A cosine function has a period of 3, a maximum value of 20, and a minimum value of 0. The function is a reflection of its parent function over the x-axis.
Which function could be the function described?
f(x)=10cos(3x)−10
f(x)=10cos(2π3x)+10 <my choice
f(x)=−10cos(2π3x)+10
f(x)=−10cos(3x)+10
There must be some typos in the above.
The correct answer is
f(x) = -10cos((2π/3)x) + 10
To determine which function could be the function described, let's analyze the given information:
1. The period of a cosine function is given by 2π/|B|, where B is the coefficient of x in the cosine function.
2. The maximum value of a cosine function is equal to its amplitude plus its vertical shift, while the minimum value is equal to the vertical shift minus the amplitude.
3. A reflection over the x-axis occurs when the amplitude is negative.
Given that the period is 3, the maximum value is 20, and the minimum value is 0, we can deduce the following:
1. The period of the cosine function is 3. Therefore, 2π/|B| = 3.
Solving this equation, we find that |B| = 2π/3.
2. The maximum value is 20, and the minimum value is 0. Since the amplitude is half the difference between the maximum and minimum values, we find that the amplitude is (20-0)/2 = 10.
The vertical shift is the average of the maximum and minimum values, which is (20+0)/2 = 10.
3. The function is a reflection over the x-axis, meaning the amplitude should be negative.
Now let's analyze the given answer choices:
a) f(x) = 10cos(3x) - 10: This function is not a reflection because the amplitude is positive.
b) f(x) = 10cos(2π/3x) + 10: This function satisfies all the given conditions. The period is 3, the amplitude is 10 (positive as required), and the function is a reflection over the x-axis due to the positive amplitude.
c) f(x) = -10cos(2π/3x) + 10: This function does not have the correct amplitude. The negative amplitude implies a reflection, but the vertical shift is also negative.
d) f(x) = -10cos(3x) + 10: This function has the correct amplitude and reflection. However, the period of 3 is not satisfied.
Therefore, the function that could be the function described is f(x) = 10cos(2π/3x) + 10, which is answer choice b).
To determine which function could be the described function, we need to compare the given information with the properties of a cosine function.
1. Period: The period of a cosine function is the length of one complete cycle. In the given problem, the period is stated as 3.
2. Reflection: A reflection of a function over the x-axis changes the sign of the function.
3. Maximum and minimum values: The maximum value of the cosine function is the amplitude added to the vertical shift, and the minimum value is the vertical shift subtracted from the amplitude.
Now, let's analyze each choice using these properties:
a) f(x) = 10cos(3x) - 10:
This function has a period of 3, which matches the given information. However, the sign in front of the cosine function is negative. Since the description states that the function is a reflection over the x-axis, the sign of the cosine function should be positive. Therefore, this choice does not match the given conditions.
b) f(x) = 10cos(2π/3x) + 10:
This function also has a period of 3 since 2π/3 is equivalent to 2π/3 * 3/1 = 2π. The sign in front of the cosine function is positive, indicating a reflection over the x-axis. This function satisfies the given period and reflection conditions, and the maximum value is 20 and the minimum value is 0 (as 10 + 10 = 20 and 10 + 10 = 0). Therefore, this choice is consistent with the description.
c) f(x) = -10cos(2π/3x) + 10:
This function has a period of 3 and a reflection over the x-axis, which match the given information. However, the maximum value is 10 and the minimum value is 20, which contradicts the provided values. Therefore, this choice is not correct.
d) f(x) = -10cos(3x) + 10:
This function has a period of 3, which matches the given information. However, the sign in front of the cosine function is negative, indicating a reflection over the x-axis. Since the given description states that the function is a reflection, the sign should be positive. Therefore, this choice does not satisfy the requirements.
Based on the analysis, the correct choice is f(x) = 10cos(2π/3x) + 10.