What is the quarter point of y=1/2sin(2x+π)

Answer 1

I've not heard of anyone officially calling it a "quarter point" but a sine wave can be broken into 4 equal portions if you consider one period as a "whole" and if you think about the graph as "starting at (0, 0)":

I think that's a little weird because who is to say that's where it "starts" but, anyway, I'll go with that assumption.

So, assuming a quarter point is the x-value at which the graph has a maximum value then we need to know the starting position as well as the period.

To do so, first put the function into standard "transformation" format. That format is:

Where "A" is the vertical stretch or "Amplitude", "B" defines the horizontal stretch (the period = 2π/B), "C" is the horizontal shift where "-" means shift right and "+" means shift left, and "D" is the vertical shift where "+" is shift up and "-" is shift down.

We can but the function into this standard format by factoring out the number in front of the "x":

y = 1/2sin[2(x + π/2)]

Now you can read that the graph is shifted from its standard starting position at (0, 0) left by π/2 and the period is 2π/2 = π.

Since the period is π then 1/4 of the period is π/4.

Since the graph is shifted left by π/2, it's "starting location" is -π/2, 0. That means the x-value of the quarter point is -π/2 + π/4 = -π/4.

The y-value of the quarter point would be the maximum value of the function defined by the amplitude. So y_max = 1/2

So by the definition used, the quarter point is (-π/4, 1/2).