Chapter 7.3, Problem 100AYU

To determine

To graph: The mentioned trigonometric function $f\left(x\right)=2\cos \left(\frac{x}{2}\right)+3$ and $g\left(x\right)=4$ on the same Cartesian plane where $x\in \left[0,4\pi \right]$ .

Explanation of Solution

Given information:

The mentioned trigonometric function $f\left(x\right)=2\cos \left(\frac{x}{2}\right)+3$ and $g\left(x\right)=4$ .

Graph:

The graph provided below is the mentioned trigonometric function $f\left(x\right)=2\cos \left(\frac{x}{2}\right)+3$ and $g\left(x\right)=4$ . The graph is plotted in the x axis and y axis real number line.

Here we can see in the mentioned graph below that the $g\left(x\right)=4$ is plotted by greencoloured line and function $f\left(x\right)=2\cos \left(\frac{x}{2}\right)+3$ is plotted by red coloured curve .The graph is plotted in the interval $x\in \left[0,4\pi \right]$ .

  

Interpretation:

The general process of graph along with the description of their traces is provided below.

Since we know that the graph of cos x is symmetric about y axis.

Therefore on the interval of $x\in \left[0,4\pi \right]$ the graph will remain on positive side of x axis.

Now if we multiply any constant to cos x then the graph will shrink from its original position and if we add some constant to the same function then the graph will release its position and will expand from its original position.

Thus the graph of function $f\left(x\right)=2\cos \left(\frac{x}{2}\right)+3$ is shown as mentioned above in the graph and the constant $g\left(x\right)=4$ is as plotted in the mentioned point in the y axis.

Therefore the graph of the function $f\left(x\right)=2\cos \left(\frac{x}{2}\right)+3$ and $g\left(x\right)=4$ is as shown above in the same Cartesian plane.