We will find the period and amplitude of the given sine function. Then, we will graph the function. Let's do these things one at a time.
Finding Period and Amplitude
Let's consider the general form of a sine function.
y= a sin bθ
Here | a| is the amplitude, | b| is the number of cycles in the interval from 0^(∘) to 360^(∘) , and 360^(∘)| b| is the period of the function. Let's now consider the given function.
y=4sin θ ⇔ y= 4 sin 1θ
In this equation we have that a= 4 and b= 1. Since |4|=4, the amplitude of the graph of the function is 4. This means that the maximum is 4 and the minimum is - 4.
Amplitude:& 4
Maximum:& 4
Minimum:& - 4
With b= 1, we know that there is 1 cycle from 0^(∘) to 360^(∘). The period of the graph of a sine function is 360^(∘)| b|. Since we already know that b= 1, we can find the period of the given function.
The period of the graph of our function is 360^(∘).
Sketching the Graph
To graph the given function, we should first identify the points where the θ-intercepts occur. These points can be identified for a sine function as shown in the table below. By substituting b=1, we can evaluate these points.
θ-intercepts
b=1
Simplify
(0,0)
(0,0)
( 0,0)
( 1/2 * 360^(∘)/b,0 )
( 1/2 * 360^(∘)/1,0 )
( 180^(∘),0)
(360^(∘)/b,0)
(360^(∘)/1,0)
( 360^(∘) ,0)
We can graph one cycle of the function by substituting calculated points and connecting them with a smooth curve. Remember that the maximum value is 4 and the minimum value is - 4. Let's do it!
Now, we can extend the graph by repeating the cycle to the left and right.
