1. **State the problem:** We are comparing the period and amplitude of two functions: the given function $f$ and the function $g(x) = \sin\left(\frac{\pi}{2} x\right)$.\n\n2. **Identify the period and amplitude of $f$:** From the graph description, $f$ completes two full cycles between $x=0$ and $x=4$. Therefore, the period of $f$ is $$\frac{4}{2} = 2.$$ The amplitude of $f$ is the maximum absolute value of $y$, which is 4.\n\n3. **Identify the period and amplitude of $g(x)$:** The function $g(x) = \sin\left(\frac{\pi}{2} x\right)$ has the form $\sin(bx)$ where $b = \frac{\pi}{2}$. The period of a sine function is given by $$\text{Period} = \frac{2\pi}{b}.$$ Substituting $b$, we get $$\text{Period of } g = \frac{2\pi}{\frac{\pi}{2}} = 4.$$ The amplitude of $g$ is the coefficient in front of the sine, which is 1.\n\n4. **Compare the periods:** The period of $f$ is 2, and the period of $g$ is 4. So, the period of $f$ is half as long as the period of $g$.\n\n5. **Compare the amplitudes:** The amplitude of $f$ is 4, and the amplitude of $g$ is 1. So, the amplitude of $f$ is 4 times as great as the amplitude of $g$.\n\n6. **Conclusion:** The amplitudes are different, and the period of $f$ is half the period of $g$. This matches option A for the period comparison but the amplitude comparison matches option C. Since the problem asks how they compare, the correct choice is:\n\n**A. The amplitudes are the same, but the period of $f$ is half as long as the period of $g$.** is incorrect because amplitudes differ.\n\n**B. The amplitudes are the same, but the period of $f$ is 4 times as long as the period of $g$.** is incorrect.\n\n**C. The periods are the same, but the amplitude of $f$ is 4 times as great as the amplitude of $g$.** is incorrect because periods differ.\n\n**D. The periods are the same, but the amplitude of $f$ is twice as great as the amplitude of $g$.** is incorrect.\n\nTherefore, none of the options perfectly describe the relationship, but based on the data, the period of $f$ is half the period of $g$, and the amplitude of $f$ is 4 times that of $g$.