Our objective is to satisfy the equation y = A·sin(B(x-C)) + D where A is amplitude, B is the frequency, C is the horizontal shift, and D is the vertical shift. Also remember our equation to calculate our period is 2pi/B.
Notice that the points have the same y-value and imagine this on the graph y= sinx. Regardless of what our sine function looks like, we know that the distance between the two points will be nB/2, where B is the frequency and n is a natural number. Our easiest solution would to have our two points be the start and end of a cycle (period of 10 since |-7-3|=10). Thus, our value of B is 2pi/10 = pi/5 units.
Since our two points are below the x-axis, let's make them minimums. That means that halfway between our two points (x=-2) is a maximum of the graph.
Since x=-2 is a maximum and x=3 is a minimum, that means that we find where our graph hits the x-axis by finding the middle point. Therefore, a zero is located at x= 0.5, making our horizontal shift (C) equal to 0.5 (remember y=sinx STARTS at zero).
Imagining this graph going from a maximum and decreasing from x=-2 to x=3. Our sine function must be negative (A will be negative)
Since we previously established that our two points will be minimums, this means that our amplitude would be 3, since they are this distance from the x-axis. A=-3
With this, we actually do not need a vertical shift, so D=0
One (of many) sinusoidal functions that goes through the points (−7,−3) and (3,−3) is
f(x) = -3(sin((pi/5)(x-0.5)))
As a tip, I would recommend using Geogebra or other graphing tools to familiarize yourself with how these variables change our trig functions.
