Polynomial functions come in various forms, from simple linear equations to complex higher-degree expressions. Each type has unique graphical characteristics, like straight lines, parabolas, or curves with multiple turning points. Understanding these shapes is crucial for analyzing function behavior.

Exponential and trigonometric functions offer a different flavor. Exponentials show rapid growth or decay, while trig functions oscillate periodically. Rational functions introduce asymptotes, and piecewise functions combine different formulas. These diverse function types expand our mathematical toolkit for modeling real-world phenomena.

Polynomial and Exponential Functions

Graphing polynomial functions

• Linear functions represented by the general form $f(x) = mx + b$ where $m$ represents the slope and $b$ represents the y-intercept produce a straight line graph
• Quadratic functions represented by the general form $f(x) = ax^2 + bx + c$ where $a$, $b$, and $c$ are constants and $a \neq 0$ produce a parabolic graph with a vertex turning point and an axis of symmetry passing through the vertex
• Cubic functions represented by the general form $f(x) = ax^3 + bx^2 + cx + d$ where $a$, $b$, $c$, and $d$ are constants and $a \neq 0$ produce a graph with one or two turning points and may have an inflection point where the concavity changes
• Higher-degree polynomial functions represented by the general form $f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$ where $a_i$ are constants and $n$ is a positive integer have a maximum number of turning points and end behavior determined by the degree of the polynomial ($n$)

Exponential and trigonometric graphs

• Exponential functions represented by the general form $f(x) = a^x$ where $a > 0$ and $a \neq 1$ produce a graph always above the x-axis for $a > 1$ (exponential growth) and always below the x-axis for $0 < a < 1$ (exponential decay)
• Logarithmic functions represented by the general form $f(x) = \log_a(x)$ where $a > 0$, $a \neq 1$, and $x > 0$ produce a graph that is the reflection of the exponential function over the line $y = x$ with a vertical asymptote at $x = 0$ and slow increase for large $x$ values
• Sine function $f(x) = \sin(x)$ produces a graph oscillating between -1 and 1 with a period of $2\pi$
• Cosine function $f(x) = \cos(x)$ produces a graph oscillating between -1 and 1 with a period of $2\pi$, phase-shifted by $\frac{\pi}{2}$ compared to the sine function
• Tangent function $f(x) = \tan(x)$ produces a graph with vertical asymptotes at odd multiples of $\frac{\pi}{2}$ and a period of $\pi$

Rational, Piecewise, and Other Functions

Properties of rational functions

• Rational functions represented by the general form $f(x) = \frac{P(x)}{Q(x)}$ where $P(x)$ and $Q(x)$ are polynomials and $Q(x) \neq 0$ have vertical asymptotes where the denominator equals zero ($Q(x) = 0$)
• Horizontal asymptotes of rational functions depend on the degrees of the numerator ($P(x)$) and denominator ($Q(x)$) polynomials
• Oblique asymptotes occur in rational functions when the degree of the numerator is one less than the degree of the denominator
• Piecewise functions defined by different formulas for different intervals of the domain produce a graph consisting of different pieces, each corresponding to a different formula, with continuity and differentiability dependent on the behavior at the endpoints of each interval

Absolute value and step functions

• Absolute value function defined as $f(x) = |x|$ produces a V-shaped graph with the vertex at the origin (0, 0)
• Step functions having a constant value over intervals of the domain produce a graph consisting of horizontal line segments with jumps at certain points
• Signum function (sign function) defined as $\text{sgn}(x) = \begin{cases} -1, & x < 0 \ 0, & x = 0 \ 1, & x > 0 \end{cases}$ produces a graph consisting of three horizontal line segments at $y = -1$, $y = 0$, and $y = 1$, with jumps at $x = 0$