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 = 2, period =2 pi , and a < 0.

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

The graph of one cycle of the sine curve with the given characteristics would look like:

Graph of sine curve with given characteristics

To write an equation for this sine curve using the form y = a sin bθ:

Since amplitude = 4, a = -4 (because a < 0)

Since period = 1/4π, we can find b using the formula:

period = 2π/b

1/4π = 2π/b

b = 8

Therefore, the equation of the sine curve is:

y = -4 sin 8θ

To understand the graph of a sine curve, you need to know its amplitude, period, and whether the amplitude is positive or negative.

In this case, the given characteristics are:
Amplitude (a) = 2
Period (T) = 2π
a < 0

The amplitude (a) represents the maximum displacement or distance from the midpoint of the graph to either the highest or lowest point. Since a < 0, we know that the graph will be reflected across the x-axis.

The period (T) represents the horizontal length of one complete cycle of the graph. Here, the period is 2π, which means that the graph will complete one cycle every 2π units in the x-direction.

To find the equation of the sine curve in the form y = a sin(bθ), we can substitute the given values of amplitude and period into the equation. Using the provided information:

a = -2 (since a < 0),
T = 2π

The equation becomes:
y = -2 sin (bθ)

Now, to determine the value of b, which represents the frequency or number of cycles in a given interval, we can use the formula: b = 2π / T.

Substituting the given period value (T = 2π) into the formula, we find:
b = 2π / (2π) = 1

Therefore, the equation of the sine curve with the given characteristics is:
y = -2 sin(θ)

The graph of this equation will have an amplitude of 2, a period of 2π, and will be reflected across the x-axis since the amplitude (a) is negative.

I shall use x instead of whatever character you chose to copy/paste.

The graph of y = a sin bx has
amplitude = |a|
period = 2π/b

so, what do you get?