Your FREE AP Precalculus Flashcards 2026 – 200+ Cards

AP Precalculus Flashcards

Function

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

Average rate of change

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.

Increasing function

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 vs concave down

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

Point of inflection

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.

Polynomial function

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.

Degree of a polynomial

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

Leading coefficient

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

End behavior of a 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.

Zero of a function

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

Multiplicity of a zero

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

Real vs complex zeros

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

Complex conjugate

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

Local (relative) maximum

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

Global (absolute) extremum

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

Even function

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

Odd function

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

Rational function

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.

Vertical asymptote

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

Hole (removable discontinuity)

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 $x = 2$.

Horizontal asymptote (degree rules)

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.

Slant (oblique) asymptote

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.

x-intercept of a rational function

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

Polynomial long division

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

Remainder theorem

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

Factor theorem

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

Rational zeros (root) theorem

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.

Quadratic formula

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

Discriminant

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

Vertex form of a parabola

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

Power function

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.

Difference of squares

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

Sum/difference of cubes

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

Domain of a function

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

Range of a function

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

Composition of functions

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

Inverse function

$f^{-1}$ 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.

One-to-one function

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

Vertical translation

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

Horizontal translation

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

Vertical dilation (stretch)

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

Horizontal dilation

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

Reflection across the x-axis

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

Reflection across the y-axis

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

Turning point

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

Behavior near a zero of even multiplicity

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

Behavior near a zero of odd multiplicity

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

Secant line

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

Domain restriction (rational)

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

Sign analysis of a polynomial

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

Complex number

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

Imaginary unit i

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

Fundamental Theorem of Algebra

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

Polynomial of best fit

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

Why rates of change matter (polynomials)

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

Asymptote vs intercept (rational)

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

Constant rate of change

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

Quadratic standard form

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

Synthetic division

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

Polynomial vs rational end behavior

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

Coterminal idea for transformations

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

Maximum number of real zeros

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

Even/odd from the formula

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

Rational inequality

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.

Reciprocal function

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

Translated parabola features

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

Zeros and factors together

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

Graphical vs analytic root-finding

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

Local behavior vs global behavior

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

Piecewise function

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

Exponential function

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

Growth factor vs growth rate

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.

Exponential growth vs decay

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

Key feature of exponential functions

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

Horizontal asymptote of an exponential

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

The number e

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.

Logarithm definition

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

Logarithm and exponential are inverses

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

Natural logarithm

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

Common logarithm

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

Product rule for logarithms

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

Quotient rule for logarithms

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

Power rule for logarithms

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

Change of base formula

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

Solving an exponential equation

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

Solving a logarithmic equation

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

Domain of a logarithmic function

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

Logarithmic scale

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.

Semi-log plot linearizes exponentials

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

Compound interest formula

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

Continuous (exponential) growth

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

Half-life

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.

Exponential vs linear growth

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

Inverse of an exponential 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 and log_b b

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

Exponential regression

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

Transformation of an exponential

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

Why logs solve for exponents

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.

Concavity of exponential growth

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

Concavity of a logarithmic function

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

Doubling time

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

Exponential function from two points

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

Exponential decay rate

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.

Negative exponent

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

Rational (fractional) exponent

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

Product of powers rule

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

Quotient of powers rule

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

Power of a power rule

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

Logarithm of 0 or a negative

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

Modeling with exponentials

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

Inverse relationship of growth/decay

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

Why e for continuous growth

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.

Combining log rules

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

Exponential unit fraction

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

Graph of y = ln x

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

Logarithmic regression

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

Exponential and logarithmic equation strategy

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

Newton's law of cooling (form)

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

Interpreting model parameters

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.

Exponential vs power function

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.

Radian measure

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

Degree-radian conversion

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

Unit circle definition of sine and cosine

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

Tangent in terms of sine and cosine

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

SOH-CAH-TOA

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

Pythagorean identity

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

Reciprocal trig functions

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

Period of sine and cosine

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

Amplitude of a sinusoid

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

Period from the formula

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

Midline of a sinusoid

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

Phase shift

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

Sine vs cosine relationship

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

Cofunction identity

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

Even/odd trig identities

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

Special angle values (30, 45, 60)

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

Reference angle

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.

ASTC (quadrant signs)

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

Inverse sine (arcsin)

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

Inverse cosine (arccos)

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

Inverse tangent (arctan)

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

Solving a trig equation

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.

Graph of 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.

Law of Sines

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

Law of Cosines

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

Polar coordinates

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

Polar-to-rectangular conversion

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

Rectangular-to-polar conversion

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

Polar function

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

Polar graph: circle

$r = a$ 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.

Polar graph: rose curve

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

Polar graph: limaçon

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

Rate of change of r in a polar graph

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

Sinusoidal model

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

Frequency vs period

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

Quadrantal angles

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.

Arc length

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

Coterminal angles

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

Why restrict inverse-trig domains

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

Sinusoid from a description

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.

Trig identity (verifying)

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.

Secant and cosecant graphs

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

Polar symmetry tests

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).

Modeling daylight or tides

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

Angle of elevation/depression

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

Why cosine leads sine

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 $\frac{\pi}{2}$.

Solving a triangle

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

Reference triangle in any quadrant

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

Period of a transformed tangent

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

Sinusoid maximum and minimum

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

Parametric function (NOT assessed)

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.

Eliminating the parameter (NOT assessed)

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.

Direction and orientation (NOT assessed)

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

Implicitly defined function (NOT 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.

Conic section (NOT assessed)

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

Ellipse standard form (NOT 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.

Hyperbola standard form (NOT assessed)

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

Vector (NOT 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.

Vector magnitude (NOT assessed)

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

Vector addition (NOT 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.

Scalar multiple of a vector (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.

Unit vector (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.

Dot product (NOT assessed)

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

Matrix (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.

Matrix multiplication (NOT 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.

Identity matrix (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.

Inverse matrix (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.

Determinant of a 2×2 matrix (NOT assessed)

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

Matrix as a transformation (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.

Linear transformation (NOT assessed)

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

Parametrically defined circle (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.

Vector components (NOT 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.

Position vs velocity vectors (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.

System of equations as a matrix (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.

Parabola as a conic (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.

Circle as a conic (NOT assessed)

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

Why Unit 4 is not on the AP Exam

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.

What the AP Exam covers (units 1–3)

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

Function transformation across types (NOT assessed)

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.

Matrix model of state change (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.