One of the big challenges my tutoring clients often face in Trigonometry class comes when they are asked to graph trigonometric functions with different amplitudes, periods, and shifts.  Even when clients understand the basic principles, they often find it hard to keep track of and condense all the information they are given into one single graph.

If you are facing these kinds of problems in your Trig class, you are not alone!  I will show you a trick that I have dubbed the “Eight Blanks” method, which is a system for organizing all the information about amplitude, period, and shifts so that drawing the actual graph of the trig function will be very easy.  I will start with sine and cosine graphs, which are the most common, and progress to graphs of other trig functions in later blog entries.

A small disclaimer: While most of my blog entries contain original methods and techniques that I created myself, in this case I cannot take credit for coming up with this method.  It was taught to me by a client, and I thought it deserved to be spread around more widely.

The base sine and cosine graphs

Before I show you the method, let’s take a moment to look at the base parent graph for sine.  If you have not yet seen in class what the base graph for sine looks like, you can learn all about it from a website such as Purplemath, Khan Academy, or MathHints.

Base sine curve. Generated by Desmos.

Even though the sine is a continuous function, there are specific points on the curve that I want to call attention to.  I refer to these specific locations as landmarks.  As I go through one whole period of the sine curve, I see three possible vertical positions for landmarks, marked by purple arrows, and five horizontal positions for landmarks, marked by green arrows.

In the vertical direction, I can first point out the centerline, the horizontal line that goes straight through the middle of the sine curve.  For the base sine graph, this line is the same as the x-axis, and I marked it with the dark purple arrow labeled “center”.  There is also the top of the crest of the curve (marked “top”), and the bottom of the trough of the curve (marked “bottom”).

Using the green arrows, I have marked out all the locations where the curve goes through one of the three vertical positions listed above, through one whole period of the sine.  I can see that the landmarks always go in the same sequence:

• Landmark 1 goes through the centerline
• Landmark 2 goes to the top position
• Landmark 3 goes through the centerline again
• Landmark 4 goes to the bottom position
• Landmark 5 goes back to the centerline

I can repeat the above process for cosine:

Base cosine curve. Generated by Desmos

Note that cosine also has a center, top, and bottom, but the sequence of landmarks is different for cosine than sine:

• Landmark 1 starts at the top 
• Landmark 2 goes through the centerline
• Landmark 3 goes to the bottom
• Landmark 4 goes through the centerline again
• Landmark 5 goes back to the top

For both sine and cosine, these five horizontal landmark positions, plus the three vertical positions, form the basis for the Eight Blanks that the method is named after.

Now we are ready to look at transformations of the sine and cosine graphs.

Transformations of sine and cosine graphs

Hopefully, your teacher has introduced to you the meaning of the various transformations that can be applied to the sine and cosine functions (and to any other function for that matter).  If not, you can read the details from a web resource such as Purplemath or MathHints.  As a summary, a transformed sine or cosine function takes the form

[math]f(x) = A sin [b(x – h)] + k[/math]       OR       [math]f(x) = A cos [b(x – h)] + k[/math]

where

• [math]A[/math] is the amplitude / vertical stretch.
• [math]b[/math] is the horizontal stretch.  This is not the period, but the period can be calculated from [math]b[/math] using the formula: [math](Period) = \frac{2\pi}{b}[/math].
• [math]h[/math] is the horizontal shift.
• [math]k[/math] is the vertical shift.

Let’s look at a concrete example.  Suppose I want to graph the function 

$$y = 3 sin [2(x + \frac{\pi}{6})] – 1$$

In this case,

• The amplitude is [math]A = 3[/math].
• The horizontal stretch is [math]b =  2[/math]; this corresponds to a period of [math]\frac{2\pi}{(2)}[/math], or [math]\pi[/math].
• The horizontal shift is [math]h = -\frac{\pi}{6}[/math] (to the left).
• The vertical shift is [math]k = -1[/math] (down).

