Question

How do you graph $ y = - \tan \left( {2x} \right) $ and include two full periods ?

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 = - \tan \left( {2x} \right) $ .
Let us substitute the value of x as $ \dfrac{\pi }{2} $ .
 $ \Rightarrow y = - \tan \left( {2 \times \dfrac{\pi }{2}} \right) = - \tan \left( \pi \right) $
 $ \Rightarrow y = 0 $
Let us substitute the value of x as $ 0 $ .
 $ \Rightarrow y = - \tan \left( {2 \times 0} \right) = - \tan \left( 0 \right) $
 $ \Rightarrow y = 0 $
Now we consider the value of x as $ \dfrac{\pi }{6} $ , the value of y is
 $ \Rightarrow y = - \tan \left( {2 \times \dfrac{\pi }{6}} \right) = - \tan \left( {\dfrac{\pi }{3}} \right) $
 $ \Rightarrow y = - \sqrt 3 $
Now we consider the value of x as $ \left( {\dfrac{\pi }{3}} \right) $ , the value of y is
 $ \Rightarrow y = - \tan \left( {2 \times \dfrac{\pi }{3}} \right) = - \tan \left( {\dfrac{{2\pi }}{3}} \right) $
 $ \Rightarrow y = - \left( { - \sqrt 3 } \right) = \sqrt 3 $
Now we draw a table for these values we have

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

We also know the nature of the graph of sine function. Hence, we can now plot the graph of the given function $ y = - \tan \left( {2x} \right) $ . The nature of the graph of a function and its slope can also be determined from the derivative of the function. 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. The period of tangent function is $ \pi $ .