Identify the graph of 𝑦 equals cotangent of 𝑥.
When we look at these three graphs, we know by their shape that they’re all representing either tangent or cotangent. To correctly identify the graph, we’ll need to know some test points. For example, what is the cotangent of 𝜋?
Two of these graphs have the cotangent of 𝜋 approaching ∞, but one of them has the cotangent of 𝜋 at zero. If you plug in the cotangent of 𝑦 on any technology, it’s going to tell you “undefined”; it does not exist at that point. We haven’t eliminated the red or the yellow graph. We can eliminate this blue graph because the point 𝜋, zero does not fall in cotangent of 𝜋. Cotangent of 𝜋 is not equal to zero.
Now, let’s zoom in a little bit closer on the red and yellow graph. Halfway between zero and 𝜋, both of these graphs are at point zero. And that means that 𝜋 over two is equal to zero in both of these graphs. Both of these graphs share all of their 𝑥-intercepts. We need another way to determine the differences. So we’re going to check the places where 𝑦 equals one. For both of these functions, what is 𝑥 equal to if 𝑦 equals one?
We assume the formula of cotangent 𝑥 in both cases. And we want to know what 𝑥-value would make the outcome one. If we take the cotangent inverse of one, we’ll find out what 𝑥 should be. The cotangent inverse of one equals 𝜋 over four. We need to look at 𝜋 over four for our 𝑥-value.
Here it is on the red graph, and here it is on the yellow graph. 𝜋 over four, one is a point on the yellow graph. There is not a point at 𝜋 over four, one on the red graph. This means that the red graph is not a graph of 𝑦 equals the cotangent of 𝑥, only the yellow graph was.