AP Precalculus Quick Review

A function's type is revealed by how its rate of change behaves: constant, steadily changing, or scaling geometrically point to linear, polynomial, or exponential behavior.

Average rate of change on $[a,b]$ is $/f(b)-f(a)b-a$. A linear function has a constant average rate of change on every interval. A degree-$n$ polynomial has constant $n$-th differences over equally spaced $x$-values (and non-constant differences of any lower order).

An increasing function (output values increasing) vs.\ an increasing rate of change (concave up, output increasing faster and faster) --- these are independent properties.

Computing $/f(a)-f(b)a-b$ inconsistently or flipping the order of subtraction between numerator and denominator, producing a sign error.

Constant 1st differences $arrow$ linear; constant 2nd differences $arrow$ quadratic; constant ratios $arrow$ exponential.

The leading term of a polynomial controls its behavior as $x → ±∞$.

For $f(x)=a_nx^n+·s+a_0$: the sign of $a_n$ and the parity of $n$ determine end behavior (4 cases). A degree-$n$ polynomial has at most $n$ real zeros (counting multiplicity) and at most $n-1$ turning points.

Degree of a polynomial vs.\ its number of terms; a zero (x-intercept) vs.\ a turning point (local max/min).

Assuming a degree-4 polynomial must cross the x-axis 4 times --- multiplicity and non-real zeros reduce the number of visible x-intercepts.

Even degree $arrow$ same-direction ends; odd degree $arrow$ opposite-direction ends; sign of $a_n$ flips which way they point.

The multiplicity of a zero determines whether the graph crosses or bounces off the x-axis there.

Non-real zeros of a real-coefficient polynomial occur in conjugate pairs $a± bi$. For $f(x)=a(x-z_1)^m_1(x-z_2)^m_2·s$ with $ m_i=n$: odd multiplicity $arrow$ graph crosses the axis; even multiplicity $arrow$ graph is tangent to the axis.

A repeated real zero (multiplicity $>1$, one crossing/bounce point) vs.\ two distinct simple zeros (two separate crossings).

Forgetting non-real zeros still count toward the total degree --- a cubic with one real zero has two complex-conjugate zeros, not two ``missing'' zeros.

Odd multiplicity crosses; even multiplicity bounces; complex zeros always travel in conjugate pairs.

Comparing the degrees of the numerator and denominator predicts the horizontal or slant asymptote.

For $f(x)=/p(x)q(x)$ with $ p = m$, $ q = n$: if $m<n$, horizontal asymptote $y=0$; if $m=n$, horizontal asymptote $y=/leading coeff.\ of pleading coeff.\ of q$; if $m=n+1$, a slant asymptote exists (found by polynomial division); if $m>n+1$, no horizontal or slant asymptote.

Horizontal asymptote (end behavior as $x→±∞$) vs.\ vertical asymptote (behavior near an excluded $x$-value).

Dividing leading coefficients when the degrees are not equal, or skipping the polynomial long division needed to find a slant asymptote.

Bottom-heavy $arrow y=0$; equal degrees $arrow$ ratio of leading coefficients; top-heavy by exactly 1 $arrow$ slant asymptote.

Zeros come from the numerator and vertical asymptotes come from the denominator --- but only after common factors are removed.

For $f(x)=/p(x)q(x)$ in lowest terms: zeros occur where $p(x)=0$ (and $q(x)0$); vertical asymptotes occur where $q(x)=0$ (and that factor did not also cancel with $p(x)$).

A denominator zero that is also a numerator zero (hole, not asymptote) vs.\ a denominator zero with no matching numerator factor (true vertical asymptote).

Listing every zero of the original (unsimplified) denominator as a vertical asymptote without first checking for cancellation with the numerator.

Numerator zero (uncancelled) $arrow$ x-intercept; denominator zero (uncancelled) $arrow$ vertical asymptote.

A factor that cancels between numerator and denominator creates a removable discontinuity, not an asymptote.

If $f(x)=/(x-c)g(x)(x-c)h(x)$ with $g(c),h(c)0$, then $f$ has a hole at $x=c$ located at $≤ft(c,\ /g(c)h(c))$, found by evaluating the reduced (simplified) function at $x=c$.

A hole (function undefined at exactly one point, graph otherwise continuous) vs.\ a vertical asymptote (function grows without bound near that $x$-value).

