Question

How do you draw the graph of $y = 1 + \sin x$ for $0 \leqslant x \leqslant 2\pi$ ?

Answer 1

Hint : A graph of a function f is the set of ordered pairs; the equation of graph is generally represented as $y = f\left( x \right)$ , where x and $f\left( x \right)$ are real numbers. We substitute the value of x and we determine the value of y and then we mark the points in the graph and we join the points.

Complete step-by-step answer :
Here, in the given question, we have to plot the graph for the given function. A graph of a function is a set of ordered pairs and it is represented as $y = f\left( x \right)$ , where x and $f\left( x \right)$ are real numbers. These pairs are in the form of cartesian form and the graph is the two-dimensional graph.
First, we have to find the value of y by using the graph equation $y = 1 + \sin x$ and then limit the portion of the graph where x lies from $0$ to $2\pi$ as we are given the condition $0 \leqslant x \leqslant 2\pi$ .
Let us substitute the value of x as $\dfrac{\pi }{2}$ .
$\Rightarrow y = 1 + \sin \left( {\dfrac{\pi }{2}} \right)$
Since we know that the value of $\sin \dfrac{\pi }{2}$ is $1$ .
$\Rightarrow y = 1 + 1$
$\Rightarrow y = 2$
Now, let us consider the value of x as $0$ .
$\Rightarrow y = 1 + \sin \left( 0 \right)$
Since we know that the value of $\sin 0$ is $0$ .
$\Rightarrow y = 1$
Now we consider the value of x as $\left( {\dfrac{\pi }{6}} \right)$ , the value of y is
$\Rightarrow y = 1 + \sin \dfrac{\pi }{6}$
Since we know that the value of $\sin \dfrac{\pi }{6}$ is $\dfrac{1}{2}$ .
$\Rightarrow y = \dfrac{3}{2}$
Now we draw a table for these values we have

x	$\dfrac{\pi }{2}$	$\left( {\dfrac{\pi }{6}} \right)$	$0$
y	$2$	$\dfrac{3}{2}$	$1$

We also know the nature of the graph of sine function. Hence, we can now plot the graph of the given function $y = 1 + \sin x$ using graphical transformation. The graph plotted for these points is represented below:

Note : The graph is plotted x-axis versus y axis. The graph is two dimensional. By the equation of a graph, we can plot the graph by assuming the value of x. We can’t assume the value of y. because the value of y depends on the value of x. Hence, we have plotted the graph of $y = 1 + \sin x$ for $0 \leqslant x \leqslant 2\pi$