Your FREE AP Precalculus Flashcards 2026 – 200+ Cards

A rule that assigns exactly one output to each input. The set of inputs is the domain; the set of outputs is the range.

Over $[a, b]$, the slope of the secant line: $\dfrac{f(b) - f(a)}{b - a}$. It measures how fast the output changes per unit input.

A function whose output rises as the input rises: if $a < b$ then $f(a) < f(b)$. Its average rate of change over any interval is positive.

Concave up: the rate of change is increasing (graph bends upward, like a cup). Concave down: the rate of change is decreasing (bends downward).

A point where a graph changes concavity (from concave up to concave down or vice versa). The rate of change switches from increasing to decreasing.

A sum of power functions with whole-number exponents: $f(x) = a_n x^n + \cdots + a_1 x + a_0$, with $a_n \neq 0$. $n$ is the degree.

The highest exponent on the variable. It bounds the number of real zeros (at most $n$) and the number of turning points (at most $n - 1$).

The coefficient $a_n$ of the highest-degree term. Together with the degree it controls the end behavior of the polynomial.

Set by the leading term. Even degree: both ends go the same way. Odd degree: ends go opposite ways. Positive leading coefficient lifts the right end up.

An input $x = a$ where $f(a) = 0$; the graph crosses or touches the x-axis there. Equivalently, $(x - a)$ is a factor.

How many times a factor $(x - a)$ repeats. Odd multiplicity: the graph crosses the x-axis. Even multiplicity: it touches and turns around (bounces).

A degree-$n$ polynomial has exactly $n$ zeros counting multiplicity (over the complex numbers). Non-real complex zeros occur in conjugate pairs $a \pm bi$.

The conjugate of $a + bi$ is $a - bi$. Non-real zeros of a real polynomial always come in conjugate pairs.

A point where a function value is greater than at all nearby inputs; the graph changes from increasing to decreasing there.

The single greatest (maximum) or least (minimum) output value over the entire domain, if it exists.

A function with $f(-x) = f(x)$; its graph is symmetric about the y-axis. Example: $f(x) = x^2$.

A function with $f(-x) = -f(x)$; its graph has 180-degree rotational symmetry about the origin. Example: $f(x) = x^3$.

A ratio of two polynomials $f(x) = \dfrac{p(x)}{q(x)}$, with $q(x) \neq 0$. Its key features come from comparing the numerator and denominator.

A line $x = a$ the graph approaches without touching, occurring where the denominator is zero but the numerator is not. Outputs blow up to $\pm\infty$.

A single missing point that appears when a factor cancels in both numerator and denominator, e.g. $\dfrac{(x-2)(x+1)}{(x-2)}$ has a hole at $x = 2$.

Compare degrees of numerator $n$ and denominator $m$. If $n < m$: $y = 0$. If $n = m$: $y = \dfrac{a_n}{b_m}$ (ratio of leading coefficients). If $n > m$: none.

Occurs when the numerator degree is exactly one more than the denominator degree. Find it by polynomial long division; the quotient line is the asymptote.

An input where the numerator equals zero but the denominator does not. Setting $p(x) = 0$ gives the x-intercepts.

A procedure to divide $p(x)$ by $d(x)$, giving a quotient and remainder. Used to find slant asymptotes and to factor.

When $p(x)$ is divided by $(x - a)$, the remainder equals $p(a)$. So $p(a) = 0$ exactly when $(x - a)$ is a factor.

$(x - a)$ is a factor of $p(x)$ if and only if $p(a) = 0$. It links zeros directly to linear factors.

Any rational zero $\dfrac{p}{q}$ of a polynomial has $p$ dividing the constant term and $q$ dividing the leading coefficient — a list of candidates to test.

For $ax^2 + bx + c = 0$: $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Solves any quadratic.

The quantity $b^2 - 4ac$. Positive gives two real roots, zero gives one repeated real root, negative gives a complex conjugate pair.

$f(x) = a(x - h)^2 + k$, with vertex $(h, k)$. $a > 0$ opens up; $a < 0$ opens down; $|a|$ sets the width.

A function of the form $f(x) = a x^p$. For positive integer $p$ it is a polynomial; power functions are the building blocks of polynomials.

$a^2 - b^2 = (a + b)(a - b)$. A go-to factoring pattern for finding zeros.

$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ and $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$.

All input values for which the function is defined. Exclude inputs that make a denominator zero or a square root negative (in the reals).