Forgetting that the original domain restriction still applies after canceling --- the simplified expression is equivalent to the original everywhere except at the hole.

Same factor top and bottom $arrow$ hole; plug $c$ into the reduced form for the hole's $y$-coordinate.

Factored form, standard form, and divided form of the same function each expose different features.

Factored form $a(x-z_1)(x-z_2)·s$ reveals zeros directly. Standard form $a_nx^n+·s+a_0$ reveals the $y$-intercept ($a_0$) and end behavior (leading term). Rewriting $/p(x)q(x)$ via long division as $quotient+/remainderq(x)$ exposes a slant asymptote.

An equivalent algebraic form of the same function vs.\ a transformation, which produces a genuinely different (shifted/stretched) function.

Sign errors while expanding a factored form or performing polynomial long division, especially with a negative leading coefficient.

Factored $arrow$ zeros; standard $arrow$ end behavior/$y$-intercept; divided $arrow$ slant asymptote.

Every transformation of a function $f$ can be tracked through one template: $g(x)=a f(b(x-h))+k$.

$a$: vertical stretch/compress, reflect over $x$-axis if $a<0$. $k$: vertical shift. $h$: horizontal shift. $b$: horizontal stretch/compress by factor $/1|b|$, reflect over $y$-axis if $b<0$.

Horizontal transformations act opposite to how they ``look'' in the equation --- $f(x-h)$ shifts right by $h$, and $f(bx)$ compresses horizontally when $b>1$ (not stretches).

Applying the horizontal stretch/shift in the same direction as the vertical one, or using the wrong sign for $h$ when reading a shift off the equation.

$a,k$ control up/down and tall/short; $h,b$ control opposite-direction left/right and squish/stretch.

Matching a numerical or contextual pattern to the right function family is a core exam skill for both MCQ and FRQ.

Constant first differences $arrow$ linear; constant $n$-th differences $arrow$ degree-$n$ polynomial; constant ratios (constant percent change) $arrow$ exponential $f(x)=a b^x$. A model must be validated against ALL given data, not just two points.

A model that merely passes through two given points vs.\ a model justified by the pattern of differences/ratios across the entire data set.

Choosing an exponential model from only the first and last data points without confirming consecutive ratios are actually constant throughout the table.

Check differences and ratios first $arrow$ let the pattern pick the function family.

Sequences are the discrete cousins of linear and exponential functions.

Arithmetic sequence: $a_n=a_1+(n-1)d$, constant common difference $d$ (discrete linear). Geometric sequence: $a_n=a_1r^ n-1$, constant common ratio $r$ (discrete exponential).

Common difference $d$ (added each step, arithmetic) vs.\ common ratio $r$ (multiplied each step, geometric).

Writing the exponent in a geometric sequence as $n$ instead of $n-1$, which shifts every term's index by one.

Add a constant $arrow$ arithmetic/linear; multiply by a constant $arrow$ geometric/exponential.

Exponential functions change by a constant percentage over equal-length intervals.

$f(x)=a· b^x$, $a0$, $b>0$, $b1$. Domain: all reals. Range: $y>0$ if $a>0$. Horizontal asymptote $y=0$. Growth if $b>1$; decay if $0<b<1$. Over any interval of length $k$, output is multiplied by $b^k$.

The percent growth/decay rate $r$ vs.\ the base $b$ itself --- a 5% growth rate means $b=1.05$, not $b=5$ or $b=0.05$.

Forgetting to add 1 to (or subtract from) a percent rate before using it as the base $b$.

Percent change $r$ $arrow$ base $b=1+r$ (growth) or $b=1-r$ (decay).

Exponent rules let you rewrite exponential expressions to compare or combine growth rates.

$b^x+y=b^xb^y$; $b^x-y=/b^xb^y$; $(b^x)^y=b^xy$; $b^-x=/1b^x$. Any base can be rewritten in base $e$: $b^x=e^x b$, with continuous growth rate $k= b$.

$(b^x)^y=b^xy$ vs.\ $b^(x^y)$ --- these are not the same expression.

Treating $(a+b)^x$ as $a^x+b^x$, or applying exponent rules across a sum instead of a product/quotient/power of the same base.

Exponent rules only combine products, quotients, and powers of the SAME base --- never sums.

Data with roughly constant ratios over equally spaced inputs is modeled by exponential regression.

