Answer:
See explanation
Step-by-step explanation:
1. For the given graph:
A. Max is at -1 and
B. Min is at -5.
C. The midline of a sinusoidal function is the horizontal center line about which the function oscillates above and below. Hence, the midline has the equation
[tex]y=\dfrac{Max+Min}{2}=\dfrac{-1+(-5)}{2}=\dfrac{-6}{2}=-3[/tex]
D. The amplitude of a sinusoidal function is one-half of the positive difference between the maximum and minimum values of a function, so
[tex]Amplitude=\left|\dfrac{-1-(-5)}{2}\right|=2[/tex]
E. The period of a periodic function is the horizontal length of one complete cycle (the distance between two consecutive maximums), then the period is
[tex]\dfrac{3\pi}{2}-\dfrac{\pi}{2}=\pi[/tex]
F. The frequency of a trigonometric function is the number of cycles it completes in a given interval. This interval is generally 2π radians (or 360º) for the sine and cosine curves. Actually,
[tex]Frequency=\dfrac{2\pi }{\pi}=2[/tex]
G. The equation of the function is
[tex]f(x)=2\sin \left(2\left(x-\dfrac{\pi}{2}\right)\right)-3[/tex]
2. For the given function [tex]f(x)=3\sin \left(\dfrac{\pi x}{2}\right)+2[/tex]
A. Max is at 5 and
B. Min is at -1.
C. The midline of a sinusoidal function is the horizontal center line about which the function oscillates above and below. Hence, the midline has the equation
[tex]y=\dfrac{Max+Min}{2}=\dfrac{5-1}{2}=\dfrac{4}{2}=2[/tex]
D. The amplitude of a sinusoidal function is one-half of the positive difference between the maximum and minimum values of a function, so
[tex]Amplitude=\left|\dfrac{5-(-1)}{2}\right|=3[/tex]
E. The period of a periodic function is the horizontal length of one complete cycle (the distance between two consecutive maximums), then the period is
[tex]|5-1|=4[/tex]
F. The frequency of a trigonometric function is the number of cycles it completes in a given interval. This interval is generally 2π radians (or 360º) for the sine and cosine curves. Actually,
[tex]Frequency=\dfrac{2\pi }{4}=\dfrac{\pi}{2}[/tex]
G. The graph of the function is attached.
