Graphing the Tangent Function: Amplitude, Period, Phase Shift & Vertical Shift

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  • 0:05 Graphing the Tangent Function
  • 1:28 Tangent on the Graph
  • 3:47 Explaining Transformations
  • 5:55 Transformations on the Graph
  • 8:08 Lesson Summary
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Lesson Transcript
Instructor: Tyler Cantway

Tyler has tutored math at two universities and has a master's degree in engineering.

Take a look at how the graph of a tangent function and how we can transform it by making a few small changes to its equation. This video shows how to graph the original function and explains its transformations.

Graphing the Tangent Function

Every child loves toys. Some like dolls. Some like action figures. Some like cartoon characters. Some children like to play with one of each all at the same time. When I was young, I liked modeling clay. I could take modeling clay and make anything. I could make a figurine, monster, or animal. Most of the time, I would start with the same shape. I would make a body, head, arms, and legs. Then, if I wanted it to be larger, I could stretch each part. If I wanted it to be smaller, I could squeeze them. If I wanted it wider or more narrow, I could adjust. If I wanted it to be upside down, I could do that too. Basically, all I had to do was be able to make the same basic form and then adjust it how I wanted.

When we look at the graph of trig functions, it works the same way. The graph for each function looks and behaves in one specific way. But, there are small changes you can make that stretch, shift, and reflect each graph. If we learn the basic tangent graph and understand the formula for it, we can easily learn how to make changes that will move it up, down, left, right, stretch it, shift it, and reflect it.

Tangent on the Graph

The tangent function looks like this:

Graph of tangent function
graph of tangent function

The formula for this graph is simply y=tan(x). On the y axis, we have the traditional number line with positive numbers and negative numbers. On the x axis, we have the measures of angles in radians. There are a few x values we want to highlight. First is zero, and it is right in the middle. As we look at the positive side of the x axis, let's look at pi/4, approximately 0.79. Let's also look at pi/2, approximately 1.57.

These points are also on the negative side of the x axis, at -pi/4 and -pi/2.

When we move along the x axis, notice what happens to the y value. On the left side of zero, tangent is negative. At x=-pi/4, tangent=-1. At x=0, tangent=0. This is the origin and the center of the graph. As x becomes positive, tangent is positive. At x=pi/4, tangent=1. If you'll notice on the left and right of the graph above, there are actually two values where the tangent graph gets more steep but never actually touches. We call this undefined.

This means there is no real value for tangent. These undefined points on the graph are at the numbers -pi/2 and pi/2. Just to make sure, if you were to type these in to your calculator, you'd see it says ERROR. As you can see, the tangent function is always getting closer and closer and closer to these values, but never actually gets there. This is the basic format of the tangent graph. The last thing to remember is that tangent doesn't just stop here.

It actually repeats this exact same shape over and over to the right and to the left. This repeating pattern is the graph of y=tan(x):

Example of repeating pattern for tangent function
repeating tangent function on graph

Explaining Transformations

Now that we understand the basic graph shape and a few key points, let's look at ways to change it.

The proper term for this is we want to make transformations to the original graph. First, let's focus on the formula. You'll see that the formula for the basic graph is simple: y=tan(x). It's just a basic function.

There are four ways we can change this graph, we show them as A, B, C, and D. These are numbers that can be multiplied or added to the original function and make specific changes.

y=Atan(Bx + C) + D

If we put a number in for A, it changes amplitude. Amplitude determines how steep or shallow the graph is.

B determines the period, or how wide or narrow the graph is.

C determines phase shift, or how the graph is shifted from left to right.

D determines the vertical shift, or how the graph is shifted up or down.

Here is the information for the basic graph:

Information for basic graph of tangent function
chart of basic information for tangent

The basic function has an amplitude of one. It has a period of pi. It has no phase or vertical shifts, because it is centered on the origin. There is one small trick to remember about A, B, C, and D. That trick is everything outside the parentheses affects the y coordinates, and you do exactly what each says. All the things inside the parentheses affect the x coordinates of the function, and they do the opposite of what they say. Let's take a look at an example.

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