Question

a what is the period of f?
b what is the midline for f?
enter a mathematical expression more...
preview
c what is the amplitude of f?
preview
d complete the function formula for f (enter \theta\ for θ.)
(1 - 3/2
preview
f(θ) = 2 sin(theta + pi)
preview

Explanation:

Step1: Find the period

The general form of a sine function is \(y = A\sin(B\theta - C)+D\). The period \(T=\frac{2\pi}{|B|}\). From the graph, we can observe that the function repeats over an interval of \(2\). So the period \(T = 2\).

Step2: Find the mid - line

The mid - line of a sine function \(y = A\sin(B\theta - C)+D\) is \(y = D\). The mid - line is the horizontal line that the graph oscillates around. Looking at the graph, the mid - line is \(y=\frac{1+( - 3)}{2}=-1\)

Step3: Find the amplitude

The amplitude \(A\) of a sine function \(y = A\sin(B\theta - C)+D\) is given by \(|A|\). The amplitude is half the distance between the maximum and minimum values of the function. \(A=\frac{1-( - 3)}{2}=2\)

Step4: Write the function formula

We know \(A = 2\), \(B=\pi\) (since \(T=\frac{2\pi}{B}=2\Rightarrow B=\pi\)), \(C =-\pi\) (phase shift), \(D=-1\). Using the formula \(y = A\sin(B\theta - C)+D\), we get \(y = 2\sin(\pi\theta+\pi)-1\)

Answer:

a. \(2\)
b. \(y=-1\)
c. \(2\)
d. \(f(\theta)=2\sin(\pi\theta+\pi)-1\)