1. The cyclical pattern of symptoms in a patient is sinusoidal in nature. When graphed as a function of intensity, g(x), and time, x in months, the function is very similar to f(x)= sin x. Intensity is ranked on a scale of 0-a, where zero represents normal (min) and ‘a’ represents full-blown symptoms(max). The amplitude of the cyclical pattern is a/2 and the equation of the axis of the curve is y=a/2.
The period of the function is a/4 months. Using f(x)= sin x as a base function, the function of the symptoms has been horizontally translated a/8 months to the right.
a. Write a possible equation for the function g(x).
b. State all the transformations of the trigonometry function.
c. Sketch a graph of the function g(x), by hand on the interval 0 <x< a/2 (use key points and the table to graph).
2. Particular areas of this cyclical pattern are noteworthy. Use the equation found in question one.
a.Determine algebraically when the patient first ranked his intensity at 0.75a.
b. Determine when the patient ranked his intensity at 0.75a for the third time. c. Determine algebraically how high the patient ranked the intensity at 12 months.
a=15
the period of sin(kx) is 2π/k
They have told you the amplitude and vertical translation. You know it will look like
y = A + Bcos(k(x-h))
a. The equation for the function g(x) can be written as g(x) = (a/2)sin[(4π/a)(x - a/8)]. This equation represents a sinusoidal function with an amplitude of a/2, a period of a/4, and a horizontal translation of a/8 units to the right.
b. The transformations of the trigonometry function are as follows:
- Amplitude: The amplitude of the function is a/2, which determines the range of the function.
- Period: The period of the function is a/4, which determines the length of one complete cycle.
- Horizontal translation: The function has been horizontally translated a/8 units to the right, shifting the entire graph to the right by that amount.
- Vertical translation: The equation of the axis of the curve is y = a/2, which is the midline of the graph. The entire graph is shifted vertically by a/2 units upwards.
c. To sketch a graph of the function g(x), you can follow these steps:
1. Determine the key points: Find the x-values where the function crosses the midline (y = a/2) and the maximum/minimum points of one complete cycle of the function.
2. Create a table of values: Choose some x-values within the given interval (0 < x < a/2) and calculate the corresponding y-values using the equation g(x) = (a/2)sin[(4π/a)(x - a/8)].
3. Plot the key points and the points from the table on a graph.
4. Connect the points smoothly to form the graph, making sure to match the sinusoidal shape.
Note: Since the value of a is not provided in the question, it is not possible to provide exact numbers for the graph and calculations.