Extra Practice Transforming cosine Functions
With amplitude (a),
period (p),
horizontal shift to the right (h),
and vertical shift up (k).
period (p),
horizontal shift to the right (h),
and vertical shift up (k).
Period:
The normal cosine function has a period of 2pi.
To find the period (p):
Find the distance between the tops of the waves. In this case, the distance is pi (~3.14...). So, the period is pi.
The normal cosine function has a period of 2pi.
To find the period (p):
Find the distance between the tops of the waves. In this case, the distance is pi (~3.14...). So, the period is pi.
Amplitude
The amplitude is the measure of the radius of the corresponding circle.
To find the amplitude (a):
Measure the distance from the top to the bottom of the wave and divide by two. In the following graph, the distance from the top of the wave to the bottom of the wave is 4. So, the amplitude is 4/2 = 2.
Horizontal Shift
To find the horizontal shift (h):
Case 1: You can see the origin. If you can see the origin, find the horizontal distance from the origin. This is your best option when you are first starting out. When you throw in a period change and a horizontal shift, things can get tricky. The distance of the highest point from 0 is pi/4. So, in this case, h=pi/4
To find the horizontal shift (h):
Case 1: You can see the origin. If you can see the origin, find the horizontal distance from the origin. This is your best option when you are first starting out. When you throw in a period change and a horizontal shift, things can get tricky. The distance of the highest point from 0 is pi/4. So, in this case, h=pi/4
Case 2: You cannot see the origin AND there is not a period change.. Find some other point on the graph where you know where either the top or bottom of the wave is supposed to be. Remember, in the cosine function, cos(0) = 1, cos(pi/2) = 0, cos(pi) = -1, cos(3pi/2) = 0, and cos(2pi) = 1. In the graph below, just for fun let's look at the the bottom of the wave. It's measuring 5pi/4 instead of just pi. How far apart are those values?
5pi/4 - pi = 5pi/4 - 4pi/4 = pi/4. So, h = pi/4 (the same as case 1).
5pi/4 - pi = 5pi/4 - 4pi/4 = pi/4. So, h = pi/4 (the same as case 1).
Vertical Shift
To find the vertical shift:
Take the average of the y-value for the top of the wave and the y-value for the bottom of the wave. In the graph above, the vertical shift is (1+-1)/2 = 0/2 = 0. In the graph below the vertical shift is (4+2)/2 = 6/2 = 3. Below, k = 3.
Take a moment to play around.
To get the pi symbol in Desmos, just write "pi". It will automatically convert the value for you.
Practice:
For each of the following graphs, find each of the values: a, p, h, and k one at a time. Put them all together at the end to get the final function.
For each of the following graphs, find each of the values: a, p, h, and k one at a time. Put them all together at the end to get the final function.
Graph 1:
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Equation 1:
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Graph 2:
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Equation 2:
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Graph 3:
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f(x) = 2*cos((2pi/pi)(cosx - pi/3)) + 1
a = 2 p = pi h = pi/3 k = 1 |
Graph 4:
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a = 0.5
p = 2pi/3 h = pi/6 k = 0.5 |
