Use graphing software to graph the functions specified.Select a viewing window that reveals the

Use graphing software to graph the functions...

Key Concepts

Graphing Software

Graphing software is a tool used to visually represent functions on a coordinate plane. It assists in understanding complex behaviors such as intercepts, asymptotes, and periodicity by providing an immediate graphical output of mathematical equations.

Trigonometric Functions

Trigonometric functions are functions derived from the relationships in a right triangle or the unit circle. They include sine, cosine, tangent, and others, and are essential in modeling periodic and oscillatory behavior in various contexts.

Function Transformations

Function transformations involve altering the graph of a function through processes like shifting, reflecting, stretching, or compressing. In trigonometric functions, these transformations can affect amplitude, period, phase shift, and vertical shift, enabling the depiction of more complex behaviors.

Period of a Function

The period of a function is the interval after which the function begins to repeat its values. For trigonometric functions like the tangent, transformations in the function's argument can alter the period, which is crucial for determining the appropriate viewing window to capture the correct number of cycles.

Vertical Asymptotes

Vertical asymptotes occur in functions where the function approaches infinity as the input nears certain values. For the tangent function, these asymptotes mark boundaries where the function is undefined, and identifying these helps in accurately graphing the behavior and determining an effective graphing window.

Transcript

00:01 All right, we want to graph four periods of y equals negative tangent of 2x.

00:06 I'm going to choose a viewing window that reveals key features.

00:10 So key features would be things like y intercepts, x intercepts, maximums, or minimums.

00:15 I've gone ahead.

00:16 I've got our graphing utility up here.

00:18 Currently, we are set just at our standard viewing window, negative 10 to 10, and from negative 10 to 10 on the y axis as well.

00:25 And we are also in radian modes.

00:27 So make sure that you are in radian modes to do this problem.

00:32 Now we're going to go ahead and turn this on, and we can see that this is way more than four periods.

00:40 So let's talk about tangent.

00:42 In general, tangent has a period of pi radiance, but if we have our period being adjusted by some value over in here, it's going to change.

00:55 And that b value in this case is 2.

00:57 So if we plug this in, the period of this function is not actually pi, it is pi over 2, which means that four periods of that would just be 2 pi.

01:11 Because if we took a period, one period is pi over 2 and multiply it by 4, we only need to be able to see 2 pi.

01:18 So my suggestion here is because we need to see 2 pi in length from left to right, this total distance on our x -axis needs to be 2 pi.

01:28 Let's put negative pi over here and positive pi over here, and that distance will be 2 -py.

01:35 So let's change that.

01:37 Let's go from negative pi to pi on that, and let's step out by, you know, pi over 4s...

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