the function $g$ is given by $g(x)=2cos(pi x)+1$. which of the following is the graph of $g$ for $0leq xleq4$?

Explanation:

Step1: Analyze the general form of cosine function

The general form of a cosine - function is $y = A\cos(Bx - C)+D$. For the function $g(x)=2\cos(\pi x)+1$, we have $A = 2$, $B=\pi$, $C = 0$, and $D = 1$.

Step2: Find the amplitude

The amplitude of the cosine function is given by $|A|$. Here, $|A|=2$. This means that the maximum value of $y = 2\cos(\pi x)+1$ is $2 + 1=3$ and the minimum value is $-2 + 1=-1$.

Step3: Find the period

The period of the cosine function $y = A\cos(Bx - C)+D$ is $T=\frac{2\pi}{|B|}$. Since $B = \pi$, then $T=\frac{2\pi}{\pi}=2$.

Step4: Evaluate at key - points

When $x = 0$, $g(0)=2\cos(0)+1=2\times1 + 1=3$. When $x=\frac{1}{2}$, $g(\frac{1}{2})=2\cos(\frac{\pi}{2})+1=2\times0 + 1=1$. When $x = 1$, $g(1)=2\cos(\pi)+1=2\times(-1)+1=-1$. When $x=\frac{3}{2}$, $g(\frac{3}{2})=2\cos(\frac{3\pi}{2})+1=2\times0 + 1=1$. When $x = 2$, $g(2)=2\cos(2\pi)+1=2\times1 + 1=3$.

The function $g(x)$ has a period of $2$, an amplitude of $2$, and a vertical shift of $1$ unit up. The graph starts at the maximum value of $y = 3$ when $x = 0$.

Answer:

A. Option Text (the graph with maximum value 3, minimum value - 1, and period 2 starting at the maximum when $x = 0$)