which function has an amplitude of 2 and a period of p?
a. f(x)=1/2cos2x
b. f(x)=2cospx
c. f(x)=1/2cospx
d. f(x)=2cos2x
Clearly it has to be either a) or d)
period: 2π/k = period
2π/k = p
k = 2π/p
I will assume you want p to be π
so k = 2
looks like d)
uggub
Well, if you're looking for a function with an amplitude of 2 and a period of p, we can rule out options a and c since they have an amplitude of 1/2. So, we're left with options b and d.
Now, let's look at the period. The period is the distance between two consecutive peaks or troughs of the function. If the period is p, that means the function repeats itself every p units. Since option d has a coefficient of 2 for x (2x), it means it will complete two cycles in the interval of p units. So, the correct choice is d.
But hey, don't worry, if you chose b, you were just 2π radians away from the right answer! Keep trying, you're on the right track!
To determine the function with an amplitude of 2 and a period of p, we can use the general form of the cosine function, which is:
f(x) = A cos(Bx)
Where A represents the amplitude and B represents the frequency (or period).
Comparing the given options:
a. f(x) = 1/2 cos(2x): The amplitude is 1/2, not 2. So, this option is not correct.
b. f(x) = 2 cos(px): The amplitude is 2, and the frequency is p. This matches the conditions given.
c. f(x) = 1/2 cos(px): The amplitude is 1/2, not 2. So, this option is not correct.
d. f(x) = 2 cos(2x): The frequency is 2, not p. So, this option is not correct either.
Therefore, the correct option is b. f(x) = 2 cos(px), which has an amplitude of 2 and a period of p.
To determine which function has an amplitude of 2 and a period of p, we need to understand the properties of sine and cosine functions.
The general form of a cosine function is given by f(x) = A * cos(Bx), where A represents the amplitude and B represents the frequency or period.
The amplitude of a function is the absolute value of the coefficient A and determines the maximum distance from the mean or midline. The period is the distance required to complete one full cycle of the function.
Let's analyze each option:
a. f(x) = 1/2cos(2x)
The amplitude of this function is 1/2, not 2.
b. f(x) = 2cos(px)
The amplitude of this function is 2, which satisfies the given condition. However, the period of this function is 2π/p, not simply p. Therefore, this option does not have a period of p.
c. f(x) = 1/2cos(px)
Similar to option (a), the amplitude of this function is 1/2, not 2.
d. f(x) = 2cos(2x)
The amplitude of this function is 2, which satisfies the given condition. Additionally, the period of this function is 2π/2 = π, which can be expressed as p. Therefore, this option has the desired properties of an amplitude of 2 and a period of p.
Based on the analysis, option d. f(x) = 2cos(2x) is the correct answer.