Drawing and Sketching of Functions

General Concepts

The process of drawing and sketching functions is a key component of graphical algebra.
This concept involves representing mathematical functions visually using a set of axes.

Linear functions have a constant rate of change and their graphs are straight lines.
Quadratic functions have a variable squared as their highest power. Their graphs typically form a parabola.
Cubic functions contain a variable raised to the power of three. Their graphs can take on a variety of shapes with turning points.
Exponential functions rise or decay rapidly and their graphs vary accordingly.
Trigonometric functions such as sine, cosine and tangent produce regular, oscillating graphs.

Determine the type of function you are dealing with. Understanding the form of the function will guide how you sketch its graph.
Identify and plot any intercepts with axes. These are points where your function crosses the x or y axis.
For quadratic functions, find the vertex of the parabola (the turning point) and plot this.
Consider the end behavior of the function as x tends to positive or negative infinity to determine the general direction of the graph.
Sketch the curve using the plotted points as a guide. Include any important features of the function.

Always consider the domain and range of the function. These will provide boundaries within which to sketch your graph.
Where necessary, use a table of values to find exact points on the graph.

Mastering the sketching of functions is immensely useful within further mathematics. It can help make abstract concepts more concrete, and can simplify the understanding of more complex mathematical relationships.