Consider the function y=7+5cos(8x)
What is the amplitude?
What is the period?
What is the largest value the graph takes?
Calculus? You have a very easy teacher.
if y=Acos(bt+p)+C
A is amplitude, b is angular frequency
and of course, b= 2PI/period
p is phase shift
C is vertical shift, or vertical bias.
To determine the amplitude, period, and largest value of the graph of the function y = 7 + 5cos(8x), we can use the general form of a cosine function:
y = A + Bcos(Cx + D)
In this case, A = 7, B = 5, C = 8, and D = 0. Now, let's explain how to find each of the requested values:
1. Amplitude:
The amplitude (A) represents the maximum displacement from the central line. For a general cosine function, the amplitude is the absolute value of B. Therefore, the amplitude of the given function is |5| = 5.
2. Period:
The period (T) of a cosine function is the distance between two consecutive peaks (or troughs) of the graph. The period can be calculated using the formula:
T = (2π) / |C|
In our function, C = 8. So, the period is:
T = (2π) / |8| = π / 4
Therefore, the period of the function y = 7 + 5cos(8x) is π / 4.
3. Largest value:
To find the largest value of the graph, we need to determine the vertical shift from the central line, which is represented by A. In our function, A = 7. Thus, the largest value the graph takes is 7 + 5 = 12.
To summarize:
- The amplitude is 5.
- The period is π / 4.
- The largest value the graph takes is 12.