Question

How do you graph $y = 2\cos \left( {2x} \right)$ ?

Answer 1

Hint : A graph of a function f is the set of ordered pairs; the equation of the 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 solution:
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 cartesian form and the graph is the two-dimensional graph.
First, we have to find the value of y by using the graph equation $y = 2\cos \left( {2x} \right)$ .
Let us substitute the value of x as $\dfrac{\pi }{2}$ .
$\Rightarrow y = 2\cos \left( {2 \times \dfrac{\pi }{2}} \right)$
$\Rightarrow y = 2\cos \left( \pi \right)$
We know that the value of $\cos \left( \pi \right)$ is $\left( { - 1} \right)$ . So, we get,
$\Rightarrow y = 2\left( { - 1} \right)$
$\Rightarrow y = - 2$
Now we consider the value of x as $\pi$ , the value of y is
$\Rightarrow y = 2\cos \left( {2 \times \pi } \right)$
$\Rightarrow y = 2\cos \left( {2\pi } \right)$
$\Rightarrow y = 2 \times 1$
$\Rightarrow y = 2$
Now we consider the value of x as $\left( {\dfrac{\pi }{4}} \right)$ , the value of y is
$\Rightarrow y = 2\cos \left( {2 \times \dfrac{\pi }{4}} \right)$
$\Rightarrow y = 2\cos \left( {\dfrac{\pi }{2}} \right)$
Now, we know that the value of $\cos \left( {\dfrac{\pi }{2}} \right)$ is zero. So, we get the value of expression as
$\Rightarrow y = 2 \times 0$
$\Rightarrow y = 0$
Now, we draw a table for these values we have,

X	$\dfrac{\pi }{2}$	$\pi$	$\left( {\dfrac{\pi }{4}} \right)$
y	$- 2$	$2$	$0$

We also know the nature of the graph of cosine function. Hence, we can now plot the graph of the given function $y = 2\cos \left( {2x} \right)$ with the help of coordinates of the point lying on it. The graph plotted for these points is represented below:

Note : The cosine function can be represented by the general equation $y = a\cos \left( {kx + \phi } \right)$ . There are various parameters in this equation such as the amplitude, period and phase shift of the sine function. The value ‘a’ is the amplitude of the cosine function $y = a\cos \left( {kx + \phi } \right)$ . The period of the cosine function can be calculated as $\left( {\dfrac{{2\pi }}{k}} \right)$ as the value of the function repeats after regular interval of $\left( {\dfrac{{2\pi }}{k}} \right)$ radians. Also, if there is a constant added in the function like $y = a\cos \left( {kx + \phi } \right)$ , then the graph of the function moves p units vertically upwards due to the upward shift.