If a sinusoidal function has a local maximum at (3,8) and the next local minimum at (7,-2),
1) What is the equation of a cosine function that has a graph characterized in the statement above
2) What is the equation of a sine function that has a graph characterized in the statement above
y = a cos (k x - b) + c
a and c are easy because c is average of 8 and -2 or 3 and a is thus 5
y = 5 cos (k x - b) + 3
cos is max of 1 at k x - b = 0
k*3-b = 0
so
b = 3 k
so y = 5 cos (k x - 3 k) + 3
cos is min of -1 at k x - b = pi
kx - 3k = pi but x = 7 here so
7 k - 3 k = pi
k = pi /4
so now
y = 5 cos pi/4( x - 3 ) + 3
or y = 5 cos (pi x/4 - 3 pi/4)
now you do it for sine wave :)
To find the equation of a cosine function with the given characteristics:
1) Each complete cycle of a cosine function consists of a local maximum, a local minimum, and a return to the local maximum position. From the given information, we know that one complete cycle of the cosine function spans from the local maximum at (3,8) to the local minimum at (7,-2).
2) First, let's determine the amplitude of the cosine function. The amplitude is half the difference between the maximum and minimum values. In this case, the amplitude would be (8 - (-2))/2 = 5.
3) Next, we need to determine the period of the cosine function, which is the horizontal length of one complete cycle. The period can be calculated by finding the difference between the x-values of the two adjacent local maxima or minima. In this case, the period would be 7 - 3 = 4.
4) The standard equation of a cosine function is given as:
y = A * cos(B * (x - C)) + D
Where:
A = amplitude
B = 2π / period
C = horizontal shift (phase shift)
D = vertical shift
5) Plugging in the values we found earlier, the equation becomes:
y = 5 * cos((2π / 4) * (x - C)) + D
6) To find the value of C, we need to determine the horizontal shift (phase shift). In this case, the first local maximum occurs at x = 3, so we substitute this value into the equation:
8 = 5 * cos((2π / 4) * (3 - C)) + D
7) Plugging in the second local maximum value, (-2,7), we get:
-2 = 5 * cos((2π / 4) * (7 - C)) + D
8) By solving these two equations simultaneously, we can determine the values of C and D. Once we have C and D, we can substitute them back into the equation to obtain the final equation of the cosine function.
To find the equation of a sine function with the given characteristics, we can follow the same steps as above, but with a few differences:
1) The sine function starts at a local minimum and reaches a local maximum at the midpoint of the period.
2) The amplitude, period, and phase shift calculations remain the same.
3) The standard equation of a sine function is given as:
y = A * sin(B * (x - C)) + D
Again, plug in the values you determined earlier for A, B, and C, and substitute them into the equation along with the given y-values to solve for D, and finally, obtain the equation of the sine function.