Question

graph the trigonometric function. y = cos x+3 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 properties of cosine function

The general form of a cosine - function is $y = A\cos(Bx - C)+D$. For the function $y=\cos x+3$, $A = 1$, $B = 1$, $C = 0$, and $D = 3$. The amplitude $|A|=1$, the period $T=\frac{2\pi}{|B|}=2\pi$, and the vertical shift is $D = 3$.

Step2: Find the maximum and minimum values

The range of the cosine function $y = \cos x$ is $[- 1,1]$. For the function $y=\cos x+3$, we add 3 to each value in the range of $\cos x$. So the minimum value of $y=\cos x+3$ occurs when $\cos x=-1$, and $y_{\min}=-1 + 3=2$. The maximum value of $y=\cos x+3$ occurs when $\cos x = 1$, and $y_{\max}=1 + 3=4$.

Step3: Determine key - points for one - cycle

For the cosine function $y=\cos x$, one - cycle is from $x = 0$ to $x = 2\pi$. The key - points for $y=\cos x$ are $(0,1),(\frac{\pi}{2},0),(\pi,-1),(\frac{3\pi}{2},0),(2\pi,1)$. For $y=\cos x+3$, the key - points are $(0,4),(\frac{\pi}{2},3),(\pi,2),(\frac{3\pi}{2},3),(2\pi,4)$.

Step4: Plot the points

Plot the maximum points $(0,4)$ and $(2\pi,4)$, the minimum point $(\pi,2)$, and the mid - line points $(\frac{\pi}{2},3)$ and $(\frac{3\pi}{2},3)$. Then, draw a smooth curve through these points to graph the function $y=\cos x+3$.

Answer:

Graph the points $(0,4),(\frac{\pi}{2},3),(\pi,2),(\frac{3\pi}{2},3),(2\pi,4)$ and draw a smooth cosine - like curve through them.