Question

How do you sketch the graph of $f\left( x \right) = {x^3} - 3{x^2}$ by plotting points?

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 $f(x)$ 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 this 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 $f(x)$ 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 $y = f\left( x \right) = {x^3} - 3{x^2}$.
Let us substitute the value of x as $0$, $1$, $2$ and $3$.
Now we consider the value of x as $0$, the value of y is
$ \Rightarrow y = {0^3} - 3{\left( 0 \right)^2}$
$ \Rightarrow y = 0 - 3\left( 0 \right)$
$ \Rightarrow y = 0$
Now we consider the value of x as $1$, the value of y is
$ \Rightarrow y = {1^3} - 3{\left( 1 \right)^2}$
$ \Rightarrow y = 1 - 3\left( 1 \right)$
$ \Rightarrow y = - 2$
Now we consider the value of x as $2$, the value of y is
$ \Rightarrow y = {2^3} - 3{\left( 2 \right)^2}$
$ \Rightarrow y = 8 - 3\left( 4 \right)$
$ \Rightarrow y = 8 - 12$
$ \Rightarrow y = - 4$
Now we consider the value of x as $3$, the value of y is
$ \Rightarrow y = {3^3} - 3{\left( 3 \right)^2}$
$ \Rightarrow y = 27 - 3\left( 9 \right)$
$ \Rightarrow y = 27 - 27$
$ \Rightarrow y = 0$
Now we draw a table for these values we have

x	$0$	$1$	$2$	$3$
y	$0$	$ - 2$	$ - 4$	$0$

The graph plotted for these point is represented below:

Note:
The graph plotted is of two dimensional. The graph is plotted in x-axis versus y axis. 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.