Question

write the equation of the trigonometric function shown in the graph.
y = \square \sin\left(\square x\
ight) + \square

Explanation:

Step1: Determine the amplitude

The amplitude \( A \) is half the distance between the maximum and minimum values of the function. The graph oscillates between \( y = 1 \) and \( y = -1 \)? Wait, no, looking at the graph, the solid line (the sine function we are finding) has a maximum of 1 and minimum of -1? Wait, no, wait the dashed line is another function. Wait, the equation is \( y = A \sin(Bx) + C \). Let's find the vertical shift \( C \) first. The midline of the sine function (the solid line) is at \( y = 0 \), because it oscillates around 0 (since it goes from -1 to 1, midline is 0). So \( C = 0 \).

Now, the amplitude \( A \): the maximum value of the solid sine wave is 1, minimum is -1, so amplitude \( A = 1 \)? Wait, no, wait maybe I misread. Wait the dashed line has a higher amplitude. Wait the solid line: let's check the period. The period of the solid sine wave: from \( 0^\circ \) to \( 180^\circ \)? Wait no, from \( 0^\circ \) to \( 360^\circ \)? Wait no, looking at the x-axis, the solid wave completes a cycle from \( 0^\circ \) to \( 360^\circ \)? Wait no, let's see the peaks. The first peak after 0 is at \( 90^\circ \), then next at \( 450^\circ \), so the period is \( 450^\circ - 90^\circ = 360^\circ \)? Wait, no, the standard sine function \( \sin(x) \) has a period of \( 360^\circ \) (or \( 2\pi \) radians). Wait, but let's check the coefficient \( B \). The formula for period \( T \) is \( T = \frac{360^\circ}{|B|} \) (when in degrees). If the period of the solid wave is \( 180^\circ \)? Wait no, let's look at the graph again. The solid wave: from \( 0^\circ \), it goes up to a peak at \( 90^\circ \), down to a trough at \( 270^\circ \), then up to a peak at \( 450^\circ \), etc. So the distance between two consecutive peaks (90° and 450°) is \( 450 - 90 = 360^\circ \), so the period \( T = 360^\circ \). For a sine function \( y = A \sin(Bx) + C \), the period \( T = \frac{360^\circ}{B} \) (in degrees). So if \( T = 360^\circ \), then \( B = \frac{360^\circ}{T} = \frac{360^\circ}{360^\circ} = 1 \)? Wait, but that would be the standard sine function. Wait but the amplitude: the solid wave has a maximum of 1 and minimum of -1, so amplitude \( A = 1 \), vertical shift \( C = 0 \). Wait, but let's confirm. Wait the equation is \( y = A \sin(Bx) + C \). Let's check the graph: at \( x = 0^\circ \), \( y = 0 \), which matches \( \sin(0) = 0 \). At \( x = 90^\circ \), \( y = 1 \), which is \( \sin(90^\circ) = 1 \), so \( A = 1 \), \( B = 1 \), \( C = 0 \). Wait, but let's check the period. The solid wave: from \( 0^\circ \) to \( 360^\circ \), it completes one cycle? Wait, at \( 360^\circ \), \( y = 0 \), which is the same as at \( 0^\circ \). At \( 180^\circ \), \( y = 0 \), which is the midpoint. So yes, the period is \( 360^\circ \), so \( B = 1 \), amplitude \( A = 1 \), vertical shift \( C = 0 \). Wait, but maybe I made a mistake. Wait the dashed line has a higher amplitude, but the solid line is the one we are finding. So putting it together: \( A = 1 \), \( B = 1 \), \( C = 0 \). Wait, but let's check again. The solid wave: maximum y-value is 1, minimum is -1, so amplitude 1. Midline at y=0, so C=0. Period is 360 degrees, so B=1 (since period T = 360/B => B=360/T=360/360=1). So the equation is \( y = 1 \sin(1x) + 0 \), so \( y = \sin(x) \). Wait, but let's confirm with the graph. At x=90°, y=1, which is sin(90°)=1. At x=270°, y=-1, which is sin(270°)=-1. At x=180°, y=0, which is sin(180°)=0. So that matches. So the amplitude is 1, B is 1, C is 0.

Step2: Confirm the values

- Amplitude \( A \): The maximum value of the sine wave is 1, minimum is -1, so \( A = 1 \).
- Period \( T \): The wave completes one cycle from \( 0^\circ \) to \( 360^\circ \), so \( T = 360^\circ \). Using the formula \( T = \frac{360^\circ}{B} \), we solve for \( B \): \( B = \frac{360^\circ}{T} = \frac{360^\circ}{360^\circ} = 1 \).
- Vertical shift \( C \): The midline of the wave is at \( y = 0 \) (since it oscillates around 0), so \( C = 0 \).

Answer:

\( y = 1 \sin(1x) + 0 \) (or simplified as \( y = \sin(x) \))