All output values the function can produce. For $f(x) = x^2$ the range is $y \geq 0$.

$(f \circ g)(x) = f(g(x))$: apply $g$ first, then $f$. Order matters — composition is generally not commutative.

$f^{-1}$ undoes $f$: $f(f^{-1}(x)) = x$. Its graph is the reflection of $f$ across the line $y = x$; a function has an inverse only if it is one-to-one.

Each output comes from exactly one input; it passes the horizontal-line test. Only one-to-one functions have inverse functions.

$f(x) + k$ shifts the graph up by $k$ (down if $k < 0$). Outputs change; inputs do not.

$f(x - h)$ shifts the graph right by $h$ (left if $h < 0$). The shift is opposite the sign inside the function.

$a \cdot f(x)$ stretches the graph vertically by factor $|a|$; if $a < 0$ it also reflects across the x-axis.

$f(bx)$ compresses the graph horizontally by factor $\frac{1}{|b|}$; if $b < 0$ it also reflects across the y-axis.

$-f(x)$ flips the graph upside down — every output changes sign.

$f(-x)$ flips the graph left-right — inputs change sign while outputs stay.

A point where a polynomial changes from increasing to decreasing or vice versa. A degree-$n$ polynomial has at most $n - 1$ turning points.

The graph touches the x-axis and turns back (bounces) without crossing, because the factor's sign does not change there.

The graph crosses the x-axis; multiplicity 1 crosses straight, multiplicity 3 flattens (inflects) as it crosses.

A line through two points on a graph, $(a, f(a))$ and $(b, f(b))$. Its slope is the average rate of change over $[a, b]$.

Exclude any input that makes the denominator zero. These excluded inputs become vertical asymptotes or holes.

Use the zeros to split the number line into intervals; test one point per interval to learn where the function is positive or negative.

A number $a + bi$ where $i = \sqrt{-1}$. $a$ is the real part; $b$ is the imaginary part.

$i = \sqrt{-1}$, so $i^2 = -1$. Powers cycle: $i, -1, -i, 1$.

Every polynomial of degree $n \geq 1$ has exactly $n$ complex zeros, counted with multiplicity.

A regression polynomial chosen to model data; AP Precalculus uses technology to fit and interpret these models.

On a polynomial, equal-length input intervals produce changing rates of change — unlike a line, whose rate of change is constant.

Intercepts come from the numerator's zeros; vertical asymptotes and holes come from the denominator's zeros (holes when a factor cancels).

Only linear functions have a constant average rate of change over every interval; this constant slope is what makes their graph a straight line.

$f(x) = ax^2 + bx + c$; the axis of symmetry is $x = -\dfrac{b}{2a}$, and the vertex lies on it.

A shortcut for dividing a polynomial by a linear factor $(x - a)$; the last entry is the remainder, equal to $p(a)$.

Polynomials run off to $\pm\infty$; rational functions approach a horizontal or slant asymptote (or $\infty$) depending on relative degrees.

Additive transformations (translations) move a graph; multiplicative transformations (dilations) scale it. Inside the function affects x; outside affects y.

A polynomial of degree $n$ has at most $n$ real zeros; the rest (if any) are non-real complex conjugate pairs.

Test symmetry algebraically: compute $f(-x)$. Equal to $f(x)$ means even; equal to $-f(x)$ means odd; neither means no symmetry.

Solve by finding zeros of numerator and denominator, marking them on a number line, and testing the sign of $\dfrac{p(x)}{q(x)}$ in each interval.

$f(x) = \dfrac{1}{x}$: vertical asymptote at $x = 0$, horizontal asymptote $y = 0$, odd, and decreasing on each side.

From $a(x - h)^2 + k$: vertex $(h, k)$, axis of symmetry $x = h$, and minimum/maximum value $k$.

Knowing all the zeros (with multiplicities) lets you write the polynomial in factored form $a(x - r_1)^{m_1}(x - r_2)^{m_2}\cdots$.

Graphs reveal approximate real zeros and turning points; the rational-zeros and factor theorems give exact zeros analytically.

Local behavior describes a graph near a point (a zero, a turning point); global (end) behavior describes it as $x \to \pm\infty$.

A function defined by different rules on different parts of its domain; evaluate by picking the rule whose interval contains the input.

$f(x) = a \cdot b^x$ with base $b > 0,\ b \neq 1$. The output is multiplied by $b$ for each unit increase in $x$.

Growth factor is the base $b$; growth rate is $b - 1$ as a percent. A factor of $1.5$ is a $50\%$ growth rate; $0.8$ is a $20\%$ decay.

