Topic 1: Polynomial and Rational Functions

AP Precalculus opens with the family of polynomial and rational functions because they encode nearly every idea you will use later: rates of change, limiting behavior, symmetry, and inverses. This article walks through average rate of change, the anatomy of polynomials (degree, leading term, end behavior, zeros, and multiplicity), the role of complex zeros, the structure of rational functions with their asymptotes and holes, techniques for polynomial and rational inequalities, function transformations, even and odd symmetry, inverse functions, and finally regression and model selection. Throughout, the goal is to move fluidly between four representations: a formula, a graph, a table of values, and a verbal description.

Average Rate of Change

The average rate of change of a function $f$ over an interval $[a,b]$ is the slope of the secant line joining the points $(a,f(a))$ and $(b,f(b))$. It is computed as $\\frac{f(b)-f(a)}{b-a}$. This single number summarizes how the output changes per unit of input across the whole interval, even when the function curves in between. For a linear function the average rate of change is constant and equals the slope. For a quadratic such as $f(x)=x^{2}$, the average rate of change over $[1,3]$ is $\\frac{9-1}{3-1}=4$, while over $[1,2]$ it is $\\frac{4-1}{2-1}=3$, showing that the rate itself increases as $x$ grows. A useful conceptual rule: a function is increasing on an interval when its average rate of change there is positive, decreasing when negative. When the average rate of change is itself increasing, the graph is concave up; when it is decreasing, the graph is concave down. Recognizing these patterns from a table lets you classify behavior without ever seeing the graph.

Polynomial Degree, Leading Term, and End Behavior

A polynomial of degree $n$ has the form $f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\\cdots+a_{1}x+a_{0}$ with $a_{n}\\neq 0$. The leading term $a_{n}x^{n}$ alone governs end behavior, because as $\\lvert x\\rvert$ grows large the highest power dominates every other term. Two features decide the ends: the parity of the degree and the sign of the leading coefficient. If $n$ is even and $a_{n}\\gt 0$, both ends rise, so $f(x)\\to +\\infty$ as $x\\to \\pm\\infty$. If $n$ is even and $a_{n}\\lt 0$, both ends fall. If $n$ is odd and $a_{n}\\gt 0$, the left end falls and the right end rises; reversing the sign reverses both. The degree also caps the number of real zeros at $n$ and the number of turning points at $n-1$. Between consecutive distinct real zeros a continuous polynomial keeps a single sign, which is what makes sign analysis possible. A handy check on a graph: count turning points and add one to get a lower bound for the degree.

Zeros, Multiplicity, and Complex Zeros

A zero of $f$ is an input $x=c$ where $f(c)=0$, corresponding to a factor $(x-c)$. The multiplicity of a zero is the exponent of its factor. Multiplicity controls how the graph meets the $x$-axis: an odd multiplicity produces a sign change (the curve crosses), while an even multiplicity produces no sign change (the curve touches and turns back, tangent to the axis). For example, $f(x)=(x-2)^{2}(x+1)^{3}$ touches at $x=2$ and crosses, flattening, at $x=-1$. Higher multiplicity also flattens the graph near the zero. The Fundamental Theorem of Algebra guarantees that a degree-$n$ polynomial has exactly $n$ zeros in the complex numbers, counted with multiplicity. Non-real complex zeros of a real polynomial always occur in conjugate pairs: if $a+bi$ is a zero, so is $a-bi$. This is why a real polynomial of odd degree must have at least one real zero, and why a quadratic with negative discriminant, such as $x^{2}+1$ with zeros $\\pm i$, never crosses the real axis.

Rational Functions: Asymptotes and Holes

A rational function is a ratio $r(x)=\\frac{p(x)}{q(x)}$ of polynomials with $q(x)$ not identically zero. Its behavior is read from the factored numerator and denominator. A vertical asymptote occurs at $x=c$ when $(x-c)$ divides the denominator but not the numerator after full cancellation; near it the outputs blow up to $\\pm\\infty$. A hole (removable discontinuity) occurs at $x=c$ when $(x-c)$ is a common factor of numerator and denominator: the factor cancels, leaving a single missing point. End behavior is set by comparing degrees. If the numerator degree is less than the denominator degree, $y=0$ is a horizontal asymptote. If the degrees are equal, the horizontal asymptote is the ratio of leading coefficients. If the numerator degree is exactly one more than the denominator, there is a slant (oblique) asymptote found by polynomial division. For instance, $r(x)=\\frac{x^{2}-1}{x-3}$ has degrees differing by one, so dividing gives the slant asymptote $y=x+3$, and a vertical asymptote at $x=3$. By contrast $\\frac{(x-1)(x+2)}{(x-1)}$ has a hole at $x=1$, not an asymptote.

Polynomial and Rational Inequalities

To solve an inequality such as $f(x)\\gt 0$ or $\\frac{p(x)}{q(x)}\\le 0$, the strategy is a sign chart. First move everything to one side so you are comparing a single expression to zero, then fully factor it. The critical numbers are the real zeros of the numerator (where the expression can equal or pass through zero) and the real zeros of the denominator (where it is undefined). These critical numbers partition the number line into intervals, and within each interval the expression keeps one sign, so testing a single point per interval reveals the sign everywhere on it. Multiplicity matters: at a factor of odd multiplicity the sign flips, while at even multiplicity the sign stays the same. When choosing which intervals to include, remember that zeros of the numerator may be included for non-strict inequalities, but zeros of the denominator are never included because the function is undefined there. For example, solving $\\frac{x-1}{x+2}\\ge 0$ gives the solution $x\\lt -2$ and $x\\ge 1$, excluding $-2$ and including $1$.

Transformations, Symmetry, and Inverses

Any function can be reshaped by transformations applied to $g(x)=a\\,f(b(x-h))+k$. The constant $k$ shifts vertically and $h$ shifts horizontally (note the inside shift is opposite in sign to its appearance). The factor $a$ scales vertically and reflects across the $x$-axis when negative, while $b$ scales horizontally by $\\frac{1}{b}$ and reflects across the $y$-axis when negative. Order matters: horizontal changes happen inside the function, vertical changes outside. Symmetry is a special structural property. A function is even when $f(-x)=f(x)$, giving a graph symmetric about the $y$-axis (for example $f(x)=x^{2}$); it is odd when $f(-x)=-f(x)$, giving symmetry about the origin (for example $f(x)=x^{3}$). An inverse function $f^{-1}$ reverses the input-output pairing, so $f(f^{-1}(x))=x$, and its graph is the reflection of $f$ across the line $y=x$. A function has an inverse that is itself a function only when it is one-to-one, which you can verify with the horizontal line test; otherwise you must restrict the domain, as with $f(x)=x^{2}$ restricted to $x\\ge 0$.

Regression and Model Selection

Real data rarely lands exactly on a clean curve, so we fit a model that captures the trend. Regression finds the function of a chosen family (linear, quadratic, cubic, higher-degree polynomial, or another type) that best matches the data, typically by minimizing the squared residuals, where a residual is the difference between an observed value and the value the model predicts. Choosing the right family is a judgment based on the shape of the scatterplot and the context: data that bends once suggests a quadratic, data with an inflection suggests a cubic, and data that levels toward a limit may suggest a rational model. A larger correlation or coefficient of determination indicates a tighter fit, but a higher-degree polynomial will always fit the sample points more closely while often predicting poorly between or beyond them, an effect called overfitting. The best model balances fit against simplicity and respects the behavior the situation demands, such as required asymptotes or sign restrictions. Always examine the residual plot: a residual pattern with no structure signals an appropriate model, while a curved residual pattern signals that the chosen family is wrong.