Question

graph the trigonometric function. $y =-\frac{5}{2}cos(\frac{3}{2}x)+1$. plot all points corresponding to minima and maxima within one cycle. within that cycle, also plot all points along the \midline\ (points whose y - coordinates are midway between the functions minimum and maximum values). then click on the graph - a - function button.

Explanation:

Step1: Recall cosine - function properties

The general form of a cosine function is $y = A\cos(Bx - C)+D$. For the function $y=-\frac{5}{2}\cos(\frac{3}{2}x)+1$, we have $A =-\frac{5}{2}$, $B=\frac{3}{2}$, $C = 0$, and $D = 1$.

Step2: Find the period

The period of the cosine function $y = A\cos(Bx - C)+D$ is given by $T=\frac{2\pi}{|B|}$. Substituting $B=\frac{3}{2}$ into the formula, we get $T=\frac{2\pi}{\frac{3}{2}}=\frac{4\pi}{3}$.

Step3: Find the maximum and minimum values

The amplitude of the function is $|A|=\frac{5}{2}$. Since $D = 1$, the maximum value of $y$ occurs when $\cos(\frac{3}{2}x)=- 1$. Then $y_{max}=-\frac{5}{2}\times(-1)+1=\frac{5 + 2}{2}=\frac{7}{2}$. The minimum value of $y$ occurs when $\cos(\frac{3}{2}x)=1$. Then $y_{min}=-\frac{5}{2}\times1 + 1=\frac{-5 + 2}{2}=-\frac{3}{2}$.

Step4: Find the mid - line

The mid - line is $y = D=1$.

Step5: Find key points

For one - cycle of the cosine function, we can find the key points. Let $\frac{3}{2}x=0,\frac{\pi}{2},\pi,\frac{3\pi}{2},2\pi$. Then $x = 0,\frac{\pi}{3},\frac{2\pi}{3},\pi,\frac{4\pi}{3}$.
When $x = 0$, $y=-\frac{5}{2}\cos(0)+1=-\frac{5}{2}+1=-\frac{3}{2}$.
When $x=\frac{\pi}{3}$, $y=-\frac{5}{2}\cos(\frac{\pi}{2})+1 = 1$.
When $x=\frac{2\pi}{3}$, $y=-\frac{5}{2}\cos(\pi)+1=\frac{5}{2}+1=\frac{7}{2}$.
When $x=\pi$, $y=-\frac{5}{2}\cos(\frac{3\pi}{2})+1 = 1$.
When $x=\frac{4\pi}{3}$, $y=-\frac{5}{2}\cos(2\pi)+1=-\frac{5}{2}+1=-\frac{3}{2}$.
We also need to find the points on the mid - line between the maxima and minima.

Answer:

To graph the function, plot the points:

• Maxima: Points where $y=\frac{7}{2}$
• Minima: Points where $y =-\frac{3}{2}$
• Mid - line points: Points where $y = 1$ within one cycle $[0,\frac{4\pi}{3}]$ and then repeat the pattern for other cycles. The key points in one cycle are $(0,-\frac{3}{2}),(\frac{\pi}{3},1),(\frac{2\pi}{3},\frac{7}{2}),(\pi,1),(\frac{4\pi}{3},-\frac{3}{2})$ and additional mid - line points can be found by taking the mid - values of $x$ between the points corresponding to maxima and minima within the cycle.