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( (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( (x)/(4 \pi ) )`

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


`y=7sin( (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( (x)/(4 \pi ) )+3`