Questions: Write an equation of the function whose graph is shown in the figure below. y=-4 cos (π/5 x)-3 y=4 sin (π/5 x)-3 y=4 cos (π/5 x)+3 y=-4 sin (π/2 x)+3

Solution Steps

The maximum value of the function is 1 and the minimum value is -7. The amplitude is half the difference between the maximum and minimum values. Amplitude \(= \frac{1 - (-7)}{2} = \frac{8}{2} = 4\)

The vertical shift is the average of the maximum and minimum values. Vertical shift \(= \frac{1 + (-7)}{2} = \frac{-6}{2} = -3\)

The graph completes one full cycle from x = 0 to x = 10. Therefore, the period is 10. The period is given by \(\frac{2\pi}{B}\), where B is the coefficient of x in the argument of the cosine or sine function. So, \(10 = \frac{2\pi}{B}\), which gives \(B = \frac{2\pi}{10} = \frac{\pi}{5}\).

Since the graph starts at a maximum value at x = 3, it resembles a cosine function. A standard cosine function starts at its maximum value at x = 0. The graph is shifted 3 units to the right. So, we can represent the function as \(y = A\cos(B(x-C)) + D\). Since the graph has been shifted 3 units right, the phase shift \(C = 3\).

Since the maximum value occurs at \(x = 3\), we have \(y = 4\cos(\frac{\pi}{5}(x-3))-3\). Also, since \( \cos(x) = \cos(-x)\), we can rewrite as \(y = 4\cos(\frac{\pi}{5}(3-x))-3\). When x = 0, \(y = 4\cos(\frac{3\pi}{5})-3 \approx 4(-0.309)-3 \approx -4.236\). The graph passes through \((0,-1)\), which we will address by checking the given options.

The graph resembles a cosine function reflected about the x-axis, shifted 3 units downwards, and has a period of 10. Consider \(y = 4\cos(\frac{\pi}{5}x) - 3\) (when x = 0, y = 1) Consider \(y = -4\cos(\frac{\pi}{5}x) - 3\) (when x = 0, y = -7) Consider \(y = 4\sin(\frac{\pi}{5}x)-3\): When x = 0, y = -3 When \(x = \frac{5}{2}\), \(y = 4\sin(\frac{\pi}{2})-3 = 4(1)-3 = 1\). The graph reaches its maximum value of 1 at \(x=\frac{5}{2} = 2.5\).

Final Answer