Question

The graph of a sinusoidal function intersects its midline at (0,2) and then has a minimum point at (3,-6) Write the formula of the function, where x is entered in radians.

Answer 1

Answer:  y = -8sin(1/3)π x + 2

Step-by-step explanation:

The equation of a cosine function is:    y = A cos(Bx - C) + D    where

• Amplitude (A) is the distance from the midline to the max (or min)
• Period (P) is the length of one cosine wave   -->   P = 2π/B
• Phase Shift (C/B) is the horizontal distance shifted from the y-axis
• Midline (D) is the vertical shift. It is equal distance from the max and min

Midline (D) = 2

(0, 2) is given as a point on the midline.  We only need the y-value.

Horizontal stretch (B) = (1/3)π

The min is located at (3,-6) --> x = 3  which is half of the Period.  Thus the period (length of one wave) is 3·2 = 6 units.

$P=\dfrac{2\pi}{B}\qquad \rightarrow \qquad 6=\dfrac{2\pi}{B}\qquad \rightarrow \qquad B=\dfrac{2}{6}\pi\qquad \rightarrow B=\dfrac{1}{3}\pi$          

Phase Shift (C) = 0

The midline is on the y-axis so there is no horizontal shift.

Amplitude (A) = 7

The distance from the min to the midline is: A = -6 - 2 = -8

Equation

Input A = -8, B = (1/3)π, C = 0, and D = 2 into the sine equation.

            y = A sin(Bx - C) + D

            y = -8sin((1/3)π x - 0) + 2

            y = -8sin(1/3)π x + 2