59. the function $h$ is given by $h(x)=-3cos(\frac{pi}{2}x)$. which of the following is the graph of $h$ for…

59. the function $h$ is given by $h(x)=-3cos(\frac{pi}{2}x)$. which of the following is the graph of $h$ for $0leq xleq8$? (a) (b) (c) (d)

Answer

Explanation:

Step1: Identify the amplitude

The general form of a cosine - function is $y = A\cos(Bx - C)+D$. For the function $h(x)=-3\cos(\frac{\pi}{2}x)$, the amplitude is $|A|$. Here, $A=-3$, so the amplitude $|A| = 3$. This means the maximum value of the function is $3$ and the minimum value is $- 3$.

Step2: Identify the period

The period of a cosine function $y = A\cos(Bx - C)+D$ is given by $T=\frac{2\pi}{|B|}$. For the function $h(x)=-3\cos(\frac{\pi}{2}x)$, $B = \frac{\pi}{2}$. Then $T=\frac{2\pi}{\frac{\pi}{2}}=4$.

Step3: Evaluate the function at $x = 0$

When $x = 0$, $h(0)=-3\cos(0)=-3\times1=-3$.

Step4: Analyze the options

Option (A): At $x = 0$, $y=-3$. The amplitude is $3$ and the period is $4$. This graph satisfies all the properties of the function $h(x)=-3\cos(\frac{\pi}{2}x)$.
Option (B): At $x = 0$, $y\neq - 3$, so this option is incorrect.
Option (C): At $x = 0$, $y = 0$, so this option is incorrect.
Option (D): At $x = 0$, $y = 3$, so this option is incorrect.