If $b > 1$ the function grows; if $0 < b < 1$ it decays. $a$ sets the initial value at $x = 0$.

Over equal-length input intervals, outputs change by equal RATIOS (a constant factor) — unlike linear functions, which change by equal differences.

$f(x) = a \cdot b^x$ has horizontal asymptote $y = 0$; adding $k$ shifts it to $y = k$.

Euler's number, $e \approx 2.718$. The base of the natural exponential $e^x$ and natural logarithm $\ln x$; it arises in continuous growth.

$\log_b y = x$ means $b^x = y$. A logarithm answers: to what power must I raise $b$ to get $y$?

$\log_b(b^x) = x$ and $b^{\log_b x} = x$. The graphs of $b^x$ and $\log_b x$ are reflections across $y = x$.

$\ln x = \log_e x$, the logarithm base $e$. $\ln e = 1$ and $\ln 1 = 0$.

$\log x = \log_{10} x$, the logarithm base 10. $\log 100 = 2$.

$\log_b(MN) = \log_b M + \log_b N$. A log of a product becomes a sum of logs.

$\log_b\!\left(\dfrac{M}{N}\right) = \log_b M - \log_b N$. A log of a quotient becomes a difference of logs.

$\log_b(M^k) = k \log_b M$. An exponent inside a log comes out as a coefficient.

$\log_b x = \dfrac{\log x}{\log b} = \dfrac{\ln x}{\ln b}$. Lets you evaluate any base on a calculator.

Take a logarithm of both sides, then use the power rule to drop the exponent. For $b^x = c$, $x = \log_b c = \dfrac{\ln c}{\ln b}$.

Rewrite in exponential form $b^x = y$, or combine logs into one and exponentiate. Always check for extraneous solutions (arguments must be positive).

$\log_b x$ is defined only for $x > 0$. The line $x = 0$ (the y-axis) is a vertical asymptote.

A scale where equal distances represent equal ratios (powers of the base), used for data spanning many orders of magnitude — e.g. the Richter and pH scales.

Graphing $\log y$ versus $x$ turns an exponential model into a straight line, making the growth factor easy to estimate.

$A = P\left(1 + \dfrac{r}{n}\right)^{nt}$: principal $P$, rate $r$, $n$ compoundings per year, time $t$.

$A = P e^{rt}$ models continuously compounded growth or decay; $r > 0$ grows, $r < 0$ decays.

The time for a decaying quantity to fall to half its value. With base $\frac{1}{2}$: $A = A_0\left(\dfrac{1}{2}\right)^{t/h}$, where $h$ is the half-life.

Linear adds a constant amount each step; exponential multiplies by a constant factor. Exponential eventually outpaces any linear model.

If $y = a b^x$, solving for $x$ gives a logarithm: $x = \log_b\!\left(\dfrac{y}{a}\right)$. Logs recover the input from the output.

$\log_b 1 = 0$ (since $b^0 = 1$) and $\log_b b = 1$ (since $b^1 = b$) for any valid base.

Fitting data to $y = a b^x$ with technology; useful when successive outputs share a roughly constant ratio.

$f(x) = a \cdot b^{x - h} + k$: $a$ stretches/reflects, $h$ shifts horizontally, $k$ shifts the horizontal asymptote to $y = k$.

The exponent is 'trapped' in $b^x$; taking a log of both sides and using $\log_b(b^x) = x$ frees it so you can solve linearly.

An increasing exponential ($b > 1$) is always concave up: its rate of change keeps increasing.

An increasing log ($b > 1$) is always concave down: it rises but its rate of change keeps decreasing.

The time for a growing quantity to double. For continuous growth $P e^{rt}$, the doubling time is $\dfrac{\ln 2}{r}$.

Two data points determine $a$ and $b$: divide the outputs to get the ratio (relate to $b$), then back-solve for the initial value $a$.

If the base is $b$ with $0 < b < 1$, the decay rate is $1 - b$ as a percent. A base of $0.85$ decays at $15\%$ per step.

$b^{-n} = \dfrac{1}{b^n}$. A negative exponent gives the reciprocal of the positive-exponent power.

$b^{m/n} = \sqrt[n]{b^m} = \left(\sqrt[n]{b}\right)^m$. Fractional exponents are roots.

$b^m \cdot b^n = b^{m+n}$. Multiplying like bases adds exponents.

