Five trigonometric questions on cycles, periods and sinusoidal functions

[Sabrina102]

Hi, I was hoping to get some help on these five questions, I've been stuck on these and any help would be greatly appreciated!

1. Inflation rises and falls in a cyclical manner. If inflation is highest at 4.8% and lowest at 1.3%, what c for the equation y = a sin(k(x + d)) + c?
2. If the moon changes from a full moon to a half moon in 13 days, how many more days does it take for it to get back to a full moon and what would the period, k, equal?
3. Explain why a circle of radius 4 cm and a circle of radius 13 m have the same number of radians?
4. Explain why when you multiply by a number greater than 1 inside the argument of a trigonometric function, the period of the function decreases and when you multiply by a number less than 1, the period increases.
5. For the function y = 4sin (pi/ 4 x - pi/2 ) - 3 list when the average rate of change is positive, negative, and zero, considering the beginning of the interval to be x = 4.

 

[MarkFL]

Hello, and welcome to MHB! (Wave)

1.) I would begin by stating:

$$-1\le\sin(k(x+d))\le1$$

$$-|a|\le |a|\sin(k(x+d))\le |a|$$

$$-|a|+c\le |a|\sin(k(x+d))+c\le |a|+c$$

And so, from the information given in the problem regarding the minimum and maximum values of inflation the sinusoid is modelling, we must have:

$$c+|a|=4.8\%$$

$$c-|a|=1.3\%$$

Add these equations, and then solve for \(c\)...what do you find?