What is the graph of one cycle of a sine curve with the given characteristics? Using the form y = a sin bθ, what is an equation for the sine curve? amplitude = 4, period =1/4 pi , and a < 0.
To understand the characteristics of the sine curve graph, we can use the equation y = a sin bθ. From the given information, we have:
Amplitude (a) = 4
Period (T) = 1/4π
a < 0
The amplitude (a) represents the maximum value the curve reaches above and below the midline. So, in this case, the graph will oscillate between a maximum value of +4 and a minimum value of -4.
The period (T) represents the distance it takes for one complete cycle of the curve. In this case, the period is given as 1/4π. Since the general formula for the period of a sine curve is T = 2π/b, we can set it equal to the given value and solve for b:
1/4π = 2π/b
To simplify, we can cancel out π on both sides of the equation:
1/4 = 2/b
Now, we can cross multiply to find the value of b:
b = 2 / (1/4) = 2 * (4/1) = 8
Therefore, the value of b is 8.
Now we have the values of both a and b. Substituting these values into the equation y = a sin bθ, we get:
y = -4 sin (8θ)
So, the equation for the sine curve is y = -4 sin (8θ). This equation gives a curve with an amplitude of 4, a period of 1/4π, and a negative value of a.