Graphing Trig Functions in PreCalculus
The graphs of trigonometric functions are usually easily recognizable — after you become familiar with the basic graph for each function and the possibilities for transformations of the basic graphs.
Trig functions are periodic. That is, they repeat the same function values over and over, so their graphs repeat the same curve over and over. The trick is to recognize how often this curve repeats and where one of the basic graphs starts for a particular function.
An interesting feature of four of the trig functions is that they have asymptotes — those notreallythere lines used as guides to the shape of a curve. The sine and cosine functions don’t have asymptotes, because their domains are all real numbers. The other four functions have vertical asymptotes to mark where their domains have gaps.
You’ll work with the graphs of trigonometric functions in the following ways:

Marking any intercepts on the xaxis to help graph the curves

Locating and drawing in vertical asymptotes for the tangent, cotangent, secant, and cosecant functions

Computing the change in the period of a function based on some transformation

Adjusting the amplitude of the sine or cosine when the basic curve has a multiplier

Making sideways moves when transformations involve horizontal translations

Moving trig functions upward or downward with vertical translations
When graphing trigonometric functions, some challenges will include

Not misreading the period of the trig function when a transformation involves a fraction

Drawing enough full cycles of the curve to show its characteristics properly

Marking the axes appropriately for the situation

Making use of intercepts when they’re helpful in a graph
Practice problems

Sketch the graph of the function: f(x) = tan(4x)
Answer:
Credit: Illustration by Thomson DigitalThe given function is f(x) = tan(4x).
Using f(x) = AtanBx, the period of the function is determined by
therefore, the graph is the standard tangent function, except with a period of

Give a rule for the equations of the asymptotes. Then sketch the graph of the function: f(x) = 4sec(5x)
Answer:
Credit: Illustration by Thomson DigitalUse g(x) = AsinB(x + C) + D, where A is the amplitude,
is the period, C represents a horizontal shift, and D represents a vertical shift. For f(x) = 4sec(5x), the period is
The multiplier 4 brings the upper curves down to 4 and the lower curves up to ‒4.
The asymptotes are found where the reciprocal of the secant, f(x) = 4cos(5x), is equal to 0: cos5x = 0 when
Solving for x, you divide each term by 5 to get
Letting k be an integer, the general rule for the equations of the asymptotes is