Answer
Blue Curve: $y = 1 + 2sin(x)$
Purple Curve: $y = 2sin(x/2) + 2cos(x/2)$
A: $(-2\pi/3, 1-\sqrt 3)$
B: $(\pi/3, 1+\sqrt 3)$
C: $(2\pi/3, 1+\sqrt 3)$
D: $(5\pi/3, 1-\sqrt 3)$
Work Step by Step
Blue Curve: We know that the curve sin(x) intersects the origin halfway between its peak and valley.
Because the given formula has 1 added to it, that same point on the curve will intersect the y-axis at y = 1. For this portion of the problem, we can ignore the amplitude of the curve.
Purple Curve: Use the process of elimination to determine that the purple curve is $y = 2sin(x/2)+2cos(x/2)$
To identify the points, set the equations of the curves equal to each other and solve for x. Then plug those x-values into one of the equations to find the y-values.
1) $1+2sin(x) = 2sin(x/2)+2cos(x/2)$
2) $1 + 2sin(x) - 2sin(x/2) - 2cos(x/2) = 0$
3) $u = x/2$ --> (use trig identities) $1 - 2cos(u) -2sin(u)+4cos(u)sin(u) = 0$
4) Factor: $(2cos(u)-1)(2sin(u)-1)=0$
5) Solve each factor: $2cos(u) - 1 = 0$ or $2sin(u)-1=0$
6) $u = \pi/3, 5\pi/3, \pi/6, 5\pi/6$
7) Substitute $u = x/2$: $x = 2\pi/3, 10\pi/3, \pi/3, 5\pi/3$
We can get rid of $x=10\pi/3$ because it is out of the given bound $(-2\pi
