Praxis Prep
Graphs of Trig Functions
By 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-cosineamplitude.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-cosineamplitude.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.
Amplitude
The 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|
Graph of Sine x - with varying
amplitudes
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.
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 of
p(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.
2
8
D. 3
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,
=
2
5
=
2 

10 5
 Horizontal Shrinking
Ex: y = cos 10 x
 Horizontal Shrinking
x
Ex: y = cos 2
=
2
 4
1
2
 Horizontal Stretching
PRS Question
Consider:
y = 3 cos 8x
What is the period of this function ?
A.

8
B.
3
8
C.

4
D.
8
PRS Question
Consider:
y = 3 cos 8x
What is the period of this function ?
A.

8
B.
3
8
C.

4
D.
8
Period = 2pi/b Here b = 8. How many cycles between [0-2pi]
TI Time
Graph 2 cycles of y = 3 cos 8x

2 

8
4
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.
Ti Time
Note that there are 8 cycles between 0 and 2π.
Ti Time
Graph 2 cycles of y = 4 sin x/3

2
 6
1
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.
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 Time
Graph the curve y = sin(2x + 1)
Let’s begin by graphing y = sin 2x
Period is π so we can capture two cycles
using 
2
Solve 2x + 1 = 0 and x = -1/2
Hence phase shift to the left by -1/2
Now graph y = sin(2x +1)
Ti Time
Now 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
Practice
1. Computer the amplitude of
y
3
sin x  cos x
3
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:
1
y  2  2 cos( x  2 )
3
Compute the amplitude:
Compute the phase shift:
How many cycles in 12π:
How many cycles in 2π: