Graphs of Sine & Cosine Functions

From Teach And Discover Wiki

There are standard forms of different types of functions.

There was the first type that we learn about in Algebra I: f(x) = mx + b

There was the second type, quadratics, that has two forms: f(x) = ax^2 + bx + c or f(x) = a(x - h)^2 + k

For these trigonometric functions, there's a standard form for them too, and each of the parts is there for a reason. I'll try to explain each of the parts and help you out with it.

For trig functions, the standard form is f(x) = d + a sin (bx - c).

There are 4 letters, or variables, that change where this function goes. You need to know what a normal sine and cosine function looks like. After that, each of the letters a, b, c, and d change it in different ways. The first way we can change the normal function is to change the "a" term. This is the "amplitude" or the height. When the sine wave starts at (0,0) and goes up, it reaches it's highest point at this number "a".

For instance, f(x) = sin (x) goes up to 1 and down to -1. The function f(x) = 2sin(x) goes up to 2 and down to -2. So on and so forth.

The letter "b" is basically the period. If you have f(x) = sin (2x), you cover two full periods in 2pi (or ~6.28 on the x-axis) as opposed to sin (3x) that will cover 3 full periods in 2pi.

The letter "c" is the horizontal shift, and is opposite (notice the bx - c, meaning the "opposite of c"). So if you are subtracting pi/2 you move to the right pi/2 and vice versa.

The letter "d" shifts the graph up and down and is pretty obvious.

NOW to answer some questions...

y=-2cos(x+pi/2)

Take a normal cosine wave. Starts at positive 1 on the y-axis and goes down, past the x-axis, to -1, then back up to positive 1. Simple...

• Now we'll change it. THIS cosine wave starts at -2 on the y-axis (notice the -2 in the "amplitude" letter "a").
• Next change: Move the entire function back to -pi/2 (notice the x + pi/2? This is "opposite" and we move it back to the left pi/2).
• So, the graph starts at (-pi/2, -2), travels up past the x-axis, then to positive 2 and back down... that's one full period. In this case, the instructions say to graph two full periods, so continue it...

y=4sin(2x+4)-2

We start with a normal sine wave. Starts at (0,0) and goes up... so on and so forth. Now for the changes:

• Amplitude is 4, so this one goes up to 4 and down to -4.
• The "2x" means that our period happens twice as fast. Instead of "up-down", this one will go "up-down-up-down."
• The "+ 4" means that the wave will now start back 4 to the left (in the negatives, opposite of +4 = -4).
• The -2 at the end is the vertical shift... after all these changes, we move the wave down to -2, so that -2 will be the axis.

My advice for all these, take a few sheets of scratch paper, and each "change" value or letter (a, b, c, or d) I would draw a quick version of each graph until you wind up with your "final draft", so to speak.