write the equation of the trigonometric function shown in the graph.\ny = \\square \\sin\\left(\\square…

Explanation:

Step1: Determine the amplitude

The amplitude ( A ) is half the distance between the maximum and minimum values. The maximum value of the sine wave (the solid line) is 1 and the minimum is -1? Wait, no, wait. Wait, the dashed line has max 3, min -1? Wait, no, the solid line: looking at the graph, the solid wave has peaks at y=1 and troughs at y=-1? Wait, no, let's check the vertical axis. The solid line: when x=0, it's at 0, then goes up to 1, down to -1, etc. Wait, the dashed line is a sine wave with amplitude 2 (from 1 to 3? Wait, no, the dashed line: at x=-90, it's at 3, x=0, it's at 1, x=90, it's at -1? Wait, no, maybe I misread. Wait, the solid line: let's find the midline first. The midline (vertical shift ( D )) is the average of max and min. For the solid line, the max is 1, min is -1, so midline is ( \frac{1 + (-1)}{2} = 0 )? Wait, no, wait the solid line: when x=0, it's at 0, then goes up to 1, down to -1, so midline is 0. The amplitude ( A ) is the distance from midline to max, so 1 - 0 = 1. So ( A = 1 ).

Step2: Determine the period and ( B )

The period of a sine function ( y = A \sin(Bx) + D ) is ( \frac{360^\circ}{|B|} ) (since we're in degrees here). Let's find the period of the solid line. From x=0 to x=180, does it complete a cycle? Wait, at x=0, it's at 0, going up. At x=180, it's at 0, going down? Wait, no, let's check the zeros. The solid line crosses the x-axis at 0, 180, 360, etc. So the period is 180 degrees? Wait, no, from 0 to 360: at x=0, 0; x=180, 0; x=360, 0. Wait, but the dashed line: its period is 360 degrees? Wait, the dashed line: at x=-180, 0; x=0, 0; x=360, 0. So dashed line has period 360, solid line: from 0 to 180, it goes up, down, so period 180? Wait, no, let's see the solid line: from x=0 to x=180, it goes from 0, up to 1, down to -1, back to 0? Wait, no, at x=90, is it at 0? Wait, no, maybe the solid line has period 180 degrees. So period ( T = 180^\circ ). Then ( T = \frac{360^\circ}{|B|} ), so ( 180 = \frac{360}{B} ), so ( B = \frac{360}{180} = 2 ). Wait, no: ( T = \frac{360}{B} ), so ( B = \frac{360}{T} ). If ( T = 180 ), then ( B = 2 ).

Step3: Vertical shift ( D )

As the midline is 0 (since max is 1, min is -1, midline is 0), so ( D = 0 ).

Wait, but let's confirm. The equation is ( y = A \sin(Bx) + D ). We have ( A = 1 ), ( B = 2 ), ( D = 0 ). Wait, but let's check with the graph. When x=0, ( y = 1 \sin(2 \cdot 0) + 0 = 0 ), which matches. At x=45, ( y = \sin(2 \cdot 45) = \sin(90^\circ) = 1 ), which is the peak. At x=90, ( y = \sin(180^\circ) = 0 ). At x=135, ( y = \sin(270^\circ) = -1 ), which is the trough. At x=180, ( y = \sin(360^\circ) = 0 ), which matches the zero at x=180. So that works.

Wait, but let's check the dashed line for comparison, but the problem is about the solid line. So putting it together: ( A = 1 ), ( B = 2 ), ( D = 0 ). So the equation is ( y = 1 \sin(2x) + 0 ).

Wait, but let's recheck the period. The solid line: from 0 to 180, it completes a full cycle (up, down, back to zero). So period is 180 degrees. So ( B = \frac{360}{180} = 2 ). Amplitude: from midline (0) to max (1) is 1, so ( A = 1 ). Vertical shift: midline is 0, so ( D = 0 ).

Answer:

( y = \boldsymbol{1} \sin\left( \boldsymbol{2} x \right) + \boldsymbol{0} )