Praxis Prep

Graphs of Trig FunctionsBy Tim

Nov. 15, 2007

Hard Problem

Sine Curve

• Dom of sine function is all real numbers. Range is [-1, 1]• Period is 2pi. Sine curve is symmetric w.r.t the origin.• X-intercepts are at 0, pi, and 2pi

Cosine Curve• Dom of cosine function is all real number. Range is [-1, 1]• Period is 2pi. Cosine curve is symmetric w.r.t the y-axis.• X-intercepts are at pi/2 and 3pi/2

Demonstration – Unit Circle and Trig curves Go to the website listed below

Select Sin and Start

http://www.intmath.com/Trigonometric-graphs/1_Graphs-sine-cosine-amplitude.php

Record values of t as the sin wave crosses the x-axis and when the wave reaches its max and min values.

Notice From Demonstration

That the shapes of the sine and cosine curves are regular (they repeat after the wheel has gone around once)? We say such curves are periodic.

The period is the time it takes to go through one cycle and then start over again.

That the sine and cosine graphs are almost identical, except shifted sideways from each other?

That in the interactive, the radius of the circle is 80 units and the curve went up to 80 units and down to -80 units on the vertical axis?

This quantity of a sine and cosine curve is called the amplitude of the graph. This indicates how much energy the graph has. Higher amplitude means greater energy.

http://www.intmath.com/Trigonometric-graphs/1_Graphs-sine-cosine-amplitude.php

PRS Praxis

The best representation to highlight periodic behavior is a

A.Line graph

B.Paragraph describing the amplitude

C.Trigonometric equation

D.Diagram of the unit circle

PRS Praxis

The best representation to highlight periodic behavior is a

A.Line graph

B.Paragraph describing the amplitude

C.Trigonometric equation

D.Diagram of the unit circle

Things that occur regularly are good candidates for modeling with trig equations. Many biological rhythms can be modeled with sine and cosine function.

AmplitudeThe a in both of the graph types

y = a sin x and y = a cos x

affects the amplitude of the graph.

The amplitude is the distance from the "resting" position (otherwise known as the mean value or average value) of the curve. In the interactive simulation, the amplitude was 80 units. In the unit circle the amplitude is 1.

Amplitude is always a positive quantity. We could write this using absolute value signs. For the curves y = a sin x and y = a cos x, amplitude = |a|

We start with y = sin x.It has amplitude = 1 and period = 2π.

Now let's look at y = 5 sin x. I have used a different scale on the y-axis.This time we have amplitude = 5 and period = 2π.

Graph of Sine x - with varying amplitudes

And now for y = 10 sin x.Now, amplitude = 10 and period = 2π.

For comparison, and using the same y-axis scale, here are the graphs of p(x) = sin x, q(x) = 5 sin x and r(x) = 10 sin x on the one set of axes.Note that the graphs have the same period (which is 2π) but different amplitude.

Graph of Cosine x - with varying amplitudes

Now let's have a look at the graph of y = cos x.

We note that the amplitude = 1 and period = 2π.

Similar to what we did with y = sin x above, we now see the graphs ofp(x) = cos x q(x) = 5 cos x r(x) = 10 cos x on one set of axes, for comparison:

PRS Question

Consider: y = 3 cos 8x

What is the amplitude of this function ?

A. 1/3 B. 30 C. D. 32

8

PRS Question

Consider: y = 3 cos 8x

What is the amplitude of this function ?

A.1/3 B. 30 C. 2pi / 8 D. 3

In the periodic function y = a cos bx

The amplitude or “height” or “strength” of the curve is denoted by a.

Graphs of y = a sin bx and y = a cos bx

The b in both of the graph types•y = a sin bx •y = a cos bx

affects the period (or wavelength) of the graph. The period is the distance (or time) that it takes for the sine or cosine curve to begin repeating again. The period is given by:

Note: As b gets larger, the period decreases.

Changing the Period

First, let's look at the graph of y = 10 cos x,

As we learned, the period is 2π

Now let's look at y = 10 cos (3x). Note the 3 inside the cosine term.

Notice that the period is different. (The amplitude is 10 in each example.) This time the curve starts to repeat itself at x = 2π/3. This is consistent with the formula for period:

Now let's view the 2 curves on the same set of axes. Note that both graphs have an amplitude of 10 units, but their period is different.

Note: b tells us the number of cycles in each 2π.

For y = 10 cos x, there is one cycle between 0 and 2π (because b = 1).

For y = 10 cos 3x, there are 3 cycles between 0 and 2π (because b = 3).

In General

If 0 < b < 1, the period of y = sin bx is greater than 2pi and represents a horizontal stretching of the graph.

If b >1, period of y = cos bx is less than 2pi and represents a horizontal shrinking of the graph.

Ex: y = sin 5x,

= Horizontal Shrinking

Ex: y = cos 10 x

= Horizontal Shrinking

Ex: y = cos

= Horizontal Stretching

5

2

510

2

2

x

4

212

PRS Question

Consider: y = 3 cos 8x

What is the period of this function ?

A.

B.

C.

D.

8

8

3

4

8

PRS Question

Consider: y = 3 cos 8x

What is the period of this function ?

A.

B.

C. Period = 2pi/b Here b = 8. How many cycles between [0-2pi]

D.

8

8

3

4

8

TI TimeGraph 2 cycles of y = 3 cos 8x

Here, b = 8, so the period is 2π/8 = π/4. To draw 2 cycles, we will need to graph from 0 to π/2 along the x-axis.

48

2

Ti Time

Note that there are 8 cycles between 0 and 2π.

Ti Time

Graph 2 cycles of y = 4 sin x/3

Keep in mind what we’re doing…Horizontal____

Remember that the period is the distance it takes the sine or cosine curve to begin repeating again.

6

312

Graphs of y = a sin(bx + c) and y = a cos(bx + c)

Here we meet the following 2 graph types:y = a sin(bx + c)y = a cos(bx + c)

Both b and c in these graphs affect the phase shift (or displacement), given by:

The phase shift is the amount that the curve is moved in a horizontal direction from its normal position.

The displacement will be to the left if the phase shift is negative, and to the right if the phase shift is positive.

There is nothing magic about this formula. We are just solving the expression in brackets for zero; bx + c = 0.

Ti TimeGraph the curve y = sin(2x + 1)

Let’s begin by graphing y = sin 2x

Period is π so we can capture two cycles using

Solve 2x + 1 = 0 and x = -1/2

Hence phase shift to the left by -1/2

Now graph y = sin(2x +1)

2

Ti TimeNow let's consider the phase shift. Using the formula above, we will need to shift

our curve by:

This means we have to shift the curve to the left (because the phase shift is negative) by 0.5. Here is the answer (in blue). I have kept the original y = sin 2x (in dotted gray) so you can see what's happening.

Vertical Translation

The final type of translation is the vertical translation caused by the constant d in the equations:

y = d + a sin (bx - c)y = d + a cos (bx - c)

Graph y = 3 cos 2x

Now graph y = 2 + 3 cos 2x

What happens?

Hard Problem

Practice1. Computer the amplitude of

Hint: We’re not looking for amplitude at a point. Amplitude refers to the strength of the curve at it’s maximum value. When do sin x and cos x have their maximum value?

2. Consider the periodic function

Compute the period: Compute the amplitude:

Compute the phase shift:

How many cycles in 12π: How many cycles in 2π:

xxy cossin3

3

)23

1cos(22 xy