Contemplating The Wave

Also see Dave's short course on trigonometry.

The sine function ... why does it look like this?

fooplot is an online graphing calculator

The cosine function ... why does it look like this?

fooplot an online graphing calculator

Graphing Trig Functionsƒ(x) = AcosB(x - C) + Dƒ(x) = AsinB(x - C) + D

The Role of Parameter A

|A|is called the Amplitude

determines the distance of the maximum and minimum values from the Sinusoidal Axis

Acontinues ...

The graph is stretched A units.

0 <| A | < 1 the graph is compressed by factor A.A < 0 the graph is reflected over the sinusoidal axis.

The role of Parameter B.B is not the period. B determines the period according to this relation:

The period is the distance along the x-axis required to make one "wave".

or

| A | > 1

The role of parameter C.C is called the phase shift. (aka the horizontal shift)C > 0 the graph is shifted to the right C unitsC < 0 the graph is shifted left C units

WARNING: watch the sign of C

The role of parameter D.D is called the vertical shift, aka average value.D determines the sinusoidal axis.D > 0 the graph shifts up D unitsD < 0 the graph shifts down D units

Sketch each of the following graphs without using a calculator ...

What else can we do to the cosine (or sine) function?

fooplot an online graphing calculator

ƒ(x) = AcosB(x - C) + Dƒ(x) = AsinB(x - C) + D

Now, put what you've learned together, graph this function ...

Graph this:

Use the questions on the following slides to practice what you've learned today.

Solve ...

where 90° ≤ x < 270°

on the interval [π, 2π]

for x ∈ R

Find the exact value of each of the following ...

Sketch each of the following graphs without using a calculator ...

Sketch the following graphs without using a calculator ...

Sketch the following graphs without using a calculator ...

Sketch the following graphs without using a calculator ...

Sketch each of the following graphs without using a calculator ...

Now, put what you've learned together, graph this function ...