Test for exponential fit by computing consecutive ratios $/y_i+1y_i$ for equally spaced $x_i$ --- near-constant ratios support an exponential model. A semi-log plot ($ y$ vs.\ $x$) of exponential data looks linear.

A model that fits the data well overall vs.\ a model that merely passes through two of the given points.

Applying the constant-ratio test to data whose $x$-values are NOT equally spaced, which invalidates the test.

Constant ratios over equal steps $arrow$ exponential; log of the data looks linear $arrow$ confirms it.

Composition chains the output of one function into the input of another.

$(f g)(x)=f(g(x))$. Domain of $f g$ is the set of $x$ in the domain of $g$ for which $g(x)$ is in the domain of $f$.

$(f g)(x)=f(g(x))$ vs.\ $(g f)(x)=g(f(x))$ --- composition is generally not commutative.

Evaluating $f(g(x))$ by multiplying $f(x)· g(x)$ instead of substituting $g(x)$ as the input to $f$.

Composition = plug the inside function's output into the outside function.

An inverse function undoes another function, swapping inputs and outputs.

$f^-1(f(x))=x$ and $f(f^-1(x))=x$ on appropriate domains. The graph of $f^-1$ is the reflection of $f$ over $y=x$. $f$ has an inverse function iff $f$ is one-to-one (passes the horizontal line test), possibly after restricting the domain.

$f^-1(x)$ (inverse function) vs.\ $/1f(x)=[f(x)]^-1$ (reciprocal) --- different objects unless the problem says otherwise.

Writing $f^-1(x)$ as $/1f(x)$, or swapping $x$ and $y$ but forgetting to actually solve for the new $y$.

Swap $x≤ftrightarrow y$, solve for $y$; graph reflects over $y=x$.

A logarithm is the exponent that produces a given output --- the inverse operation of exponentiation.

$y=_b x b^y=x$, with $b>0$, $b1$, $x>0$. Domain $x>0$; range all reals; vertical asymptote $x=0$. $_b b=1$, $_b 1=0$. Natural log: $ x=_e x$.

Domain of a logarithm ($x>0$) vs.\ domain of the exponential it inverts (all reals) --- swapped because inverse functions exchange domain and range.

Trying to evaluate or graph $_b(x)$ for $x≤0$, or placing the vertical asymptote at $x=1$ instead of $x=0$.

Log undoes exponent; domain $x>0$; vertical asymptote at $x=0$, not $y=0$.

Log properties convert products/quotients/powers into sums/differences/coefficients, and back.

$_b(mn)=_bm+_bn$; $_b≤ft(/mn)=_bm-_bn$; $_b(m^p)=p_bm$; change of base $_ba=/ a b=/ a b$.

$_b(mn)=_bm+_bn$ (product splits) vs.\ $_b(m+n)$, which does NOT simplify to $_bm+_bn$.

``Simplifying'' $_b(m+n)$ as $_bm+_bn$, or dropping the base when changing bases.

Log of a product/quotient/power splits apart; log of a sum never splits.

Periodic behavior repeats at fixed intervals, and radians measure angles as arc length per radius.

Period = length of one complete repeating cycle. Radian measure $=/arc lengthradius$. Conversions: $radians=degrees×/180$, $degrees=radians×/180$; $2$ radians $=360^$.

Period (length of one cycle) vs.\ frequency (cycles per unit, $frequency=1/period$).

Leaving the calculator in the wrong angle mode (degrees vs.\ radians), producing answers off by a large, nonsensical factor.

$2$ rad $=360^$; always check the calculator's angle mode first.

Sine and cosine are defined as the $y$- and $x$-coordinates of a point on the unit circle at angle $$.

$=x$, $=y$, $=/yx=/$. Key exact values at $=0,/6,/4,/3,/2$. $$ is undefined where $=0$.

The reference angle (acute angle to the $x$-axis, gives magnitude) vs.\ the actual angle $$ (determines sign and quadrant).

Dropping the correct sign of sine/cosine/tangent in Quadrants II, III, IV after finding only the reference-angle magnitude.

All Students Take Calculus: All/Sin/Tan/Cos positive in Quadrants I/II/III/IV respectively (ASTC).

The graphs of sine and cosine are smooth periodic waves oscillating about a midline.

$y= x$ and $y= x$: period $2$, amplitude 1, midline $y=0$, domain all reals, range $[-1,1]$. Identity: $ x=≤ft(x+/2)$.

