Adv. Algebra help needed asap! Please help!

Question

For a sinusoidal function in the form y = A sin Bx + C ory = A cos Bx + C, the period is 1, the maximum is 2, the equation of the midline is y = –4, the y-intercept is –10, and A < 0.

What is the equation of the function?

Answer 1

 

Let's deal with each of those pieces of information in turn.

 

 

 

The period is 1. This will help us figure out the B term (the horizontal stretch). In general:

 

Period = B/(2π)

 

So:

 

1 = B/(2π)

2π = B

 

 

 

The equation of the midline is y = -4. This will help us figure out the C term (the vertical shift). Normally, the midline is at y = 0, so we've shifted the function down 4. That means:

 

C = -4

 

 

 

The maximum is 2. Now that we know the vertical shift C, this will help us figure out the A term (the vertical stretch). This is also called the amplitude, and it's just the maximum height above the midline (or depth below the midline) that the function reaches.

 

A = Maximum - Midline

A = 2 - (-4)

A = 2 + 4

A = 6

 

This also tells us that our minimum value is -4 - 6 = -10.

 

 

 

A < 0. This tells us that we should add a negative sign onto our A-value. It also indicates a reflection over the x-axis.

 

A = -6

 

 

 

The y-intercept is -10. This will help us figure out whether to use sine or cosine. Normally, sine is at its midline value at x = 0 and cosine is at its maximum value at x = 0. Remember, though, that our function has been reflected over the x-axis. So, sine will still be at its midline value, but cosine will be at its minimum. Since our minimum is -10 and our function is at its minimum at x = 0, we'll want to use the cosine function.

 

 

 

Let's put that all together:

 

y = Acos(Bx) + C

y = -6cos(2πx) - 4