Find the formula for each of the functions graphed below

9) The period of this graph is 10 because the curve repeats every 10 units. The amplitude is 4 because that is the distance from the midline (y = 0) to the top/bottom of a curve. In order to find the equation, you need to start out with the most basic equation there is. This graph is more similar to a sin(x) graph, so we will use this:

f(x) = sin(x)

The amplitude is basically the the vertical stretch or shrink. Adding the amplitude to the function will make it:

f(x) = 4 sin(x)

To find the horizontal stretch or shrink, we have to apply the period, which is 10. The horizontal stretch/shrink is
(2pi)/(horizontal shrink or stretch) = period
--> 2pi / ? = 10
--> ? = pi/5
Therefore, the horizontal shrink/stretch is pi/5. Adding this to the equation:

f(x) = 4 sin(pi/5)(x)

Next, we use the phase shift. The graph is shifted over one to the left from a sin graph, so this is the new equation.

f(x) = 4 sin(pi/5)(x + 1) <--final equation

10) The period of this graph is 6. The amplitude is 3. The vertical stretch/shrink = amplitude, and the horizontal stretch/shrink = pi/3. The horizontal phase shift is 1 to the left.

f(x) = 3 sin(pi/3)(x + 1)

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