Note that [math]h[/math] really is [math]-\frac{\pi}{6}[/math] and not [math]\frac{\pi}{6}[/math], as [math]h[/math] is supposed to be the number that is subtracted from [math]x[/math].  Again, if you are not familiar with how these transformations work, there are websites such as MathHints or Purplemath to help you become familiar with them. 

We are now ready to fill out the Eight Blanks.

The Eight Blanks method for sine

To use the Eight Blanks method, I want to draw five blanks in a horizontal row, and the other three blanks in a vertical column to the right:

Let’s look at the set of three blanks first.  As I mentioned earlier, these three blanks represent the three vertical positions in the sine and cosine graphs, namely the centerline, top and bottom. 

• The middle blank represents the vertical position of the centerline.  Since the centerline is shifted by the same vertical offset that shifts the rest of the sine or cosine curve, the vertical position of the centerline is just the vertical offset.  So I place the value of [math]k[/math] in this space.
• The top blank represents the top vertical position.  This is one amplitude in height above the centerline, so I place the value of [math]k + A[/math] here.
• The bottom blank represents the bottom vertical position.  This is one amplitude in height below the centerline, so I place the value [math]k – A[/math] here.

Now we are ready for the row of five horizontal blanks.  In each of these spaces, I put the horizontal position of each of the five landmarks for the sine or cosine curve:

• The first space is the first landmark and represents the horizontal position where I should start drawing my curve.  I want to start drawing at the position of the horizontal shift, so I place the value of [math]h[/math] in this space.
• The last (fifth) space represents the fifth landmark, which is exactly one period horizontal distance from the first landmark position.  So the value to place here is the initial position plus one period.
• The third space is at the midpoint between the first and fifth landmarks and so should be the average of the first and fifth spaces.  Similarly, the second space should be the average of the first and third, and the fourth space should be the average of the third and fifth.

Let’s try this for the example above.  As a reminder, the function to graph was

$$y = 3 sin [2(x + \frac{\pi}{6})] – 1$$

I’ll start with the three vertical blanks:

• The middle blank (centerline) is where I put the value of [math]k[/math], so I write [math]-1[/math] in this space.
• I am supposed to place [math]k + A[/math] in the top blank, so I write [math](-1) + (3) = 2[/math] here.
• Finally, I want to place [math]k – A[/math] in the bottom blank, so I write [math](-1) – (3) = -4[/math] in this location.

My Eight Blanks look like this so far:

Now I will put values in the five horizontal blanks:

• In the first blank, I am supposed to place the value of [math]h[/math], so I write [math]-\frac{\pi}{6}[/math] here.
• Now I add the period to the value in the first blank to get the value for the fifth blank.  I remember that the period was [math]π[/math], so I compute [math](-\frac{\pi}{6}) + π = \frac{5\pi}{6}[/math] for the value here.
• The third blank takes the average between the first and fifth blanks, so I calculate [math](-\frac{\pi}{6} + \frac{5\pi}{6}) / 2 = \frac{\pi}{3}[/math] and place that value in this space.
• The second blank is the average between the first and third blanks.  I will compute [math](-\frac{\pi}{6} + \frac{\pi}{3}) / 2 = \frac{\pi}{12}[/math] and place it in this blank.
• Finally, the fourth blank is the average between the third and fifth blanks, so I will place [math](\frac{\pi}{3} + \frac{5\pi}{6}) / 2 = \frac{7\pi}{12}[/math] as the value for this space.

Most of my clients tell me that calculating all those averages is the most difficult part of the method.  But it is worth it, I promise!  My final set of Eight Blanks, with all spaces filled, looks like this:

Now is where the magic happens!

