Question

A sinusoidal function whose period is 1/2, maximum value is 10, and minimum value is -4, has a y-intercept of 3. What is the equation of the function described?

Answer 1

Period of sin wave = 1/2
Period of parent sine function is 2π.

This means the x value of sine function (sin(x)) has to be divided by 4π to get a period of 1/2 because 2π/4π = 1/2

Thus, our function will be of the form: $sin( \frac{x}{4 \pi })$

Maximum Value = 10 
Minimum Value = -4

Amplitude of the function = (Maximum Value - Minimum Value)/2
So, amplitude of the function is 7 

Thus, the function will be of the form:

$7sin( \frac{x}{4 \pi } )$

The function has a y-intercept other than zero, this means the function will be of the form:

$y=7sin( \frac{x}{4 \pi } )+k$

The y-intercept is 3. This means for x=0, y=3. Using the values, we get:

$3=7sin(0)+k \\ \\ 3=k$

Thus the equation of the sinusoidal function becomes:

$y=7sin( \frac{x}{4 \pi } )+3$