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 = 1/4pi, and a < 0.
The graph of one cycle of the sine curve with amplitude 4 and period 1/4pi is shown below:
[asy]
import TrigMacros;
size(200);
xmin=-pi/2;
xmax=pi/2;
ymin=-4.5;
ymax=4.5;
xaxis(-pi/2, pi/2, Ticks("$$\frac{\pi}{8}", 1.57), Arrows);
yaxis(-4.5, 4.5, Ticks("$$4", 1), Arrows);
real f(real x) {return -4*sin(8*x);}
draw(reflect((0,0),(1,1))*graph(f,-pi/16,pi/16),red);
draw(reflect((0,0),(1,1))*graph(f,pi/16,3pi/16));
draw(reflect((0,0),(1,1))*graph(f,3pi/16,5pi/16));
draw(reflect((0,0),(1,1))*graph(f,5pi/16,7pi/16));
draw(reflect((0,0),(1,1))*graph(f,7pi/16,9pi/16));
[/asy]
One cycle of the sine curve goes from its maximum value of -4 to its minimum value of 4, then back to its maximum value, so it oscillates between these two y-values. It completes one oscillation in 1/4pi units of θ, so it completes 8 cycles in 2pi units of θ (the standard period for the sine function).
Using the given information, we can write the equation of this sine curve as:
y = -4 sin (8θ)
To graph a sine curve with the given characteristics, we need to determine the values of amplitude (a), period (b), and whether a is less than 0.
Given:
Amplitude (a) = 4
Period (b) = 1/4π
a < 0
The general equation for a sine curve is y = a sin(bθ).
1. Find the value of b:
Period (T) = 2π/b
1/4π = 2π/b
b = 8
2. Write the equation using the given values:
Since a < 0, we can substitute a with -4:
y = -4sin(8θ)
The graph of this sine curve will have an amplitude of 4, a period of 1/4π, and the curve will be inverted since a is less than 0.