Polynomial and Rational Functions: Zeros and Graphs

Core Concept

Zeros ↔ x-Intercepts ↔ Factors

For any function $y = f(x)$:

• An x-intercept occurs where $y = 0$, i.e., where $f(x) = 0$. The solutions to $f(x) = 0$ are called the zeros of $f$.
• The y-intercept is the output $f(0)$ — plug in $x = 0$.

The Factor Theorem connects zeros to algebra:

If $p(a) = 0$, then $(x - a)$ is a factor of $p(x)$. Conversely, if $(x - a)$ is a factor, then $p(a) = 0$.

So if you know a zero, you know a factor — and vice versa. A polynomial of degree $n$ can have at most $n$ real zeros, hence at most $n$ x-intercepts.

Quick example. Suppose $p(x) = (x - 2)(x + 5)(x - 7)$.

• Zeros (x-intercepts): set each factor to zero → $x = 2,\ -5,\ 7$.
• y-intercept: $p(0) = (0-2)(0+5)(0-7) = (-2)(5)(-7) = 70$.

Reading backward: if you're told $p(3) = 0$, you immediately know $(x - 3)$ divides $p(x)$.

Reading Intercepts from a Table

If a table lists $(x, p(x))$ pairs:

• Any row with $p(x) = 0$ gives an x-intercept (and a factor via the Factor Theorem).
• The row with $x = 0$ gives the y-intercept directly as $p(0)$.

Graphs of Simple Rational Functions

A rational function has the form $r(x) = \dfrac{N(x)}{D(x)}$.

For a simple case like $r(x) = \dfrac{x - a}{x - b}$:

• x-intercept: set the numerator equal to zero → $x = a$ (provided $a \neq b$).
• Vertical asymptote: set the denominator equal to zero → $x = b$ (the graph approaches but never touches this vertical line).
• y-intercept: $r(0) = \dfrac{0 - a}{0 - b} = \dfrac{-a}{-b} = \dfrac{a}{b}$ (provided $b \neq 0$).

The graph of a simple rational function like $\dfrac{x-a}{x-b}$ has two branches that curve around the vertical asymptote — it never crosses $x = b$.