Question

How do you graph $$y=4\cos x+4$$ ?

Answer 1

Hint: We are given a function of cosine and we have to plot this function on the graph. We have to first convert the given function to its standard form and then compare it with the standard equation for plotting any function on the graph, this way we can find out how the graph of the given function should be plotted. The standard form of the cosine equation is $y=A\cos (Bx+C)+D$ .

Complete step-by-step answer:
On comparing $$y=4\cos x+4$$ with the standard form $y=A\cos (Bx+C)+D$ , we get –
$A=4,B=1,C=0andD=4$
Thus, the given cosine function has a vertical shift of 4, so it has peak values at -4+4 and 4+4, that is, it oscillates between 0 and 8 and the given cosine function completes one oscillations between 0 and $2\pi$ , but for plotting the graph, we have to find out the period of the function. Period of this cosine function is –
$$\begin{gathered} p={\displaystyle \frac{2\pi }{B}} \\ \Rightarrow p={\displaystyle \frac{2\pi }{1}}=2\pi \end{gathered}$$
That is after every $2\pi$ radians the given function repeats the oscillation; it completes 1 oscillation in the interval 0 to $2\pi$ .

Note: We know that the general form of the cosine function is $y=A\cos (Bx+C)+D$ where
A is the amplitude, the peak values of a function are known as its amplitude.
B is the frequency, the number of oscillations that a function does in a fixed interval is known as its frequency.
C and D tell us the horizontal and vertical shift of a function respectively. There is no horizontal or vertical shift in the given function as the value of C and D is zero for the given function.