4.5
TLW graph sine curves.
Values of Sine and Cosine
X
0
90
180
270
360
sin x
0
1
0
-1
0
cos x
1
0
-1
0
1
The points in the table are critical values; intercepts, max and min.
For this course please set your calculator to degrees.
Graph of y = sin x
Graph of y =cos x
Amplitude and Period
• Amplitude: half the distance between max and min; what does
the graph “waffle” between.
• Period: distance on horizontal axis to complete one crest and
one trough
Changing Amplitude
• y=a sinx
or y=a cos x
• “a” changes the amplitude.
• Graph y=2sinx and y=2cosx
– What’s the period?
– What’s the amplitude?
• What happens if a < 0?
Identify the amplitude, period and any
reflections.
• y=3sinx
• y=-2cos x
• y=.5sinx
Changing the Period
•
•
•
•
•
Normally the period is 360⁰, but it can be adjusted.
y=sin bx
or y=cos bx
The b adjusts the period.
Take a look at y = sin 2x and compare it to y = sinx
Do the same for y = cos 2x and y = cos x
• How do we find the period?
– Period =
360
𝑏
If you change the period, then other things
must change.
•
•
If you change the period then you change the intercepts, maximum
and minimum points.
Steps to adjust your critical values:
1. Interval length between critical points =
𝑝𝑒𝑟𝑖𝑜𝑑
4
This cuts the graph into four equal intervals .
2.
3.
4.
5.
Add interval length to 1st point to get 2nd point
Add 2nd point + interval = 3rd point
Do it again
And again, now you have all 5 points.
Give It A Go!
• y = sin .5x
• y = 2 cos 3x
Vertical Shift
• y=sin x +c
•
•
•
•
•
y=cos x +c
Examine the graphs:
y = sin x +1
y = cos x + 1
y =sin x -2
y =cos x -2
Summary
• y = a sin bx + c OR y = a cos bx + c
– a changes amplitude
– b changes period
– c moves the graph up and down
Putting It Together
• Graph: y = 2sin 2x -1
• Graph: y = 0.5cos 0.5x +2
• Steps:
1. Find amplitude,
period and interval
2. Graph what you
know
3. Move up or down
if necessary
Graph This!
• y = -3 cos x -2
Find the equation.
(180, 1.5)
90 180 270
360 450 540 630 720
(540, -4.5)
Find the equation
90
180 270
360 450
540 630
720
Evaluate
• Find f(30) when f(x) = 2sinx – 0.5
• Find g(120) when g(x) = 0.5cos 2x +5
Solve
• 2sinx .5 = 1, 0 ≤ 𝑥 ≤ 450
• .5cos 2x + 1 = .5, 0 ≤ 𝑥 ≤ 360
Negative Domain
• Graph y = sinx, −360 ≤ 𝑥 ≤ 0
• Graph y = cosx, −360 ≤ 𝑥 ≤ 0
Solve
• .4sin 0.5x + 1 = .75, −702 ≤ 𝑥 ≤ 0
• 2cos x – 1 = -2, −360 ≤ 𝑥 ≤ 0
•
•
The height of the tide in a local harbor was measured. It was found
that the water rose 3 m above the mean sea level and fell 3 m below
the mean sea level in one complete cycle over a 12-hour period.
Plot the graph over an 18- hour period
Time
0
3
6
9
12
15
18
H(t)
•
•
•
Assuming that the height of water with respect to mean sea level can
be modeled by a sine function, determine the amplitude, period and
the amount of vertical shift for this function.
Determine the equation of the sine function in the form H(t)=asin bt +
c, with bt measured in degrees and where t is in hours. H(t) = height
with respect to mean sea level
If the mean sea level occurred at midnight, use the equation of the
sine function to determine the height of the tide at 8 a.m.
• The air temperature (℃) for a 24-hour period at a particular
location can be modeled by the function T=25-5cos10t, where
t is the number of hours after midnight and 10t is measured in
degrees.
a. Complete the table of values for the temperature function.
Time
0
4
8
12
16
20
24
Temp.
a. Draw the graph of T for 0° ≤ 𝑡 ≤ 24°
b. Use the graph to estimate
i.
ii.
iii.
iv.
The temperature at 10 am
The time interval when the temperature was greater than 28℃
The increase in temperature between 10 am and midday
The time of day when the temperature was at its maximum
Assignment
• P.215 #1-3 every other one of a-i, 4-7