f(x) is a sinusoidal curve with maximum at f(0)=10 and next minimum at f(6)=0. Find the equation

Question

I need help please

Answer 1

First, let's figure out the vertical shift and vertical stretch (amplitude).

 

Since the sine wave has its maximum at 10 and its minimum at 0, its middle is at (10+0)/2 = 10/2 = 5, their average. That means the sine wave has been shifted up 5.

 

Since the sine wave reaches 10 - 5 = 5 units above its middle, its amplitude is 5. That's a vertical stretch of 5.

 

 

Now, let's work on the sine wave's horizontal shift and horizontal stretch (period).

 

Normally, the sine wave reaches its first maximum at x = π/2, but here, it happened at x = 0. That means the sine wave's maximum happened π/2 early, so it's been shifted left π/2.

 

Since the sine wave has a maximum at x = 0 and a minimum at x = 6, its period is 6(2) = 12. That's how long it will take for the sine wave to reach its maximum again. The stretch coefficient for the period, b, is:

 

(2π)/b = period

(2π)/b = 12

2π = 12b

(2π)/12 = b

π/6 = b

 

That's a horizontal stretch of π/6.

 

 

Let's take all that information and put it into the sine's equation. A normal sine wave has the equation:

 

y = sin(x)

 

A transformed sine wave has the equation:

 

y = asin(bx + c) + d

 

where a = vertical stretch (amplitude), b = horizontal stretch (period), c = horizontal shift (positive is left, negative is right), and d = vertical shift (positive is up, negative is down).

 

For us, a = 5, b = π/6, c = π/2 (shift left), and d = 5 (shift up). Plug those numbers in to get your equation:

 

y = 5sin((π/6)x + π/2) + 5