Solution of Equations and Transcendental Functions by Graphical Methods

Transcendental Functions

Transcendental functions are functions that are not algebraic, i.e., they cannot be expressed as finite combinations of algebraic operations.
They include exponential functions (base e, base 10), logarithmic functions (natural and common logs), and trigonometric functions (sine, cosine, tangent).
The key characteristic to remember is that a transcendental function is any function that does not satisfy a polynomial equation.
Transcendental functions can be manipulated algebraically, similar to polynomials. For example, you can add, subtract, multiply, divide, and compose them.

Graphs of Exponential and Logarithmic Functions

Exponential functions have the general form y = a(e)^x where a ≠ 0, e > 0. The base, e, is the constant approximately equal to 2.71828.
The graph of an exponential function always passes through the point (0, 1) and never touches the x-axis, resulting in a horizontal asymptote at y=0.
Logarithmic functions have the general form y = log_a(x) where a > 0 and a ≠ 1. The base, a, is typically e (the natural logarithm) or 10 (the common logarithm).
The graph of a logarithmic function always passes through the point (1, 0) and has a vertical asymptote at x=0.

Graphs of Trigonometric Functions

Trigonometric functions include sine, cosine and tangent. These functions are periodic, repeating their values in regular intervals, known as their periods.
The sine function has a period of 2π, passing through the origin (0,0), with local maximum at (π/2, 1) and local minimum at (3π/2, -1).
The cosine function also has a period of 2π, but starts at a maximum (0,1), with another maximum at (2π, 1) and minimum at (π, -1).
The tangent function has a period of π, and contains asymptotes at odd multiples of π/2.

Solution of Equations using Graphs

The roots of an equation can be found by locating where the graph of the function intersects the x-axis (y=0).
For transcendental functions, exact roots may not be easily calculable. In such cases, estimate the solutions by reading the roots off the graph, or use graphical methods such as iteration or the Newton-Raphson method.
For equations involving two functions, the intersection of their graphs provides the solutions, because at these points, the y-values (and hence the values of the functions) are equal.

Practise drawing graphs of these functions and using them to solve equations. The more self-assured you are with these concepts, the more adept you will be at tackling these types of problems.