6.4 : Graphs of Trigonometric Functions

In this section, we will

1. revisit previous sections for important concepts

2. graph the basic trig. functions (based on the definitions fromprev. sections)

3. define periodicity of a graph and other properties (e.g.,amplitudes)

4. graph transformed trig. functions

Definition of trig functions and the ratios of the sides

Unit circle definition

x = cos θ, y = sin θ

Example 1) Identify the corresponding (x , y) on the U.C. if θ = π6

Example 2) Find (x , y) if θ = 11π6

Fill in the blanks

The shape of cos t for t ∈ [0, 2π)

For the graph of cos t, simply follow the x coordinates of thepoints below on the U.C. as t increases from 0 to 2π.

Characteristics of f (t) = cos t

I Domain :

I Range :

I Minimum :

I Maximum :

I Zeros :

The shape of sin t for t ∈ [0, 2π)

For the graph of sin t, simply follow the y coordinates of the pointsbelow on the U.C. as t increases from 0 to 2π.

Characteristics of f (t) = sin t

I Domain :

I Range :

I Minimum :

I Maximum :

I Zeros :

Let’s graph tan t

Since tan t = yx

Graph of tan t

I Domain :

I Range :

I Asymptotes:

Periodicity

DefinitionA function f is said to be periodic if there is a positive number psuch that

f (x + p) = f (x)

for all x in the domain of f . The smallest such number p is calledthe period of p

Question 1: For f (t) = cos t what is its period? What aboutf (t) = sin t, tan t?

Amplitude

Definition (amplitude)

For a periodic function such as sine or cosine, the average value =M+m

2 and the amplitude = M−m2 , where M = max, and m = min.

Question 1: For f (t) = cos t, what are its avg. value andamplitude?Question 2: What about sin t?

Symmetry: Even or Odd

DefinitionA function f is odd if f (−x) = −f (x) for all x in domain of f . Afunction f is said to be even if f (−x) = f (x) for all x in thedomain.

Examples) Determine whether the function is even or odd.

1. f (x) = x2. f (x) = x2

3. f (x) = x3

4. f (x) = sin x5. f (x) = cos x6. f (x) = tan x

Graphs of cosecant, secant, and cotangent functions

Ideas:

1. csc t = 1sin t and sec t = 1

cos t

2. From the graphs of the sine and cosine function, we can drawcsc t and sec t.

3. If t → 0+ (i.e t goes to 0 from the positive direction), then1t →∞?

4. If t → 0− (i.e. t goes to 0 from the negative direction),

5. From the graph of tangent function, we can draw the graph ofcot t.

Recall csc t = 1sin t and graph csc t.

Here’s the graph of the sine function:

Graph of csc t

I Domain :

I Range :

I Period :

I Symmetry :

I Asymptotes:

Recall sec t = 1cos t and graph sec t.

Here’s the graph of the cosine function:

Graph of sec t

I Domain :

I Range :

I Period :

I Symmetry :

I Asymptotes:

From tan t to cot t

Graph of cot t

I Domain :

I Range :

I Period :

I Symmetry :

I Asymptotes:

Graphing Transformed Trig functions: f (t) = A cos(Bt)

We will vary A and B and see what properties of f (t) = cos(t)changes.

Question 1: Draw a sketch of f (t) = 2 cos t. (i.e. A=2, B=1)

What changes?

Graphing f (t) = A cos(Bt) with A < 0

Let A < 0 and see what characteristics of f (t) = cos(t) change.

Question: Draw a sketch of f (t) = − cos t.

What changes?

Graphing f (t) = A cos(Bt) with B 6= 1

Say we want to graph f (t) = cos(2t). (That is, A= 1, B =2.)(Table 4.6)

So what changed?

Period FormulaPeriod formula: Given f (t) = A cos(Bt) or f (t) = A sin(Bt),

p =2π

|B|Note: For f (t) = A tan(Bt) ⇒ p = π

t .Examples) Find the period for each of the following functions.

1. f (t) = cos(t)

2. f (t) = sin(t)

3. f (t) = cos(2t)

4. f (t) = sin(−2t)

5. f (t) = cos(12 t)

6. f (t) = sin(−12 t)

7. f (t) = 3 cos(2t)

8. f (t) = −3 tan(−2t)

Graphing y = A cos(Bt) when both A and B are not 1Draw a sketch of y = −2 cos(.5t).Plan:

1. Draw y = cos(.5t)2. Draw y = 2 cos(.5t)3. Draw y = −2 cos(.5)t by flipping it about the t-axis.

ExerciseDraw a sketch of y = −3 sin(2t).Plan:

1. Draw y = sin 2t2. Draw y = 3 sin 2t3. Draw y = −3 sin t by flipping it about the t-axis.

Graph of tan(34t)

Graph of sec(2t)

Graph of sec(2πt)

Vertical Shift

Rule: let k > 0. Then,

1. The graph of y = f (x) + k shifts the graph of y = f (x), kunits upward.

2. The graph of y = f (x)− k shifts the graph of y = f (x), kunits downward.

Example) Let’s try with f (x) = x2 + k, where k = 1 and k = −1.

Graphing y = A sin(Bt) + D

Plan:

1. Start with the graph of y = sin t.

2. Graph y = A sin(Bt).

3. Shift y = A sin(Bt) either up or down D units.

Example: Graph y = 2 sin(12 t) + 1.

Horizontal Translation

Rule: let k > 0. Then,

1. The graph of y = f (x + k) shifts the graph of y = f (x), kunits to the left.

2. The graph of y = f (x − k) shifts the graph of y = f (x), kunits to the right.

Example) Let’s try with f (x) = x2

y = A sin(Bt + C ) + D

Plan:

1. First write y = A sin(Bt +C ) +D as y = A sin(B(t + CB )) +D.

2. Graph y = A sin(Bt).

3. Shift y = A sin(Bt), CB units either to the left or right.

4. Shift y = A sin(B(t + CB )), D units either up or down.

Example: Graph y = sin(2t + π) + 1

y = A cos(Bt + C ) + D

Let’s graph y = cos(2t − π)− 1.Plan:

1. First, note that y = cos(2t − π)− 1 =

2. From the graph of y = cos t, graph y = cos 2t.

3. Shift y = cos 2t ( ) units to left/right.

4. Shift y = cos(2t − π) ( ) units up/down.

Summary: Standard form vs. Shifted form, and more

DefinitionGiven y = A sin(Bt), translations can be written as

Standard From Shifted form

y = A sin(Bt + C ) + D y = A sin

(B

(t +

C

B

))+ D

Here, C is called the phase angle, CB gives the horizontal shift (or

phase shift), and D vertical shift. The functions have amplitude|A|, period 2π

B .

Class Exercise 1 - Analyzing the transformation of asinusoidal function

Given y = 2 sin(π4 t + 3π

4

)+ 6, identify

I Max

I Min

I Center

I Amplitude

I Period

I Horizontal shift (or phase shift)

I Vertical Shift

Class Exercise 2

Graph the following functions (without a calculator)

1. y = sin(2t + π)− 1

2. y = 3 cos(t − π/4)

Class Exercise 3

Given y = 2 cos(π2 t + π

4

)− 3, identify

I Max

I Min

I Center

I Amplitude

I Period

I Horizontal shift (or phase shift)

I Vertical Shift

HW for Ch 6.4 (pg. 515)

14, 19, 20, 28, 30, 31, 37, 38, 46Show work to get credit. For instance, for problems 28 ∼ 46,identify amplitude, period, horizontal/vertical shifts.