Amplitude (half the vertical distance between max and min) vs.\ the maximum value itself (max $=$ amplitude $+$ midline).

Reading amplitude directly off the maximum $y$-value when the midline is not $y=0$, instead of computing $/-2$.

Amplitude $=/-2$; midline $=/+2$.

Any periodic wave can be written as a transformed sine or cosine and matched to a real context.

$f(x)=a(b(x-c))+d$: $|a|=$ amplitude, period $=/2|b|$, $c=$ phase shift, $d=$ midline. To build a model from data: $d=/+2$, $a=/-2$, period from consecutive maxima, then $c$ from a known max/min/zero.

Period $/2|b|$ vs.\ the coefficient $b$ itself --- a ``faster'' wave has a LARGER $b$ but a SMALLER period.

Solving for $b$ by setting $b=$ period instead of period $=/2b$, or misplacing the sign of the phase shift (the template uses $x-c$).

$b=/2period$; midline $=$ average of max/min; amplitude $=$ half their difference.

The remaining trig functions are ratios or reciprocals of sine and cosine, each with its own period and asymptotes.

$=/$ (period $$, asymptotes where $=0$); $=/=/1$; $=/1$; $=/1$. Secant/cosecant are undefined exactly where cosine/sine equal zero.

$=/1$ (reciprocal of cosine) vs.\ $^-1$ (inverse cosine function) --- the ``$-1$'' notation means two different things.

Writing $=^-1$, or assigning tangent/cotangent asymptote locations to secant/cosecant.

Tan/cot period $$; sec/csc period $2$; a reciprocal function's asymptotes sit exactly at its parent's zeros.

Inverse trig functions require restricted domains on the originals to be genuine, one-to-one functions.

$ x=^-1x$: range $≤ft[-/2,/2]$. $ x=^-1x$: range $[0,]$. $ x=^-1x$: range $≤ft(-/2,/2)$.

$^-1x$ (inverse sine, restricted range) vs.\ $/1 x= x$ (reciprocal) --- same superscript notation, different meaning.

Reporting an inverse-trig answer outside its restricted range (e.g., a Quadrant III angle for $$, whose range never leaves Quadrants I/IV).

Arcsin/arctan output in $≤ft[-/2,/2]$; arccos outputs in $[0,]$.

Because trig functions are periodic, equations generally have infinitely many solutions described by a general formula.

Solve $=k$ (or cos/tan) by finding all reference-angle solutions in $[0,2)$, then add $2 n$ (or $ n$ for tangent) for the general solution. Inequalities: find boundary solutions, then test the sign on each sub-interval of one period.

Solutions restricted to a given interval (finite list) vs.\ the general solution (infinite family, $+2 n$).

Giving only one solution per period when two exist (e.g., $=k$ typically has two solutions per $2$ cycle).

Sin/cos equations usually give 2 solutions per period; don't forget $+2 n$ for ``all solutions.''

Identities let you rewrite trig expressions into equivalent forms to simplify, solve, or match a target expression.

Pythagorean identity: $^2+^2=1$, giving $^2+1=^2$ and $1+^2=^2$. Double-angle: $2=2$, $2=^2-^2$. Even/odd: $(-)=$, $(-)=-$.

$^2$, meaning $()^2$, vs.\ $(^2)$ --- exponent placement changes the operation entirely.

Assuming $(A+B)= A+ B$ (false) instead of applying the actual sum identity.

$^2+^2=1$ is the master identity everything else is built from.

Polar coordinates locate points by distance and direction from the origin instead of horizontal/vertical displacement.

Conversions: $x=r$, $y=r$, $r^2=x^2+y^2$, $=/yx$. Common curves: circle $r=a$; rose $r=a(n)$ or $a(n)$; cardioid/lima con $r=a± b$ (or $$).

A negative $r$ (point plotted opposite the terminal ray, effectively adding $$ to $$) vs.\ a negative $$ (angle measured clockwise) --- both valid, but they move the point differently.

Plotting $r=a± b$ without checking whether $r$ goes negative for some $$, which reflects that point through the pole.

$x=r,\ y=r$; negative $r$ flips the point through the origin.

How $r$ changes with respect to $$ describes whether a polar curve is expanding, contracting, or looping back toward the pole.