$\dfrac{b^m}{b^n} = b^{m-n}$. Dividing like bases subtracts exponents.

$(b^m)^n = b^{mn}$. Raising a power to a power multiplies the exponents.

Undefined — the argument of a logarithm must be positive. This is why log equations require checking for extraneous solutions.

Use exponential models for processes that grow or shrink by a fixed percent: populations, radioactive decay, drug clearance, compound interest.

Decay with base $b$ is growth with base $\frac{1}{b}$ run backward; reflecting an exponential across the y-axis swaps growth and decay.

As the number of compounding periods $n \to \infty$, $\left(1 + \frac{r}{n}\right)^{n} \to e^{r}$; $e$ captures the limit of continuous compounding.

Use product, quotient, and power rules to condense several logs into one ($\log_b$ of a single expression) before solving, then exponentiate.

Outputs of an exponential over a unit interval differ by the factor $b$; over an interval of length $t$, by $b^t$.

Passes through $(1, 0)$, increasing and concave down, with a vertical asymptote at $x = 0$ and no horizontal asymptote.

Fitting data to $y = a + b\ln x$; appropriate when a quantity rises quickly then levels off.

Isolate the exponential or log, then apply its inverse (log to undo an exponential, exponentiation to undo a log) and check the domain.

An exponential model $T = T_s + (T_0 - T_s)e^{-kt}$: temperature approaches the surroundings $T_s$ exponentially over time.

In $y = a b^x$: $a$ is the value when $x = 0$ (initial amount) and $b$ is the per-unit growth/decay factor — read meaning, not just numbers.

In a power function $x^p$ the base varies and the exponent is fixed; in an exponential $b^x$ the base is fixed and the exponent varies.

An angle measured by the arc length it subtends on a unit circle. A full circle is $2\pi$ radians $= 360^\circ$.

Multiply degrees by $\dfrac{\pi}{180}$ to get radians; multiply radians by $\dfrac{180}{\pi}$ to get degrees.

For an angle $\theta$, the point on the unit circle is $(\cos\theta, \sin\theta)$: cosine is the x-coordinate, sine the y-coordinate.

$\tan\theta = \dfrac{\sin\theta}{\cos\theta}$, the slope of the terminal ray; undefined where $\cos\theta = 0$.

Right-triangle ratios: $\sin = \dfrac{\text{opp}}{\text{hyp}},\ \cos = \dfrac{\text{adj}}{\text{hyp}},\ \tan = \dfrac{\text{opp}}{\text{adj}}$.

$\sin^2\theta + \cos^2\theta = 1$ for every angle $\theta$; it follows directly from the unit circle.

$\csc\theta = \dfrac{1}{\sin\theta},\ \sec\theta = \dfrac{1}{\cos\theta},\ \cot\theta = \dfrac{1}{\tan\theta}$.

Both $\sin$ and $\cos$ repeat every $2\pi$ radians. $\tan$ repeats every $\pi$.

For $y = a\sin(b(x - c)) + d$, the amplitude is $|a|$ — half the distance between the maximum and minimum.

For $y = a\sin(bx)$, the period is $\dfrac{2\pi}{|b|}$; a larger $|b|$ compresses the wave.

The horizontal line $y = d$ about which the wave oscillates; $d$ is the vertical shift of $y = a\sin(b(x - c)) + d$.

The horizontal shift $c$ in $y = a\sin(b(x - c)) + d$; it moves the wave left or right without changing its shape.

Cosine is a sine shifted left by $\frac{\pi}{2}$: $\cos\theta = \sin\!\left(\theta + \dfrac{\pi}{2}\right)$.

Complementary angles satisfy $\sin\theta = \cos\!\left(\dfrac{\pi}{2} - \theta\right)$ and likewise for the other cofunction pairs.

Cosine is even: $\cos(-\theta) = \cos\theta$. Sine and tangent are odd: $\sin(-\theta) = -\sin\theta$, $\tan(-\theta) = -\tan\theta$.

$\sin 30^\circ = \frac{1}{2}$, $\sin 45^\circ = \dfrac{\sqrt{2}}{2}$, $\sin 60^\circ = \dfrac{\sqrt{3}}{2}$; cosines are the reverse.

The acute angle between the terminal ray and the x-axis. Trig values at any angle equal those of its reference angle, up to sign by quadrant.

All trig positive in QI; only Sine in QII; only Tangent in QIII; only Cosine in QIV ('All Students Take Calculus').

