Essay on module 4

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04.06 Graphs of Other Trigonometric Functions

The Standard Form of the Tangent Function
The standard equation for the tangent graph is y = a tan(bx - c). When a > 0, the value of y increases as the value of x increases, and when a < 0, the value of y decreases as the value of x increases.

About the Graph of the Tangent Function
Previously, we learned that a unique feature of the tangent function is that tan(-x) = -tan(x) and therefore, the graph of y = tan(x) is symmetric to the origin. Also, tan(x) is undefined when cos(x) = 0. This occurs when x has a value . As x approaches from the left, tan(x) approaches ∞ and as x approaches from the right, tan(x) approaches -∞. These produce vertical asymptotes in the graph.

Also, since the function repeats in intervals of , the vertical asymptotes occur at each interval. This can be restated as vertical asymptotes occur when x =. The domain of the tangent function is all real numbers with the exception of those which equal .

Summary of How to Graph the Tangent by Hand
1 Solve bx - c = 0, to find the phase shift and the starting point.
2 Find the new period by evaluating /b.
3 Find the first asymptote by adding /2b to the starting point. Add the new period (/b) to find each subsequent asymptote.
4 Find the x-intercepts by solving the equation tan (bx - c) = 0. Repeat the x-intercepts every /b units. Notice that the x-intercepts are always mid-way between the asymptotes, and that the first one is at the same place as the starting point.
5 Find the y-intercept by evaluating f(0)
Let's look at an example: Graph y = 2 tan (3x + )
1 We find the phase shift to be -/3, so the starting point of the graph is (-/3, 0).
2 The new period will be /3.
3 The first asymptote will be -/3 + /6 = -/6 and then each subsequent one will appear /3 units from there (/6, /2, 4/6, etc.).
4 An x-intercept is at -/3, and every /3 units before and after that (-2/3, -/3 , 0, /3, etc.).
5 f(0) = 0 so the y-intercept is (0, 0)

We plot those points and draw the tangent curve.

Test your skills

For each of the following tangent curves, list the new period, the phase shift, equation of the vertical asymptotes, and the intercepts.
1 y = 4 tan ( 2x - )

2 y = tan (x/2 + )

3 y = 0.5 tan ( 4x + /2)
Check Your Answers

Graph of the Cotangent Function
The cotangent graph is similar to the tangent graph in that the period is , the domain is all real numbers except for
Show More

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