Back To CourseMath 103: Precalculus
12 chapters | 94 lessons | 10 flashcard sets
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Tyler has tutored math at two universities and has a master's degree in engineering.
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.
The tangent function looks like this:
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):
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:
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.
If we looked at the formula y=-2tan(4x - pi) +1, what transformations took place? Let's do the transformations outside the parentheses first. Everything outside the parentheses is going to change how we look at the y coordinates of all the points. It affects only the vertical part of the graph.
We see that A=-2. Not only did the graph get twice as steep, it also is negative. This means that instead of starting negative on the left and increasing as it goes to the right, it's the opposite. It starts high on the left and keeps decreasing as we move to the right. We call this a reflection.
We can also see from the formula that D=1. D is a vertical shift, and since we add one, this means the graph shifts up one unit.
Now, let's take a look at the factors inside the parentheses. Numbers inside the parentheses affect the x coordinates of the graph, or the horizontal aspect of the graph. Remember, these do the opposite of what you think. B=4. B represents how the period changes for the graph. Since this is multiplied by a positive four, we remember to do the opposite. This actually makes the period smaller, or we can say the period is pi/4.
Notice how we did the opposite. Now, let's take a look at C. C=pi, but notice it is subtracted. Since C controls the phase shift (left or right shift) we want to do the opposite. That means we shift the entire graph to the right by pi units.
Sometimes, you'll see all four transformations. Sometimes, you'll just see a few. But, knowing how to change amplitude, period, phase shift, and vertical shift will help you understand how to change tangent functions into graphs and vice versa.
Let's take a look at the four ways we can transform a tangent function and graph. A changes the amplitude, or how steep the graph is along the y axis. B changes the period, or how wide one pattern of the graph is. C changes the phase shift, or how the graph is shifted to the left or the right. D controls the vertical shift, or how the graph is shifted up or down.
Everything outside the parentheses affects the vertical part of the graph, and we do what they say. If it is multiplied by two, it is twice as steep. If three is subtracted, we shift down three. Everything inside the parentheses affects the horizontal part of the graph, and we do the opposite of the commands. If x is divided by two, the graph actually gets wider by a factor of two. If C=pi/2, even though it is positive, we do the opposite. This means we shift to the left by pi/2 units.
If you have these down, you can spot a tangent graph in no time. Or, you could start with the same basic tangent graph each time and make the transformations necessary to make it what you want. Regardless, just like playing with modeling clay, you can always start with the basic form and do any stretches and shifts to make it look how you want.
Following this lesson, you should be able to:
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Back To CourseMath 103: Precalculus
12 chapters | 94 lessons | 10 flashcard sets