“Graphing cosine is as easy as transforming the parent cosine graph y=cosx around.”
From geometry and
trigonometry, you have studied that the cosine of an angle is the
opposite side over the hypotenuse side. Also, you learn from the unit
circle and references angles that sin(300) = ½, or cos
= -1, etc… Don't remember? Here's a quick recap.
=>
Alright, now on to graphing y = cosx
Plot the points and graph….
x
|
0
| pi/2 | |||
Cos(x)
|
1
|
0
|
-1
|
0
|
1
|
 Here is the basic cosine graph: y=cos x.
It’s
also known as the “parent graph of cosx”.
·
The amplitude,
the distance from x-axis and the “hump”, is 1.
·
The period, the
part unique part of the graph that was repeated, is
o From this, notice
that at each fourth of the period – interval, where we plot our point. So
each interval is 2pi/4, which is pi/4
o Cosine graph
characteristic: 1)stay, move up by amplitude
2)go right by interval, stay
3)go right by interval, move down by amp.
4)go right by interval, stay
5)go right by
interval, move up by amp.
This is the parent graph for cos(x). But here's is a more general
form: y = acosx(bx - h)+k
________________________________________________________________________
"a"
symbolizes the vertical stretch of
the graph; it can also be used to tell the amplitude or height of the “hump”.
Graphs of
y=cos(x) vs. y =2cos(x) vs. y= -cosx
 |
0
|
pi/2
|
pi
|
3pi/2
|
2pi
|
|
2Cos(x)
|
2
|
0
|
-2
|
0
|
2
|
-Cos(x)
|
-1
|
0
|
1
|
0
|
-1
|
**Do you see how when a = 2, the vertical
stretch is 2 so the amplitude change to 2? It's true! However, when a = -1, the
vertical stretch is -1, causing the graph to “flip” over the x-axis.
"b"
symbolizes the horizontal stretch of
the graph, a change in b would affect the period. To FIND the new period, take 2pi and divide by b to get the new period.
· Graphs of y=cos(x) vs. y=cos(2x)
x
|
0
|
pi/4
|
pi/2
|
3pi/4
|
pi
 |
Sin(2x)
|
1
|
0
|
-1
|
0
|
1
|
**When b = 2 (increased), the period is 2pi/2 = pi (decreased) Because the period change to pi, The new interval is period/4 = pi/4
"h" in the equation is the horizontal shift of
the graph; it helps identify how many units the graph shifts, and whether it’s right
of left
·
Graphs of
y=cos(x) vs. y =cos(x - pi/2)
x
|
pi/2
|
pi
|
3pi/2
|
2pi
|
5pi/2
|
Sin(x)
|
1
|
0
|
-1
|
0
|
1
|
**Because h changed into pi/2,
the graph shifts pi/2 unit to the right
k
represents the vertical shift of the
graph; it helps tell how many units the graph shift, and whether it move up or
down.
·
Graphs of y=cos(x) vs. y=cos(x) +1
x
|
0
|
pi/2
|
pi
|
3pi/2
|
2pi
|
Sin(x)
|
2
|
1
|
0
|
1
|
2
|
**Since k became (1), the graph will
shift 1 unit up.
____EXAMPLE GRAPH!!!____________________________________________
Graph y = 2cos (pi (x-1))-1
1) Remind yourself to always check for all of the
changes first
i.
a => 2, the amplitude is 2
ii.
b => pi, so period is (2pi)/pi = 2, interval is 2/4 = 1/2
iii.
h => 1 the graph is shifted right by 1
iv.
k
=> -1, the graph is shifted down by 1
2) Place your starting point from the origin to the new location shifting it 1 unit down and 1 unit right. (1, -1)
3) Next, use
the cos characteristic to plot the rest of the points. Interval and amplitude
are very very helpful in this stage.
i.
Stay, go up 2
ii.
Move right by ½ , stay
iii.
Move right by ½ , go
down by 2
iv.
Move right by ½ , stay
v.
Move right by ½ , go up
by 2
** Remember, the basic characteristic of a cosine graph
is shifting right by interval, moving UP by amp. Why is it move DOWN here? Well
simple, because it's a negative amplitude, instead of going up, we go down.
4) Draw the
graph by connecting the points & check if the amplitude and the period
match that of the original problem.
x
|
1
|
3/2
|
3
|
5/2
|
3
|
y
|
1
|
-1
|
-3
|
-1
|
1
|
-TrigInADay
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