Trig Functions: Easy Points on the Math Test

Introduction

Haven't had pre-calc? Don't quite remember graphs of trigonometric functions? If your math score is above 21, I'd recommend learning this. It's not technically difficult. Trigonometric functions are virtually guaranteed to show up on the ACT®. Studying this page is a valuable use of most students' time.

Consider this example:

If y = 5 sin(x) + 2, what is the amplitude of the function?

If I want to know what the amplitude is, I need to know how the different parts of this function interact. This question and questions like it can be solved quickly if you know the different components of the function.

Y = A sin(B(x - C)) + D

• A = Amplitude (the difference between the highest and lowest points) / 2
• B = Period (the horizontal distance it takes for one complete cycle)
• C = Horizontal shift (how far left or right the graph is moved)
• D = Vertical shift (how far up or down the entire graph is moved)

Y = sin(x)

This is just sin(x) with nothing else added. The graph starts going up through the origin. The graph goes up to 1, then down to -1 before repeating. I'll reference this graph to see how the shape of the function changes as I manipulate the other parts. This is my base function that I'll compare others to.

Y = 2sin(x)

This graph has A = 2. It looks a lot like the previous graph. The only difference is that it goes up to 2, then down to -2. That's because the amplitude has been changed. The peaks and the troughs are equivalent to the positive and negative "A" value. This demonstrates how changing the A value affects the graph's height.

I might be able to solve the question above at this point. Let's keep going for now. I'll come back to it at the end.

Y = sin(2x)

In this function, I've manipulated the B. You'll notice that the peaks are much closer together. The increase in B causes the wave to complete more cycles in the same horizontal distance.

In math terms, the period of a sine function is 2π/B. Don't worry about memorizing this for the ACT® Math section. Just remember that as B increases, the period decreases, and vice versa.

Consider this – which is bigger? 2/1 or 2/2? 2/1 = 2. 2/2 = 1. If the denominator represents my B, then it makes sense that increasing the B causes the period to decrease. Leaving out the π for demonstration purposes, I increase the B from 1 to 2. In the first example, 2/1, the period is 2. Then I increase the B to 2. That gives me a period of 2/2, or 1, which is smaller than 2. That's how increasing the B in my equation decreases the period.

In terms of these graphs, increasing the B will make the waves packed much closer together. Decreasing the B will space the waves out.

Y = sin(x - π)

It looks like y = sin(x), but it goes down through the origin. The ACT® is likely to ask you whether this function has been translated right or left. Since the function has a negative, students tend to think the graph has been translated to the left. That is wrong.

The ACT® Math usually tries to trick students this way. The equation is Y = A sin(B(x - C)) + D. Notice the C is already negative. So if I put a negative value in for C, the equation would read y = A sin(B(x + C)) + D. But that's not the graph I have above.

I have y = sin(x - π). That means that the π was positive. This goes back to my negative rules. A positive and a negative make a negative.

The graph was translated to the right. The same origin in y = sin(x) is now just shifted over to the right. To be clear, the origin looks the same on the left side. There's no way to tell based on the graph whether the function was translated to the left or the right. I only have the C value to go by to make that determination.

This is a common trick question on the ACT® Math section. They may not even give you graphs – they may say "you have a graph with a positive C. which way is it translated, left or right?" or something of that nature.

Y = sin(x) + 2

This one is pretty straightforward – increasing the D moves the entire function. It doesn't change anything else. It just takes the function I already had, then just moves its origin up two units.

Now instead of starting at (0, 0), the function starts at (0, 2). The amplitude is still 1. The function goes up to 3 and down to 1.

Y = cos(x)

This is the graph of a cosine function. It looks a lot like a sine function. The major difference is that the cosine function starts at the amplitude, then does the same thing the sine function does. Whereas the sine function started at (0, 0) and then went up to its amplitude of 1, the cosine function starts at 1, then goes down to -1.

For the purposes of the ACT® Math section, it has all of the same rules as the sine function, except for where it starts. Remember, cosine is just sine shifted to the left by π/2.

Further Study

The ACT® Math almost never asks about graphs of tangent functions. Here's what they look like so you know. But seldom do I ever see the ACT® Math. Moreover, there may be questions on the unit circle as well. But those are few and far between.