Sketch the graph of the function (Include two full periods. Find one complete cycle. Show your work. y=-3+5cos πt/12
there are graphing sites online
fooplot is one
To sketch the graph of the function y = -3 + 5cos(πt/12), we need to follow a few steps:
Step 1: Determine the period of the function.
The period of a cosine function is given by 2π divided by the absolute value of the coefficient of t inside the cosine function. In this case, the coefficient is π/12. So, the period T is given by:
T = 2π / (π/12) = 2π * (12/π) = 24
Step 2: Determine the amplitude of the function.
The amplitude of a cosine function is the absolute value of the coefficient in front of the cosine function. In this case, the coefficient is 5, so the amplitude is 5.
Step 3: Find key points and shape of the graph.
The general shape of the cosine function is a wave that oscillates between its maximum and minimum values. The maximum value is equal to the amplitude, which is 5, and the minimum value is equal to the negative of the amplitude, which is -5.
To find key points, we can divide the period T into four quarters:
- From t = 0 to t = T/4, we will find the graph at its maximum value, 5.
- From t = T/4 to t = T/2, we will find the graph decreasing from the maximum value, 5, to the middle value, -3.
- From t = T/2 to t = 3T/4, we will find the graph at its minimum value, -5.
- From t = 3T/4 to t = T, we will find the graph increasing from the minimum value, -5, to the middle value, -3.
Now, let's calculate these key points.
i) At t = 0: y = -3 + 5cos(π(0)/12) = -3 + 5cos(0) = -3 + 5 = 2
So, our first key point is (0, 2).
ii) At t = T/4: y = -3 + 5cos(π(T/4)/12) = -3 + 5cos(π/3) = -3 + 5(1/2) = -3 + 2.5 = -0.5
Our second key point is (T/4, -0.5).
iii) At t = T/2: y = -3 + 5cos(π(T/2)/12) = -3 + 5cos(π/6) = -3 + 5(√3/2) = -3 + 5√3/2
Our third key point is (T/2, -3 + 5√3/2).
iv) At t = 3T/4: y = -3 + 5cos(π(3T/4)/12) = -3 + 5cos(π/2) = -3 + 5(0) = -3 + 0 = -3
Our fourth key point is (3T/4, -3).
v) At t = T: y = -3 + 5cos(π(T)/12) = -3 + 5cos(π) = -3 - 5 = -8
Our fifth key point is (T, -8).
Step 4: Sketch the graph.
Using the key points we calculated, we can plot the graph of the function. The graph will repeat the shape of one complete period within two full periods. So, we can draw our graph based on the key points and repeat it accordingly.
Here is the sketch of the graph of the function (y = -3 + 5cos(πt/12)) for one complete cycle within two full periods:
Remember, the x-axis represents values of t, and the y-axis represents values of y. The graph starts at (0, 2) and repeats the pattern every 24 units on the x-axis. The amplitude is 5 in both the positive and negative y-directions, and the graph oscillates between -8 and 2.