$\sin^{-1} x$ returns the angle in $\left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right]$ whose sine is $x$; the restricted range makes it a function.

$\cos^{-1} x$ returns the angle in $[0, \pi]$ whose cosine is $x$.

$\tan^{-1} x$ returns the angle in $\left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)$ whose tangent is $x$.

Find the reference solution, then use periodicity to write all solutions, e.g. $\theta = \theta_0 + 2\pi k$ for sine/cosine and $+\pi k$ for tangent.

Has vertical asymptotes where $\cos\theta = 0$ (at $\frac{\pi}{2} + \pi k$), period $\pi$, and passes through the origin, increasing on each branch.

$\dfrac{\sin A}{a} = \dfrac{\sin B}{b} = \dfrac{\sin C}{c}$; used to solve triangles given two angles and a side, or two sides and a non-included angle.

$c^2 = a^2 + b^2 - 2ab\cos C$; used when you know two sides and the included angle, or all three sides.

A point given as $(r, \theta)$: $r$ is the distance from the pole (origin) and $\theta$ the angle from the polar axis.

$x = r\cos\theta$ and $y = r\sin\theta$. These project the polar point onto the x- and y-axes.

$r = \sqrt{x^2 + y^2}$ and $\theta = \tan^{-1}\!\left(\dfrac{y}{x}\right)$, choosing $\theta$ by the quadrant of $(x, y)$.

A relationship $r = f(\theta)$ giving radius as a function of angle; tracing $\theta$ sweeps out curves like circles, roses, and spirals.

$r = a$ is a circle of radius $a$ centered at the pole; $r = a\cos\theta$ or $r = a\sin\theta$ gives a circle through the pole.

$r = a\cos(n\theta)$ or $a\sin(n\theta)$ traces a rose with $n$ petals if $n$ is odd, or $2n$ petals if $n$ is even.

$r = a \pm b\cos\theta$ (or with $\sin$); the ratio $\frac{a}{b}$ decides whether it has an inner loop, a dimple, or is convex.

Where $r$ is increasing, the curve moves away from the pole; where $r$ is decreasing, it moves toward the pole.

$y = a\sin(b(x - c)) + d$ models periodic real-world data — tides, daylight hours, sound — using amplitude, period, phase shift, and midline.

Frequency is the number of cycles per unit; it is the reciprocal of the period. A short period means a high frequency.

Angles $0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}$ land on the axes; their sine/cosine are 0 or $\pm 1$, making them anchor points of the unit circle.

On a circle of radius $r$, an angle of $\theta$ radians subtends arc length $s = r\theta$.

Angles that share the same terminal ray, differing by full rotations: $\theta + 2\pi k$. They have identical trig values.

Trig functions are not one-to-one, so their domains are restricted before inverting; this gives each inverse a single principal output value.

Build $a$ from half the max-minus-min, $d$ from the average of max and min, $b$ from $\frac{2\pi}{\text{period}}$, and $c$ from a known peak.

Transform one side using Pythagorean, reciprocal, quotient, or even/odd identities until it matches the other side; never move terms across an unproven equals sign.

$\sec x$ and $\csc x$ have vertical asymptotes where their reciprocal functions are zero, and U-shaped branches reaching out to $\pm\infty$.

Replace $\theta$ with $-\theta$ (symmetry about the polar axis), $\pi - \theta$ (about the line $\theta = \frac{\pi}{2}$), or $r$ with $-r$ (about the pole).

Periodic phenomena are sinusoidal: amplitude is the swing from average, period is the cycle length, and the midline is the average value.

The angle above (elevation) or below (depression) horizontal; right-triangle trig solves the unknown side or height.

On the unit circle, $\cos\theta$ reaches its maximum a quarter-turn before $\sin\theta$ does, so cosine's graph is sine shifted left by $\frac{\pi}{2}$.

Find all unknown sides and angles using the Law of Sines, the Law of Cosines, or right-triangle ratios, depending on what is given.

Drop a perpendicular to the x-axis to form a right triangle with the reference angle; the quadrant sets the signs of the coordinates.

For $y = \tan(bx)$, the period is $\dfrac{\pi}{|b|}$; tangent's base period is $\pi$, not $2\pi$.

For $y = a\sin(b(x-c)) + d$, the maximum is $d + |a|$ and the minimum is $d - |a|$.

A pair $x = f(t),\ y = g(t)$ tracing a curve as the parameter $t$ varies. Unit 4 — taught in the course but not tested on the AP Exam.

