmetic (add a common difference) from geometric (multiply a common ratio) sequences and their linear/exponential function cousins.
• Exponential = repeated multiplication, so it eventually outgrows any linear or polynomial model.
• The logarithm is the inverse of the exponential ($\log_b c=a\Leftrightarrow b^a=c$); use it to solve $b^x=k$.
• Apply the log laws (product, quotient, power) and the change-of-base formula; the argument of a log must be positive.
• On a semi-log plot an exponential model becomes a straight line.

A relationship is periodic 周期性 if its output pattern repeats at regular input steps. The period 周期 is the smallest positive $k$ with $f(x+k)=f(x)$ for all $x$. You can build the whole graph by copying a single cycle 周期段, and estimate the period by finding how far apart the pattern repeats. Within each cycle a periodic function still has intervals of increase/decrease and maxima/minima.

sin and cos wave between -1 and 1; tan breaks at 90 and 270 degrees

An angle is in standard position 标准位置 when its vertex is at the origin and its initial ray lies on the positive $x$-axis. Its radian 弧度 measure is the arc length on a unit circle. For the point $P$ where the terminal ray meets a circle of radius $r$:

Naming the sides of a right triangle from the angle

On a circle of radius $r$, $x=r\cos\theta$ and $y=r\sin\theta$. Exact values at the special angles ($0,\tfrac{\pi}{6},\tfrac{\pi}{4},\tfrac{\pi}{3},\tfrac{\pi}{2},\dots$) come from isosceles right and equilateral triangle 等边三角形 geometry, with signs set by the quadrant 象限 of the angle.

On the unit circle 单位圆 ($r=1$), $\sin\theta$ is the $y$-coordinate and $\cos\theta$ is the $x$-coordinate. As $\theta$ increases, both oscillate 振荡 smoothly between $-1$ and $1$ with period $2\pi$. Sine starts at $0$ (rising); cosine starts at $1$. They are the same wave shifted by $\tfrac{\pi}{2}$.

y = sin x and y = cos x are smooth waves between -1 and 1

A sinusoidal function 正弦型函数 is any transformation of sine (or cosine). Its key features:

• Period and frequency 频率 are reciprocals; $\sin\theta$ has period $2\pi$.
• Amplitude 振幅 = half the distance between the maximum and minimum output.
• Midline 中线 $y=d$ = the average of the maximum and minimum (the horizontal center line).

The graph alternates concave up and concave down each half-cycle.

Worked example. For $f(\theta)=3\sin(2\theta)+1$: the amplitude is $3$, the period is $\dfrac{2\pi}{2}=\pi$, and the midline is $y=1$. So the maximum output is $1+3=4$ and the minimum is $1-3=-2$.

The general sinusoid is

To model periodic data (tides, daylight, temperature): read the period from where the pattern repeats, get the amplitude and midline from the max and min, and set the phase shift so a peak lands at the right input. Technology fits a sinusoidal regression. Such a model is only trustworthy over its contextual domain.

$\tan\theta=\dfrac{\sin\theta}{\cos\theta}$ is the slope of the terminal ray. Its slope values repeat every half-revolution, so its period is $\pi$. It has vertical asymptotes where $\cos\theta=0$ (at $\theta=\tfrac{\pi}{2}+k\pi$), and it is always increasing between consecutive asymptotes, switching concavity at each zero.

The inverse trigonometric functions 反三角函数 – arcsine 反正弦, arccosine, arctangent – reverse the trig functions, so they take a ratio and return an angle. Because sine, cosine, and tangent repeat, their domains must be restricted (e.g. sine to $[-\tfrac{\pi}{2},\tfrac{\pi}{2}]$) before an inverse can exist.

Use inverse trig functions to solve trig equations. Because the functions are periodic, there are usually infinitely many solutions – write the general solution by adding multiples of the period, then keep those in the required (often contextual) domain.

Worked example. Solve $2\sin\theta=1$ on $[0,2\pi)$. Then $\sin\theta=\tfrac12$, and on the unit circle sine is positive in the first and second quadrants, so $\theta=\dfrac{\pi}{6}$ or $\dfrac{5\pi}{6}$. Over all reals you would add $2\pi k$ to each.

These are the reciprocals 倒数 of the main three:

sec is the reciprocal of cos, with asymptotes where cos is zero

The Pythagorean identity 毕达哥拉斯恒等式 comes from $x^2+y^2=1$ on the unit circle:

Polar coordinates 极坐标 locate a point by $(r,\theta)$ – distance $r$ from the origin at angle $\theta$. Convert with

Polar coordinates give a point by its distance r and angle theta

The graph of $r=f(\theta)$ is all points $(r,\theta)$ as $\theta$ sweeps around. Restricting the domain of $\theta$ draws only part of the curve. As $\theta$ increases, $r$ grows or shrinks, tracing spirals, circles, roses, and limaçons.

A cardioid r = 1 + cos theta, sketched directly in polar form

As $\theta$ increases, the distance from the origin $r$ changes:

• $r$ positive and increasing (or negative and decreasing) $\Rightarrow$ moving away from the origin;
• $r$ positive and decreasing (or negative and increasing) $\Rightarrow$ moving toward the origin.

Where $r$ switches between increasing and decreasing, the distance reaches a relative extreme. The average rate of change of $r$ with respect to $\theta$, $\dfrac{\Delta r}{\Delta \theta}$, estimates how fast the curve moves in or out over an interval.

• Use radians and the unit circle: $\cos\theta$ and $\sin\theta$ are the coordinates, always between $-1$ and $1$.
• For a sinusoid $a\sin(b(\theta+c))+d$: $|a|$ is the amplitude, $\tfrac{2\pi}{|b|}$ the period, $d$ the midline, $c$ the phase shift.
• Model periodic data by reading period, amplitude, and midline from the max and min.
• Inverse trig functions need a restricted domain and return an angle; trig equations have infinitely many solutions (add the period).
• Convert polar↔rectangular with $x=r\cos\theta$, $y=r\sin\theta$.

Unit 4 is part of the course but is not assessed on the AP Exam's free-response section; 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.

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).

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.

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.

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.

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.

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 $x=h+A\cos t,\ y=k+B\sin t$.

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.

Adding vectors by the triangle law

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 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.

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).

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.

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

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.

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.

• 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 on the AP exam's free-response, but it underpins later courses.