graph $g(x)=5cosleft(\frac{pi}{2}x - \frac{3pi}{2}\right)-2$ in the interactive widget. note that one…

graph $g(x)=5cosleft(\frac{pi}{2}x - \frac{3pi}{2}\right)-2$ in the interactive widget. note that one moveable point always defines an extremum point in the graph and the other point always defines a neighbouring intersection with the mid - line.

graph $g(x)=5cosleft(\frac{pi}{2}x - \frac{3pi}{2}\right)-2$ in the interactive widget. note that one moveable point always defines an extremum point in the graph and the other point always defines a neighbouring intersection with the mid - line.

Answer

Explanation:

Step1: Identify the general form of cosine function

The general form of a cosine - function is $y = A\cos(Bx - C)+D$, where for $g(x)=5\cos(\frac{\pi}{2}x-\frac{3\pi}{2}) - 2$, $A = 5$, $B=\frac{\pi}{2}$, $C=\frac{3\pi}{2}$, and $D=-2$.

Step2: Find the amplitude

The amplitude is given by $|A|$. So, $|A| = 5$.

Step3: Find the period

The period $T$ of a cosine function $y = A\cos(Bx - C)+D$ is $T=\frac{2\pi}{|B|}$. Here, $T=\frac{2\pi}{\frac{\pi}{2}}=4$.

Step4: Find the phase - shift

The phase - shift is $\frac{C}{B}$. So, $\frac{C}{B}=\frac{\frac{3\pi}{2}}{\frac{\pi}{2}} = 3$.

Step5: Find the mid - line

The mid - line is $y = D$. So, the mid - line is $y=-2$.

Step6: Find the maximum and minimum values

The maximum value of $y = A\cos(Bx - C)+D$ is $y_{max}=A + D$ and the minimum value is $y_{min}=-A + D$. $y_{max}=5+( - 2)=3$ and $y_{min}=-5+( - 2)=-7$.

Step7: Plot key points

We know that for the cosine function $y=\cos x$, key points are $(0,1),(\frac{\pi}{2},0),(\pi,-1),(\frac{3\pi}{2},0),(2\pi,1)$. For $y = 5\cos(\frac{\pi}{2}x-\frac{3\pi}{2})-2$, when $\frac{\pi}{2}x-\frac{3\pi}{2}=0$, $x = 3$ and $y=3$ (maximum point). When $\frac{\pi}{2}x-\frac{3\pi}{2}=\frac{\pi}{2}$, $x = 4$ and $y=-2$ (mid - line intersection). When $\frac{\pi}{2}x-\frac{3\pi}{2}=\pi$, $x = 5$ and $y=-7$ (minimum point). When $\frac{\pi}{2}x-\frac{3\pi}{2}=\frac{3\pi}{2}$, $x = 6$ and $y=-2$ (mid - line intersection). When $\frac{\pi}{2}x-\frac{3\pi}{2}=2\pi$, $x = 7$ and $y=3$ (maximum point).

To graph the function $g(x)=5\cos(\frac{\pi}{2}x-\frac{3\pi}{2})-2$:

  1. Draw the mid - line $y = - 2$.
  2. Mark the maximum points at $y = 3$ and minimum points at $y=-7$ with a period of $4$.
  3. Plot the points based on the phase - shift and the key - points of the cosine function and connect them with a smooth curve.

Since this is a graphing problem and not a problem with a single numerical answer, the steps above provide a guide to graph the function. If you are using an interactive widget, you would use the information about amplitude, period, phase - shift, mid - line, maximum and minimum values to place the moveable points as described in the problem statement.

Answer:

Follow the steps above to graph the function $g(x)=5\cos(\frac{\pi}{2}x-\frac{3\pi}{2})-2$ in the interactive widget. Place one moveable point at an extremum (either maximum $y = 3$ or minimum $y=-7$) and the other at a neighbouring intersection with the mid - line $y=-2$.