Solve one parametric equation for $t$ and substitute into the other to get a rectangular relation $y$ in terms of $x$. Unit 4 — not AP-Exam assessed.

As $t$ increases, a parametric curve is traced in a specific direction (orientation), shown with arrows. Unit 4 — not AP-Exam assessed.

A relation like $x^2 + y^2 = 25$ that defines $y$ in terms of $x$ without solving for $y$. Unit 4 — taught but not assessed.

A curve from slicing a cone: circle, ellipse, parabola, or hyperbola, each with a standard equation. Unit 4 — not AP-Exam assessed.

$\dfrac{(x-h)^2}{a^2} + \dfrac{(y-k)^2}{b^2} = 1$, centered at $(h, k)$. Unit 4 — taught but not tested on the AP Exam.

$\dfrac{(x-h)^2}{a^2} - \dfrac{(y-k)^2}{b^2} = 1$, with two branches and asymptotes. Unit 4 — not AP-Exam assessed.

A quantity with both magnitude and direction, written $\langle a, b \rangle$ or with an arrow. Unit 4 — taught but not on the AP Exam.

The length $\|\langle a, b \rangle\| = \sqrt{a^2 + b^2}$. Unit 4 — taught but not AP-Exam assessed.

Add component-wise: $\langle a, b \rangle + \langle c, d \rangle = \langle a + c,\ b + d \rangle$ (the tip-to-tail rule). Unit 4 — not assessed.

$k\langle a, b \rangle = \langle ka, kb \rangle$: scales length by $|k|$ and reverses direction if $k < 0$. Unit 4 — not assessed.

A vector of magnitude 1; divide any vector by its magnitude to get the unit vector in its direction. Unit 4 — not AP-Exam assessed.

$\langle a, b \rangle \cdot \langle c, d \rangle = ac + bd$; zero means the vectors are perpendicular. Unit 4 — taught but not assessed.

A rectangular array of numbers; AP Precalculus uses matrices to represent and transform data and to model systems. Unit 4 — not AP-Exam assessed.

Combine rows of the first matrix with columns of the second by dot products; order matters and dimensions must match. Unit 4 — not assessed.

A square matrix with 1s on the diagonal and 0s elsewhere; multiplying by it leaves a matrix unchanged. Unit 4 — taught but not assessed.

A matrix $A^{-1}$ with $A A^{-1} = I$; used to solve linear systems and reverse transformations. Unit 4 — not AP-Exam assessed.

$\det\!\begin{bmatrix} a & b \\ c & d \end{bmatrix} = ad - bc$; a zero determinant means no inverse. Unit 4 — not assessed.

Multiplying a vector by a matrix rotates, scales, reflects, or shears it in the plane. Unit 4 — taught but not on the AP Exam.

A mapping represented by a matrix that preserves lines and the origin; built from scaling, rotation, reflection, and shear. Unit 4 — not assessed.

$x = r\cos t,\ y = r\sin t$ traces a circle of radius $r$ as $t$ runs over $[0, 2\pi)$. Unit 4 — not AP-Exam assessed.

Horizontal and vertical parts: a vector of magnitude $v$ at angle $\theta$ has components $\langle v\cos\theta,\ v\sin\theta \rangle$. Unit 4 — not assessed.

A position vector locates a moving object; a velocity vector gives its speed and direction of motion. Unit 4 — taught but not assessed.

Write $A\mathbf{x} = \mathbf{b}$ and solve with the inverse: $\mathbf{x} = A^{-1}\mathbf{b}$ (when $A^{-1}$ exists). Unit 4 — not assessed.

The set of points equidistant from a focus and a directrix; opens around its axis of symmetry. Unit 4 — taught but not AP-Exam assessed.

$(x - h)^2 + (y - k)^2 = r^2$: all points a fixed distance $r$ from center $(h, k)$. Unit 4 — not assessed.

Unit 4 (parametric, vectors, matrices, conics, implicit relations) is part of the course framework but, per College Board, is NOT assessed on the AP Precalculus Exam.

The AP Precalculus Exam assesses Units 1–3 only — polynomial/rational, exponential/logarithmic, and trigonometric/polar functions. Unit 4 is taught locally, not tested.

The same shift/scale/reflect rules apply to parametric and matrix-defined transformations, connecting Unit 4 back to Units 1–3. Unit 4 — not assessed.

Repeated matrix multiplication models step-by-step change (e.g. transition models), iterating a state vector. Unit 4 — taught but not AP-Exam assessed.