Average rate of change of a polar function on $[_1,_2]$ is $/r(_2)-r(_1)_2-_1$. Increasing $r$ means the curve moves farther from the pole as $$ increases on that interval; decreasing $r$ means it moves closer.

The rate of change of $r$ (distance from the pole) vs.\ the rate of change of $x$ or $y$ (rectangular coordinates) --- generally different values at the same $$.

Interpreting a decreasing $r$ as the whole curve ``shrinking,'' when it may just be spiraling closer to the pole over one $$-interval.

Treat $r()$ like any function of $$ --- same average-rate-of-change formula, new variable name.

Parametric equations describe both coordinates of a moving point as separate functions of a shared parameter.

$x=x(t)$, $y=y(t)$ trace a path as $t$ varies. Eliminate the parameter by solving one equation for $t$ and substituting into the other to get a rectangular equation. Direction of motion is found by evaluating the path at increasing $t$-values.

The parametric curve's shape (rectangular path) vs.\ its motion (direction and speed), which a rectangular equation alone does not show.

Eliminating the parameter and losing information about direction or domain restriction --- a rectangular equation can trace the same curve without showing which part is covered or in which direction.

Two functions of one parameter $arrow$ one path; eliminate $t$ for the equation, keep $t$ for direction.

Not every curve is a function of $x$; conic sections are classic implicitly defined relations.

Circle: $x^2+y^2=r^2$. Ellipse: $/x^2a^2+/y^2b^2=1$. Hyperbola: $/x^2a^2-/y^2b^2=1$. These fail the vertical line test and are not functions of $x$ unless solved and restricted to one branch.

A conic section (relation, generally not a function) vs.\ a function graph --- most conics require splitting into two function pieces (upper/lower or left/right branch) to graph with a function-only tool.

Trying to enter a full circle or ellipse equation directly into a function grapher without solving for $y$ and splitting into $±$ branches.

Conics fail the vertical line test $arrow$ split into $±$ branches to graph as functions.

A vector encodes both magnitude and direction, and vector arithmetic combines these geometrically.

For $ v= v_x,v_y$: magnitude $\| v\|=sqrtv_x^2+v_y^2$, direction angle $=^-1≤ft(/v_yv_x)$ (quadrant-adjusted). Addition: $ a_1,a_2+ b_1,b_2= a_1+b_1,a_2+b_2$. Scalar multiplication: $k v_x,v_y= kv_x,kv_y$.

A vector (magnitude AND direction) vs.\ a scalar (magnitude only) --- adding vectors is not the same as adding their magnitudes unless they point the same direction.

Computing $\| v\|$ by adding the components instead of using the Pythagorean/distance formula, or finding the direction angle without checking the correct quadrant.

$\| v\|=sqrtv_x^2+v_y^2$; add vectors component-by-component, not magnitude-by-magnitude.

A vector-valued function pairs a parameter with a vector, modeling position or motion that changes over that parameter.

$ v(t)= x(t),y(t)$ traces the same path as the parametric curve $(x(t),y(t))$. Vector-valued functions add, scale, and combine component-wise just like constant vectors.

A vector-valued function $ v(t)$ (an infinite family of vectors, one per $t$) vs.\ a single fixed vector (one magnitude/direction).

Evaluating $ v(t_1)+ v(t_2)$ as if it were ordinary function composition, instead of adding the two resulting vectors component-wise.

Vector-valued function = parametric equations packaged as $ x(t),y(t)$.

Matrices organize numbers into rows and columns and can represent linear transformations of the plane.

Addition/subtraction is entrywise (same dimensions required). Scalar multiplication multiplies every entry. Matrix multiplication requires columns of the first matrix to equal rows of the second. For $pmatrixa&b c&d $, determinant $=ad-bc$. Rotation, reflection, and dilation matrices act on $pmatrixx y $ to transform the plane.

Matrix multiplication (generally NOT commutative, $AB BA$) vs.\ real-number multiplication (commutative).

Multiplying matrices in the wrong order, or attempting to multiply two matrices whose inner dimensions don't match.

Rows $×$ columns must match to multiply; $2×2$ determinant $=ad-bc$.

Unlike some AP exams, the AP Precalculus Exam does not hand students a printed formula/reference sheet --- every rule below must be committed to memory.

This is the backbone skill behind the FRQ ``modeling a non-periodic context'' and ``modeling a periodic context'' prompts.

Knowing which skill each section rewards changes how you should approach the problem.