which of the following is an equation for the graph? a. $2 \\sin(x + \\frac{\\pi}{4}) - 1$ b. $2 \\sin(x…

Explanation:

Step1: Analyze the amplitude

The general form of a sine function is ( y = A\sin(B(x - C)) + D ), where ( |A| ) is the amplitude. Looking at the graph, the maximum value is ( 1 ) and the minimum is ( -3 ). The amplitude ( A=\frac{1 - (-3)}{2}=2 ), so ( A = 2 ), which matches all options.

Step2: Analyze the vertical shift ( D )

The midline (vertical shift) is the average of the maximum and minimum values. Midline ( D=\frac{1+(-3)}{2}=-1 ). So the function should have ( + D=-1 ), which means we can eliminate options C and D (they have ( +1 ) and ( +1 ) respectively).

Step3: Analyze the phase shift

Now we have options A and B left: ( 2\sin(x+\frac{\pi}{4})-1 ) and ( 2\sin(x - \frac{\pi}{4})-1 ). The general phase shift is ( C ) (for ( y = A\sin(B(x - C))+D )). Let's check the starting point or a key point. The standard ( \sin x ) has a zero crossing at ( x = 0 ), but here, let's check the graph at ( x = -\frac{\pi}{4} ) (for phase shift analysis). Alternatively, let's look at the graph's behavior. The graph seems to have a phase shift to the left (since compared to ( 2\sin x-1 ), the graph is shifted left). The formula ( y = 2\sin(x + \frac{\pi}{4})-1 ) has a phase shift of ( -\frac{\pi}{4} ) (left by ( \frac{\pi}{4} )), and ( y = 2\sin(x - \frac{\pi}{4})-1 ) has a phase shift of ( \frac{\pi}{4} ) (right by ( \frac{\pi}{4} )). Looking at the graph, when ( x = -\frac{\pi}{4} ), let's see the value. Alternatively, check the point where the sine function crosses the midline. The midline is ( y=-1 ). For the standard ( \sin x ), crossing midline at ( x = 0 ), but here, the crossing at midline ( ( y=-1 )) should be shifted. Let's take the option A: ( 2\sin(x+\frac{\pi}{4})-1 ). Set ( y=-1 ), then ( 2\sin(x+\frac{\pi}{4})-1=-1 \implies \sin(x+\frac{\pi}{4}) = 0 \implies x+\frac{\pi}{4}=k\pi \implies x = k\pi-\frac{\pi}{4} ). For ( k = 0 ), ( x=-\frac{\pi}{4} ), which seems to match the graph's crossing at midline. For option B: ( 2\sin(x - \frac{\pi}{4})-1=-1 \implies \sin(x - \frac{\pi}{4}) = 0 \implies x - \frac{\pi}{4}=k\pi \implies x = k\pi+\frac{\pi}{4} ), which doesn't match the graph's crossing. So the correct option is A.

Answer:

A. ( 2 \sin\left(x + \frac{\pi}{4}\right) - 1 )