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Before we graph the tangent function, we're going to look at some general graphing techniques.
Specifically we're going to focus on the locations of zeroes and asymptotes.
These ideas are usually covered when graphing rational functions, so we won't go into the
full detail here.
Suppose we have a function of the form f(x) = N(x)/D(x), where we've canceled out all
the common factors.
The zeroes of f(x) correspond to the zeros of the numerator N(x) and the vertical asymptotes
correspond to the zeros of the denominator D(x).
For example, consider the graph of f(x) = (x-2)(x+3)/(3(x+1)^2).
The numerator is 0 at x = 2 and x = -3, which is where we have zeros of the function.
And the denominator is 0 at x = -1, which is where we have a vertical asymptote.
Since tan(x) = sin(x)/cos(x), we can apply these ideas to get a reasonable sketch of
the graph.
We will start with the zeros.
The zeros of tan(x) will be the zeros of sin(x).
These happen at 0, +- pi, +- 2 pi, +- 3 pi, and so on.
So we will plot these zeros for the tangent function.
The asymptotes of tan(x) will be at the zeros of cos(x).
These happen at +- pi/2, +- 3 pi/2, +- 5 pi/2, and so on.
We will draw a vertical dashed lines to represent the vertical asymptotes for the tangent graph.
To determine whether we are approaching positive or negative infinity at the asymptotes, we
need to think about whether the sine and cosine functions are positive or negative on each
interval.
Because the sine and cosine functions are periodic with period 2 pi, we will just need
to look at the intervals between 0 and 2 pi and can repeat from there.
These sketches allow us to quickly see whether each function is positive or negative.
For the four subintervals and the sine function we get positive, positive, negative, negative.
For the four subintervals of the cosine function we get positive, negative, negative, positive
And then the pattern continues for both of these.
The first interval is from 0 to pi/2.
Both functions are positive here so we know that the graph will approach positive infinity
on the left side of pi/2.
On the interval pi/2 to pi, the sine is positive but cosine is negative, which means that the
tangent will be negative, and so the graph will approach negative infinity from the right
side of pi/2.
We can continue this logic to fill out the rest of the graph.
From this information we can get a rough sense of the shape of the tangent function.
Notice that the tangent function is periodic with period pi instead of period 2 pi.
In fact the fundamental period for tangent is (-pi/2, pi/2).
We would say that this is one branch of the tangent function.
Here's the chart of key values for the tangent function notice that the values at +- pi/2
are undefined.
This is just a reminder that you have asymptotes there.
Once we have the key values we can graph the transform versions of the tangent functions
using the same techniques as we used for the sine and cosine functions.
The only difference is that when determining our fundamental period we will set the argument
equal to +- pi/2 because that's the fundamental period for the tangent function.
Example: Graph y = 2 tan(x/2 -pi) - 2.
We set the argument equal to +- pi/2 and solve for x.
This shows us that the fundamental period is (pi, 3 pi).
The midpoint of this interval is 2 pi, and the midpoints of the new intervals are 3 pi/2
and 5 pi/2.
We can now take the y coordinates and substitute to get the transformed values.
We multiply the base values by 2 and then subtract 2.
This gives us points and asymptotes that we can plot and use to sketch the graph.
Lastly, we can use periodicity to graph the other branches of the function.
These reference points don't correspond to well-defined features as they do for the sine
and cosine functions.
Because of this, it is especially important that they are graphed carefully and labeled
so that it's clear that you know what you're doing.
In the next video, we will talk about graphing reciprocals of functions and use that to graph
the cotangent function.
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