Question

which function describes the graph below? graph of a curve options: y = 8cos(x) + 3; y = 4cos(x) + 3; y = 4sin(x) + 3

Explanation:

Step1: Analyze the general form of trigonometric functions

The general form of a sinusoidal function is \( y = A\sin(Bx - C) + D \) or \( y = A\cos(Bx - C) + D \), where \( |A| \) is the amplitude, \( \frac{2\pi}{|B|} \) is the period, \( \frac{C}{B} \) is the phase shift, and \( D \) is the vertical shift.

Step2: Determine the vertical shift (D)

From the graph, the midline (vertical shift) is at \( y = 3 \), so \( D = 3 \).

Step3: Determine the amplitude (|A|)

The distance from the midline to the maximum (or minimum) value is the amplitude. Looking at the graph, the maximum value is 7 and the midline is 3, so the amplitude \( |A| = 7 - 3 = 4 \). So \( A = \pm 4 \).

Step4: Determine the type of function (sine or cosine)

The graph has a maximum at \( x = 0 \) (or near \( x = 0 \))? Wait, looking at the graph, when \( x = 0 \), let's check the value. Wait, the graph at \( x = 0 \) seems to be at the midline? Wait, no, maybe I misread. Wait, the options are \( y = 8\cos(x)+3 \), \( y = 4\cos(x)+3 \), \( y = 4\sin(x)+3 \). Wait, the amplitude we found is 4, so \( A = 4 \). Now, check the shape: the graph starts at a maximum? Wait, no, let's see the options. The cosine function \( y = A\cos(x)+D \) has a maximum at \( x = 0 \) (when \( A>0 \)). The sine function \( y = A\sin(x)+D \) has a value of \( D \) at \( x = 0 \). Looking at the graph, at \( x = 0 \), what's the value? Wait, the graph at \( x = 0 \) – let's see the y - axis. The graph seems to have a maximum or minimum? Wait, the options: \( y = 4\cos(x)+3 \) has amplitude 4, vertical shift 3. Let's check the amplitude: if \( A = 4 \), then the maximum is \( 4 + 3 = 7 \), minimum is \( - 4+3=-1 \). The graph's maximum is 7 (since midline is 3, amplitude 4: 3 + 4 = 7), minimum is 3 - 4=-1. Now, the shape: the cosine function \( y = 4\cos(x)+3 \) starts at \( x = 0 \) with \( y = 4(1)+3 = 7 \), which matches the graph's maximum at \( x = 0 \) (or near \( x = 0 \))? Wait, the graph in the image – let's see, the top of the graph is at y = 7, midline at 3, so amplitude 4. The function \( y = 4\cos(x)+3 \) has amplitude 4, vertical shift 3, and at \( x = 0 \), \( y = 7 \), which is the maximum. The sine function \( y = 4\sin(x)+3 \) at \( x = 0 \) is \( y = 3 \), which is the midline. The graph at \( x = 0 \) – looking at the image, the graph at \( x = 0 \) (the y - axis) seems to be at the maximum (since the top of the curve is near x = 0). So it's a cosine function with amplitude 4. So the function is \( y = 4\cos(x)+3 \).

Answer:

\( y = 4\cos(x)+3 \) (the second option: \( y = 4\cos(x)+3 \))