The five horizontal values are five equally spaced horizontal positions over one period.  They represent the horizontal positions of the five landmarks that I discussed in the beginning.  I know what the vertical positions are for the five landmarks for sine, so I can use the vertical values to figure out the (x, y) coordinates for each of the points:

• Landmark 1 is supposed to be at the centerline, so the coordinates are (horizontal position 1, centerline) = [math](-\frac{\pi}{6}, -1)[/math].
• Landmark 2 is supposed to be at the top, so the coordinates are (horizontal position 2, top) = [math](\frac{\pi}{12}, 2)[/math].
• Landmark 3 goes back to the centerline, so its coordinates are (horizontal position 3, centerline) = [math](\frac{\pi}{3}, -1)[/math].
• Landmark 4 goes to the bottom, so its coordinates are (horizontal position 4, bottom) = [math](\frac{7\pi}{12}, -4)[/math].
• Finally, Landmark 5 goes back to the centerline again; its coordinates are (horizontal position 5, centerline) = [math](\frac{5\pi}{6}, -1)[/math].

I can plot these five points and then connect all the dots to generate the curve.

Transformed sine graph. Generated by Desmos

The Eight Blanks method for cosine

Let’s try another example, this time with cosine.  Suppose I want to graph the function

$$y = \frac{3}{2} cos [\frac{\pi}{3} (x – 2)] + \frac{5}{2}$$

As before, I decide what all the parameters are for this case.

• The amplitude is [math]A = \frac{3}{2}[/math].
• The horizontal stretch is [math]b = \frac{\pi}{3}[/math]; this corresponds to a period of [math]\frac{2\pi}{\pi / 3}[/math], or [math]6[/math]. Oddly, this time the period does not have a factor of [math]\pi[/math] in it!
• The horizontal shift is [math]h = 2[/math] (to the right).
• The vertical shift is [math]k = \frac{5}{2}[/math] (up).

Again, I’ll start with the three vertical blanks:

• The middle blank (centerline) is the value of [math]k[/math], so I write [math]\frac{5}{2}[/math] here.
• The top blank takes [math]k + A[/math], so I write [math](\frac{5}{2}) + (\frac{3}{2}) = 4[/math].
• The bottom blank takes [math]k – A[/math], so I write [math](\frac{5}{2}) – (\frac{3}{2}) = 1[/math].

Now I will set the five horizontal blanks:

• The first blank takes the value of [math]h[/math], so I write [math]2[/math] here.
• Now I add the period to the value in the first blank to get the value for the fifth blank. The period is 6, so I compute [math](2) + (6) = 8[/math] for the value here.
• The third blank takes the average between the first and fifth blanks, so I calculate [math](2 + 8) / 2 = 5[/math] and place that value in this space.
• The second blank is the average between the first and third blanks, and I will compute [math](2 + 5) / 2 = \frac{7}{2}[/math] for this blank.
• Finally, the fourth blank is the average between the third and fifth blanks, so I will place [math](5 + 8) / 2 = \frac{13}{2}[/math] as the value for this space.

Note that none of the horizontal positions are multiples of [math]\pi[/math], and although that is a little weird, there is nothing wrong with it and it is all correct.

Here is the complete set of Eight Blanks:

Now I plot points, but I have to use the landmark pattern for cosine this time, and not sine:

• Landmark 1 is supposed to be at the top, so the coordinates are (horizontal position 1, top) = [math](2, 4)[/math].
• Landmark 2 is supposed to be at the centerline, so the coordinates are (horizontal position 2, centerline) = [math](\frac{7}{2}, \frac{5}{2})[/math].
• Landmark 3 goes to the bottom, so its coordinates are (horizontal position 3, bottom) = [math](5, 1)[/math].
• Landmark 4 goes back to the centerline, so its coordinates are (horizontal position 4, centerline) = [math](\frac{13}{2}, \frac{5}{2})[/math].
• Finally, Landmark 5 goes back to the top again; its coordinates are (horizontal position 5, top) = [math](8, 4)[/math].

And here is the graph:

Transformed cosine graph. Generated by Desmos

If you follow this method consistently, graphing transformed trig functions is a snap.

What about tangent and cotangent?  We will look at those in Part 2.
I hope you find my post helpful!  If you would like more help for any level of Math, I would love to work with you on 1:1 tutoring.  Please see my website at https://andrewjeungtutoring.com .