Sketch the graph of the function (Include two full periods. Find one complete cycle. y=10cos πx/6
x= 2x3xy. Y=23.xy=9.5
To sketch the graph of the function y = 10cos(πx/6), we need to understand the characteristics of the cosine function and how to represent the given behavior on a graph.
The general form of the cosine function is y = A*cos(Bx + C) + D, where A, B, C, and D are constants that affect the amplitude, frequency, phase shift, and vertical shift of the graph, respectively. In our case, A = 10, B = π/6, C = 0, and D = 0.
1. Amplitude (A): The amplitude determines the height of the function. In this case, A = 10, so the graph will oscillate between -10 and 10 on the y-axis.
2. Frequency (B): The frequency of a function determines how many complete cycles are completed over a specific unit on the x-axis. In our case, B = π/6, which means one complete cycle is completed in 2π / (π/6) = 12 units on the x-axis.
3. Phase shift (C): The phase shift determines any horizontal shift in the graph. Here, C = 0, so there is no phase shift.
4. Vertical shift (D): The vertical shift determines any upward or downward shift of the graph. D = 0, so there is no vertical shift.
Given these characteristics, we can draw the graph:
- Start by identifying the domain and range of the function. Since cosine oscillates indefinitely, the domain of our function is all real numbers. The range will be between -10 and 10 in this case.
- Plot two full periods on the coordinate plane. Since one cycle is completed in 12 units, we can choose any two points that are 12 units apart and complete the graph between them. For simplicity, let's choose x = 0 and x = 12.
* At x = 0, y = 10cos(π*0/6) = 10cos(0) = 10.
* At x = 12, y = 10cos(π*12/6) = 10cos(2π) = 10.
- Plot the two points (0, 10) and (12, 10) on the graph.
- Notice that the graph starts at its maximum point (peak) at x = 0, reaches its minimum point (valley) at x = 6, returns to the maximum point at x = 12, and repeats this pattern infinitely. The graph has a symmetrical pattern around the x-axis.
- Now, we can plot other key points that lie on the graph.
* At x = 3, y = 10cos(π*3/6) = 10cos(π/2) = 0. So, plot the point (3, 0) on the graph.
* At x = 6, y = 10cos(π*6/6) = 10cos(π) = -10. So, plot the point (6, -10) on the graph.
* At x = 9, y = 10cos(π*9/6) = 10cos(3π/2) = 0. So, plot the point (9, 0) on the graph.
- Connect these points smoothly with a smooth curve. Make sure the graph maintains its symmetrical shape around the x-axis.
You have now sketched the graph of the function y = 10cos(πx/6). It should resemble a cosine wave oscillating between -10 and 10, with two full periods shown on the graph.