Sunday, October 23, 2011
Practice for graphing sine waves.
If you want to visualize the graphs you are asked to draw, the website Wolfram|Alpha has an instruction you can type called "plot". Here is the example of plot 3sin(2x/3).
Here are some examples for practice without graphs. You can do that yourself on Wolfram|Alpha. The answers are in the comments.
a] f(x) = 2 + 4sin(x/2)
period = ______
amplitude = _______
maximum of f is ______, reached when x = _______.
minimum of f is ______, reached when x = _______.
b] k(x) = 4 + 2cos(3x)
period = ______
amplitude = _______
maximum of k is ______, reached when x = _______.
minimum of k is ______, reached when x = _______.
c] q(x) = 1 - 3sin(5x)
period = ______
amplitude = _______
maximum of q is ______, reached when x = _______.
minimum of q is ______, reached when x = _______.
Answers in the comments.
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a] f(x) = 2 + 4sin(x/2)
ReplyDeleteperiod = 2pi/(1/2) = 4pi.
amplitude = 4.
Sine reaches max when x = pi/2, but the function is sin(x/2), so x/2 = pi/2 when x = pi. The value will be 2 + 4 = 6.
maximum of f is 6, reached when x = pi.
There are other values of x where f(x) = 6.
Sine reaches min when x = -pi/2 or 3pi/2, but the function is sin(x/2), so x/2 = -pi/2 when x = -pi or x/2 = 3pi/2, so x = 3pi. The value will be 2 - 4 = -2.
minimum of f is -2, reached when x = -pi.
minimum of f is -2, reached when x = 3pi.
There are other values of x where f(x) = -2.
b] k(x) = 4 + 2cos(3x)
period = 2pi/3
amplitude = 2
Cosine reaches a maximum when x = 0, so k(0) = 4 + 2 = 6. It also reaches a max at x = 2pi, so 3x = 2pi when x = 2pi/3
maximum of k is 6, reached when x = 0.
maximum of k is 6, reached when x = 2pi/3.
Cosine reaches its minimum at -pi or pi, among other values, and that minimum will be 4 - 2 = 2. This happens at x = pi/3 or x = -pi/3.
minimum of k is 2, reached when x = pi/3.
minimum of k is 2, reached when x = -pi/3.
c] q(x) = 1 - 3sin(5x)
period = 2pi/5
amplitude = 3
Because sine is multiplied by -3, q(x) will reach a minimum when sin(5x) = 1 and a maximum when sin(5x) = -1. The lowest value is 1 - 3 = -2 and the highest value is 1 + 3 = 4.
maximum of q is 4, reached when x = pi/10.
maximum of q is 4, reached when x = -3pi/10.
minimum of q is -2, reached when x = 3pi/10.
minimum of q is -2, reached when x = -pi/10.