Final answer:
The period of -3sin(1/3x) +4 is 6π units.
Explanation:
In order to find the period of a trigonometric function, we need to first understand what the period represents. The period of a function is the length of one complete cycle, or the distance between two consecutive maximum or minimum points on the graph. In this case, we are dealing with a sine function, which has a period of 2π.
To find the period of -3sin(1/3x) +4, we can use the formula T = 2π/|b|, where b is the coefficient of x in the function. In this case, b = 1/3, so the period is T = 2π/|1/3| = 2π * 3 = 6π units.
To further understand why this is the period, we can break down the function into its individual components. Starting with the inner function, 1/3x, we can see that the coefficient of x is 1/3. This means that the graph of the function will be compressed horizontally, with a period of 2π/|1/3| = 6π. This is because the graph will complete one cycle in a smaller distance, resulting in a shorter period.
Next, we have the -3sin(x) term. The negative sign in front of the sine function means that the graph will be reflected over the x-axis. This results in the maximum and minimum points being switched, but does not affect the period.
Finally, we have the +4 term, which shifts the entire graph upwards by 4 units. This does not affect the period either, as it only changes the y-values of the points on the graph.
Combining all these factors, we can see that the period of -3sin(1/3x) +4 is 6π units, which is the length of one complete cycle on the graph. This means that the graph will repeat itself every 6π units, or every 6π radians.
In conclusion, the period of -3sin(1/3x) +4 is 6π units. This is determined by the coefficient of x in the inner function, which compresses the graph horizontally, and is not affected by any other terms in the function. It is important to understand the concept of period in trigonometric functions, as it helps us to accurately graph and analyze these types of functions.