Question

How do you graph \[y = {x^2} - 4\] ?

Answer 1

Hint: Here we have to plot a graph. In this given equation by giving the values to the x like 0, 1, 2, 3, … simultaneously we get the values of y with respect to the x value. After getting the x and y values, write the coordinates of the given equation in the form of (x,y), by using the coordinates construct the required graph of the given equation.

Complete step-by-step answer:
Given equation in the form of linear equation in the form of two variables x and y
Consider the equation \[y = {x^2} - 4\]
Now, by giving the x values … -3, -2, -1, 0, 1, 2, 3, … to the above equation simultaneously we get the values of y
When we substitute the value of x=-3, then
 \[ \Rightarrow y = {\left( { - 3} \right)^2} - 4\]
 \[ \Rightarrow y = 9 - 4\]
 \[\therefore y = 5\]
Therefore, co-ordinate \[\left( {x,y} \right) = \left( { - 3,5} \right)\]
When we substitute the value of x=-2
 \[ \Rightarrow y = {\left( { - 2} \right)^2} - 4\]
 \[ \Rightarrow y = 4 - 4\]
 \[\therefore y = 0\]
Therefore, co-ordinate \[\left( {x,y} \right) = \left( { - 2,0} \right)\]
When we substitute the value of x=-1
 \[ \Rightarrow y = {\left( { - 1} \right)^2} - 4\]
 \[ \Rightarrow y = 1 - 4\]
 \[\therefore y = - 3\]
 Therefore, co-ordinate \[\left( {x,y} \right) = \left( { - 1, - 3} \right)\]
When we substitute the value of x=0
 \[ \Rightarrow y = {\left( 0 \right)^2} - 4\]
 \[ \Rightarrow y = 0 - 4\]
 \[\therefore y = - 4\]
 Therefore, co-ordinate \[\left( {x,y} \right) = \left( {0, - 4} \right)\]
When we substitute the value of x=1
 \[ \Rightarrow y = {\left( 1 \right)^2} - 4\]
 \[ \Rightarrow y = 1 - 4\]
 \[\therefore y = - 3\]
 Therefore, co-ordinate \[\left( {x,y} \right) = \left( {1, - 3} \right)\]
When we substitute the value of x=2
 \[ \Rightarrow y = {\left( 2 \right)^2} - 4\]
 \[ \Rightarrow y = 4 - 4\]
 \[\therefore y = 0\]
Therefore, co-ordinate \[\left( {x,y} \right) = \left( {2,0} \right)\]
When we substitute the value of x=3, then
 \[ \Rightarrow y = {\left( 3 \right)^2} - 4\]
 \[ \Rightarrow y = 9 - 4\]
 \[\therefore y = 5\]
Therefore, co-ordinate \[\left( {x,y} \right) = \left( {3,5} \right)\]
And so on …
Hence by substituting the value of x we have determined some of the values or points we use to plot the graph.
The coordinates can be written in table as :

\[x\]	\[ - 3\]	\[ - 2\]	\[ - 1\]	\[0\]	\[1\]	\[2\]	\[3\]
\[y\]	\[5\]	\[0\]	\[ - 3\]	\[ - 4\]	\[ - 3\]	\[0\]	\[5\]
\[\left( {x,y} \right)\]	\[\left( { - 3,5} \right)\]	\[\left( { - 2,0} \right)\]	\[\left( { - 1, - 3} \right)\]	\[\left( {0, - 4} \right)\]	\[\left( {1, - 3} \right)\]	\[\left( {2,0} \right)\]	\[\left( {3,5} \right)\]

Hence, the graph of the given linear equation \[y = {x^2} - 4\] represent the parabola is given by

Note: The question belongs to the concept of graph. By comparing the given equation to the equation of a line we calculate the slope and intercept. Or by choosing the value of x we can determine the value of y and then plotting the graphs for these points we obtain the result.