Find a possible formula for the trigonometric function whose values are in the following table (note the values in the table may be slightly rounded):
x 0.00 0.17 0.33 0.50 0.67 0.83 1.00 1.17 1.33 1.50 1.67 1.83 2.00
g(x) 4.00 5.50 6.60 7.00 6.60 5.50 4.00 2.50 1.40 1.00 1.40 2.50 4.00
SKETCH A GRAPH !!
add all g values and divide by 13 to get mean
high = 7-mean
low = mean -1
amplitude A = (hing - low)/2
y = A sin (w t + phi)
w T = 2 pi in one period from 7 to 1 and back up so use twice the time from 7 down to 1 for T and w = 2 pi/T
To find a possible formula for the trigonometric function, we can look for patterns in the given values. From the table, we can observe that the values of g(x) seem to oscillate between a maximum and a minimum value.
Let's consider the maximum value of g(x), which occurs at x = 0.50, 1.50, and 2.00. These values are 7.00, 1.00, and 4.00, respectively. These maximum values form a pattern that decreases by 3.00 each time.
Similarly, let's consider the minimum value of g(x), which occurs at x = 0.00, 1.00, and 2.00. These values are 4.00, 2.50, and 4.00, respectively. These minimum values form a pattern where the first and third values are the same, and the middle value is different.
Based on these observations, a possible formula for the given trigonometric function g(x) could be:
g(x) = A + B * cos(C * x + D)
Where A is the average of the maximum and minimum values (3.50), B is the amplitude of the oscillation (3.00), C is a constant that controls the period of the oscillation, and D is a phase shift.
To determine the constant C in the formula, we can calculate the difference between the x-values of two consecutive maximum values (0.50 to 1.50 or 1.50 to 2.00) or two consecutive minimum values (0.00 to 1.00 or 1.00 to 2.00). The difference between these x-values is 1.00. Since the period of a cosine function is given by 2π / |C|, we have:
2π / |C| = 1.00
Solving for C, we get:
|C| = 2π
Choosing C = 2π, we have:
g(x) = 3.50 + 3.00 * cos(2π * x + D)
To determine the phase shift D, we can observe that the maximum value occurs at x = 0.50, which corresponds to a phase angle of 0 radians. This means that for D = 0, the maximum value aligns with x = 0.50. Therefore, D = 0.
The final formula for the trigonometric function can be written as:
g(x) = 3.50 + 3.00 * cos(2π * x)
To find a possible formula for the trigonometric function g(x) using the given table, we can analyze the pattern in the values.
Looking at the values in the table, we can observe that g(x) increases and decreases in a periodic manner. This suggests that g(x) might be a sinusoidal function.
Since the values of x in the table are increasing in equal increments, we can assume that g(x) is some form of the sine or cosine function.
To determine which trigonometric function and what specific formula might fit the given data, we can calculate the sinusoidal transformation parameters:
1. Amplitude (A): By examining the maximum and minimum values of g(x) in the table, we can deduce that the amplitude is 3 (half the difference between the maximum and minimum values of g(x)).
2. Period (P): The period can be calculated by finding the difference between two consecutive x values with the same g(x) value. In this case, it repeats every 0.33 units (0.67 - 0.33 = 0.34, which is close to two times the average increment of x).
3. Phase shift (ϕ): To find the phase shift, we need to determine the horizontal shift of the function. In this case, the function reaches its maximum value at x = 1.00. Therefore, the phase shift is 1.00 units to the right.
4. Vertical shift (D): Looking at the minimum and maximum values of g(x), it is clear that the function is symmetric about y = 4.00. Therefore, there is no vertical shift; D = 4.00.
With the calculated values for A, P, ϕ, and D, we can write the formula for the trigonometric function g(x):
g(x) = A * sin((2π/P) * (x - ϕ)) + D
Plugging in the values we obtained:
g(x) = 3 * sin((2π/0.33) * (x - 1.00)) + 4.00
This formula should give you a possible representation of the trigonometric function g(x) based on the given table.
SKETCH A GRAPH !!
add all g values and divide by 13 to get mean
high = 7-mean
low = mean -1
amplitude A = (hing - low)/2
y = A sin (w t + phi)
w t = 2 pi in one period from 7 to 1 and back up so w T = pi from 7 to 1