find the equation for the cosine function that has an amplitude of 3/5, a period of 3pi/2 and a y-intercept of 5.
amplitude of 3/5:
3/5 cos(x)
period of 3pi/2:
3/5 cos (4/3 x)
y-intercept of 5:
22/5 + 3/5 cos (4/3 x)
http://www.wolframalpha.com/input/?i=22%2F5+%2B+3%2F5+cos+%284%2F3+x%29
Well, well, well, let's get mathematical, shall we? Since you're looking for a cosine function, we can start with the general form:
y = A * cos(Bx + C) + D
Now, you mentioned some specific details we need to incorporate. Our amplitude is 3/5, which means A = 3/5. The period is 3π/2, so we have B = 2π/(3π/2) = 4/3. Lastly, the y-intercept is 5, so D = 5.
So, our equation becomes:
y = (3/5) * cos((4/3)x) + 5
Voila! There you have it. Now go out there and graph that cosine function with style!
To find the equation for the cosine function with the given conditions, we can use the general form of a cosine function:
y = A*cos(Bx + C) + D
Where A represents the amplitude, B represents the frequency, C represents the phase shift, and D represents the vertical shift.
We are given:
Amplitude (A) = 3/5
Period = 3π/2
The amplitude is defined as half the distance between the maximum and minimum values of the function. In this case, the maximum value is (A + D) = (3/5 + 5) and the minimum value is (D - A) = (5 - 3/5).
So, the amplitude is ((3/5 + 5) - (5 - 3/5)) / 2 = (33/5)/2 = 33/10.
Since the function has a period of 3π/2, the frequency can be calculated using the formula:
Frequency (B) = 2π / Period = 2π / (3π/2) = 4/3.
The y-intercept is given as 5, so D = 5.
Now we have:
A = 33/10
B = 4/3
D = 5
Finally, let's determine the phase shift (C). Since the cosine function is not shifted left or right, the phase shift (C) is 0.
Replacing the corresponding values, the equation for the cosine function is:
y = (33/10)*cos((4/3)*x) + 5
To find the equation for the cosine function with the given properties, we can use the general form of a cosine function:
f(x) = A * cos(Bx + C) + D
Where:
A represents the amplitude,
B represents the frequency (related to the period),
C represents the phase shift,
and D represents the vertical shift or the y-intercept.
In this case, we are given:
Amplitude (A) = 3/5
Period = 3π/2
Y-intercept (D) = 5
1. Amplitude: The amplitude is the absolute value of A, which equals 3/5.
2. Period: The period is determined by the formula 2π/B, where B is the coefficient of x inside the cosine function. In this case, the period is 3π/2, so we have the equation:
2π/B = 3π/2
To solve for B, we can cross multiply:
2π * 2 = 3π * B
Simplifying, we get:
4π = 3πB
Divide both sides by 3π to isolate B:
4π / 3π = B
4/3 = B
So, B = 4/3.
3. Y-intercept: The y-intercept (D) is given as 5.
Now that we have all the values, we can write the equation:
f(x) = (3/5) * cos((4/3)x + C) + 5
The only missing value is C, which represents the phase shift. However, since a cosine function does not have a phase shift when the y-intercept is specified, C can be left as an arbitrary constant.