Transcendental Functions

Types and Behaviors

1. Exponential Function

Form:

• $a$: amplitude, scales the curve vertically;
• $b$: growth or decay rate, defines how quickly the function increases or decreases;
• $c$: horizontal shift, moves the curve left or right;
• $d$: vertical shift, moves the graph up or down.

Behavior:

• Increases rapidly when $b > 0$;
• Decreases towards zero when $b < 0$;
• Always positive for all $x$;
• Passes through the point $(c, a + d)$;
• Domain: $(-\infty, \infty)$;
• Range: $(d, \infty)$ if $a > 0$, or $(-\infty, d)$ if $a < 0$.

Use case: modeling population growth, radioactive decay, and compound interest.

2. Logarithmic Function

Form:

• $a$: amplitude, vertically stretches or compresses the curve;
• $b$: base, determines the growth or decay rate;
• $c$: horizontal shift, moves the graph left or right;
• $d$: vertical shift, moves the graph up or down.

Behavior:

• Defined only for $x > c$;
• Increases slowly as $x$ grows;
• Approaches negative infinity near $x = c$;
• Passes through the point $(c + 1, d)$;
• Domain: $(c, \infty)$;
• Range: $(-\infty, \infty)$.

Use case: measuring data with multiplicative change, such as pH, sound intensity, or earthquake magnitude.

3. Trigonometric Function

Form:

where $\text{trig}$ can be $\sin$, $\cos$, or $\tan$.

• $a$: amplitude, controls the height of the wave;
• $b$: cycles, defines how many oscillations occur within a period;
• $c$: horizontal shift, moves the wave left or right;
• $d$: vertical shift, moves the graph up or down.

Behavior:

• Sine and cosine: oscillate periodically between $-a + d$ and $a + d$;
• Tangent: repeats every $\pi$ and has vertical asymptotes at $x = \frac{\raisebox{1pt}{$\pi$}}{\raisebox{-1pt}{$2b$}} + n\pi/b$;
• All are periodic and continuous within their domains;
• Domain and range:
  • $\sin(x), \cos(x)$: domain $(-\infty, \infty)$, range $[d - a, d + a]$;
  • $\tan(x)$: domain $\mathbb{R} \setminus \left\{ {\frac{\raisebox{1pt}{$\pi$}}{\raisebox{-1pt}{$2b$}} + n\pi/b} \right\}$, range $(-\infty, \infty)$.

Use case: modeling cycles and oscillations in signal processing, physics, and engineering.