Now that we know how to graph the sine and cosine functions, we can learn how to graph the tangent and cotangent.
In this post, I will go over the basic graphs for the tangent and cotangent and the ways that we can manipulate them.
How to Graph the Tangent Function
Let’s take a look at what our base tangent function looks like.
y = \text{tan}x
Let’s break this graph down.
The first thing you might notice are the dotted lines on both sides of the graph. These dotted lines denote vertical asymptotes and are separated by a period of the graph. Asymptotes are imaginary lines on a graph that our function approaches but never actually touches.
The next thing I would like to point out is that this is an increasing function.
The final thing I would like to point out about the graph of tangent is that the period of the function is not π/2, but π. We start at -π/2, and go all the way to π/2.
Now let’s look at the inverse function:
y=-\text{tan}x
As expected, this graph is much like the graph before, but instead of being an increasing function, this is now a decreasing function. Our period is still π.
How to Graph the Cotangent Function
y=\text{cot}x
One of the differences between the graph of tangent and cotangent is that the cotangent is shifted to the right and doesn’t pass through 0 anymore. We still have two vertical asymptotes and we still have a period of π. Another difference between the two functions is that the graph of cotangent is a decreasing function instead of an increasing function.
Now let’s take a look at the graph of the inverse function:
y= -\text{cot}x
Here we can see that the inverse of the function is exactly as we would expect. We still have a period of π, and instead of being a decreasing function, we now have an increasing function.
More examples
So we have seen the basic functions and their inverse, let’s take a look at some more examples of how we can further manipulate these graphs.
Lets graph this equation:
y = \text{tan}2x
In this graph, we have a horizontal shrink of a factor of 1/2 because we have a b of 2. So, we divide π by 2 to get a period of π/2. Now, we know that the period has to extend to both sides, not just one, so divide that further in half to get your vertical asymptotes on both sides at -π/4 and π/4.
Let’s try another graph!
y = -3\text{tan}\frac{1}{2}x
Here we have a decreasing function because we know that the function is negative 3 tan, and also our amplitude is 3. Our period is 2π because we have stretched the period by a factor of 2. If you don’t understand this concept, refer back to this post where I go over how you get the period of a function.
Conclusion
And that my friends is how we manipulate the graphs of tangent and cotangent! In this post we saw what the graphs are supposed to look like and how we can manipulate the graphs. The way we manipulate the graphs is very similar to the way that we manipulate the graphs of sine and cosine, but we have two very looking graphs from sine and cosine.
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Thank you!