Graphing

The last thing you need to explore in trigonometry is graphing the functions y = sin x and y = cos x. These pages will NOT replace your notes on this topic. The examples presented here are meant to be a refresher of what you have learned in class. It is also assumed that you are comfortable working with transformations on quadratic functions.

The Basic Graph of y = sin x (from [0, 360°])

If you forget what the basic graph of ANY function looks like, you can always use a table of values to spark your memory. Let's make a table of values below, with the five important points for y = sin x listed.

Placing these points on a graph gives us:

Now you should connect those points with a smooth curve to arrive at the basic y = sin x graph:

This graph does in fact continue on, and on, and on... We were told to show the graph in the domain of [0, 360]. What you see above is normally all that you need to work with when you are showing transformations on the basic graph. A graph of y = sin x with more of the domain shown is shown below. (Remember, this graph has a domain of and as such continues to the left and right forever.)

Before we move on to transformations, we need to look at some terminology:

Terminology

There are certain aspects of the graph of y = sin x (and y = cos x) which have specific 'names' which you must know. Each term is described below, and then pictured on the graph shown under the definitions:

• Period: The distance on the x-axis after which the graph starts to repeat it's y-values. You can see this as a repeating 'shape' on the graph. The graph of y = sin x has a period of 360°.
  
  
• Maximum: The highest point on the y-axis reached by the graph. In the case of y = sin x, the maximum is 1.
  
  
• Minimum: The lowest point on the y-axis reached by the graph. In the case of y = sin x, the minimum is -1.
  
  
• Amplitude: The distance from the x-axis to EITHER the minimum or the maximum on the graph. (Since this is a distance, it will always have a positive value.) The amplitude of y = sin x is 1.

All of these are shown on the graph below:

Transformations on y = sin x

The transformations on y = sin x are the same as what you have seen when you worked on transformations of quadratic functions. The standard form of y = sin x with transformations is shown below:

where:
a is the vertical stretch (multiply all y-values by a). This is also the value of the amplitude.
h is the horizontal shift. If h is negative, move left h units: If h is positive, move right h units.
k is the vertical shift. If k is negative, move down k units: If k is positive, move up k units.

Example 1

Sketch the graph of y = 3sin(x - 90°) + 1

Solution

We will work on this transformation by transformation. First, draw the graph of y = sin x with the five important points clearly marked:

Notice that we have shown more of the graph (we have extended our axes). This is because we will need room to move the graph around. Now we will look at the three transformations in order.

1. a is three. That means that we have to multiply all of the y-coordinates on the graph by 3. The five important points (which are (0,0),(90,1),(180,0),(270,-1),(360,0)) will become: (0,0),(90,3),(180,0),(270,-3),(360,0). The new graph looks like:

This is the graph of y = 3sin x.

2. h is positive 90°. (If you don't remember why it is positive, check the lesson on transformations of quadratic functions.) That means we must move the graph right 90° (which corresponds to 2 ticks on the graph we are using. There is a problem however, in that that moves our rightmost point off of the right edge of the domain we are looking at on the graph (past 360°). We also need to 'bring in' a point from the left (left of 0°). Below we show the movement that must be made, and the resulting graph.

Move the points like this:

to get this:

This is the graph of y = 3sin(x - 90°).

3. The last transformation is the k value. In this example k is 1, which means we need to move EVERY point UP 1. This results in our final graph:

This is the graph of y = 3sin(x - 90°)+1 (your final graph).

The Basic Graph of y = cos x

Now we will look at the basic graph of y = cos x. Again, we will make a table of values showing our 5 'important' points.

Now you join these points using a smooth curve - just like the graph of sin x. The basic graph (with the important information labelled) is shown below:

You can see from the graph that the period of y = cos x is 360°, the amplitude is 1, the maximum is 1 and the minimum is -1. As a matter of fact, the graph of y = cos x is just the graph of y = sin x shifted 90° to the left!

Transformations on y = cos x

Again, you will see that all of the same transformations you have already learned apply to all types of graphs.

Example

Sketch the graph of y = -2cos(x) - 3.

Solution

Again, start with the basic graph of y = cos x drawn on a grid that gives you a little 'breathing room'.

This is the graph of y = cos x.

Now apply the transformations in order:

1. In this example, a is -2. This means that you should multiply each y - coordinate from your original graph by -2. The five important points (which are (0,1),(90,0),(180,-1),(270,0),(360,1)) will become: (0,-2),(90,0),(180,2),(270,0),(360,-2). The new graph looks like the following:

This is the graph of y = -2 cos x.

2. There is no h value in this example, so we can move on to the k value. In this example, k is -3. This means that we must move EVERY point down 3 units. The final graph appears as follows:

This is the graph of y = -2cos x - 3 (your final graph).

Please remember that this is just a refresher of the graphing skills that you learned in class. There is no replacement for the notes from the classes on this topic, or the examples that your teacher covered during that time. Please insure that you have all the notes from this topic. Now you can try a couple sample graph questions, or head back to the Trigonometry main page.