Identify the maximum and minimum values of the function y = 8 cos x in the interval [–2π, 2π]. Use your understanding of transformations, not your graphing calculator.
seriously?
cos x is max of 1 when x = -2pi , zero and +2 pi
it is -1 when = -pi and + pi
To find the maximum and minimum values of the function y = 8 cos x in the interval [–2π, 2π], we need to understand the properties of the cosine function.
The cosine function has a maximum value of 1 when the angle is 0° or 2π radians (cos(0) = cos(2π) = 1). It has a minimum value of -1 when the angle is 180° or π radians (cos(180°) = cos(π) = -1).
In the given function y = 8 cos x, the coefficient of cosine is 8. This value only scales the amplitude (height) of the cosine function and does not affect the maximum and minimum values.
Therefore, in the given interval [–2π, 2π], the maximum value of y = 8 cos x is 8, and the minimum value is -8.
To identify the maximum and minimum values of the function y = 8 cos x in the interval [–2π, 2π], we need to understand the nature of the cosine function and how it is affected by transformations.
The cosine function, by default, has a maximum value of 1 and a minimum value of -1. However, when we introduce a coefficient in front of the cosine function, it affects the amplitude (vertical stretch or compression) of the graph.
In this case, we have y = 8 cos x, which means the graph of y = cos x is stretched vertically by a factor of 8. Hence, the maximum value of this function will be 8, and the minimum value will be -8.
Now, to determine the interval [–2π, 2π], we need to find the x-values within this range that correspond to the maximum and minimum values of y.
To find the maximum value, we need to find the x-value where y is at its peak, which occurs when the cosine function has a value of 1. Taking into account the stretch factor of 8, we can set up the equation:
8 cos x = 1.
Dividing both sides by 8, we get:
cos x = 1/8.
To solve for x, we take the inverse cosine (also known as arccosine) of both sides:
x = arccos(1/8).
Using a calculator (since we are not allowed to use a graphing calculator for this solution), we find that x ≈ 1.444.
So, one maximum value is at x = 1.444. However, we still need to consider the range [–2π, 2π].
Since the cosine function is a periodic function, it repeats its pattern every 2π units. Therefore, the maximum value at x = 1.444 will repeat again after an interval of 2π.
To find the minimum value, we follow a similar process. We need to find the x-value where y is at its minimum, which occurs when the cosine function has a value of -1. Again, considering the stretch factor of 8, we set up the equation:
8 cos x = -1.
Dividing both sides by 8, we get:
cos x = -1/8.
Taking the inverse cosine of both sides, we find:
x = arccos(-1/8).
Again, using a calculator, we find that x ≈ 1.972.
So, one minimum value is at x = 1.972. Like the maximum values, the minimum value will repeat after an interval of 2π, due to the periodic nature of the cosine function.
In conclusion, the maximum values of the function y = 8 cos x in the interval [–2π, 2π] occur at x = 1.444 and x = 1.444 + 2π. The minimum values occur at x = 1.972 and x = 1.972 + 2π.