Graphing Trigonometric Functions Examples

Graph y = 3sin (2x) – 4.

We already know what a basic sine curve (y = sin x) looks like, and that is our basis for this graph. Looking at the differences between the equations will help us figure out what we need to change in the graph. Let's start at the left and work our way to the right.

The first difference we see is the 3. This number is our amplitude, so the maximum and minimum values are each 3 units away from the midline. A graph of y = 3sin x looks like this:

Now let's take our y = 3sin x and turn it into y = 3sin (2x). The 2 tells us that our period is, or π. We can fit a whole cycle within 0 to π on the x-axis. Scrunching up our curve to fit gets us:

Now all we are missing is the -4 that is tacked on at the end. That tells us that our midline drooping down 4. The basic sine curve has a midline at the x-axis (y = 0). We need to take our curve and move it 4 units down:

This is it, the graph of y = 3sin (2x) – 4.

Graph y = cos(2x – π) + 1.

A journey of a thousand miles begins with a single step. We don't plan on going nearly that far, but we still need a place to start, and that's with the basic cosine function, y = cos x. It's the same as y = sin x, only shifted  to the left.

The period of a cosine function is , and our value of B is 2, so the period of this function is π. This means we can fit one complete cycle into 0 to π on the x-axis. In the basic cosine curve, one complete cycle fits into 0 to 2π, just like its bro sine. We need to cram twice as much function into our new curve. Put on some gloves, 'cause this could get messy.

Next up, let's look at the horizontal shift with , or . Your parents always said that two wrongs don't make a right, but two negatives do make a positive. And in this case, it means we actually move the function to the right. Your folks may be wise, but they don't know everything.

We are almost there. All that is left is the + 1 at the end, which tells us that the midline is at y = 1. Move it up, so we can move out.

Now we finally have the graph of y = cos(2x – π) + 1.

What is the equation for the sine function graphed here?

We know the sine function is in the form of y = Asin (Bx + C) + D, we just need to figure out the values of A, B, C, and D. Grab your deerstalker cap and magnifying glass, and let's get to the bottom of this mystery.

The midline, D, is the easiest variable to track down. The maximum value of the function is at y = 0 and the minimum value is at y = -4. Midway between them is the midline (whoda thunk), and it is at y = -2. This means D = -2.

Next up is A, for apple, adamantium, and amplitude. That last one is how far the maximum and minimum values are from the midline. We now know they are 2 away, so A = 2.

Whooah, we're halfway there! Whooah, livin' on a prayer. Take my hand and we'll make it I swear.

The period of the function only depends on B, while the amount of horizontal movement depends on both B and C, so we'll take a look at the period next. It is the length of a complete cycle of the curve. Looking at our curve, we see that it takes 2π to get back to the beginning of the curve, so our period is 2π. If the period is 2π/B, B must be 1.

Moving on to B, we know that the period of our function is , so if we know the period, we know B. Looking at our curve, we see that it takes 2π to from one peak to the next, so our period is 2π. If the period is , and it is, B must be 1.

Now that we're looking at the horizontal position of the graph, it looks a lot like cosine. The maximum on the y-axis is at x = 0, while the basic sine function has a maximum at . This shows a horizontal shift of  to the left. That means that  (the amount of horizontal movement) equals . Let's solve for C:

We've tracked down all the values. A = 2, B = 1, C = , and D = -2, and we can plug these in to the equation y = Asin (Bx + C) + D to get: