The GRAPH of an equation in two variables x and y is the set of all points in the xy-plane whose coordinates are members of the solution set of the equation

So the Graph of an equation is just all those points in the xy-plane which when plugged into the x and y satisfy the equation. So with this in mind, we will now freely use the words SKETCH or PLOT the GRAPH of an equation,. This will mean that you are trying to give a picture of the set that contains the solutions of a given equation. This is where algebra is being converted to geometry!! Here is an example:

Example#2

Sketch the graph of

Solution:-

Now here we are looking for all those ordered pairs (a,b) in the xy-plane which when substituted for the x and the y in the equation satisfy it. Well an easy way to find such numbers is by picking values for x, plugging them into the equation and noting the resulting y value. This pair of x and y will DEFINITELY satisfy the equation because it came out of the equation. Well, this can be done forever, because there are infinitely many real numbers that I can square!!!!

So the solution set of

has infinitely many elements, and we cant write them all down. And if we cant write them all, we certainly can sketch them on the xy-plane!! What to do?? Well, we don't have to get them all. Let's get a few and sketch them and see if we can get somewhere. Here are the values I will choose for x, the corresponding y values, and the resulting pair (x,y) that satisfies the equation

When I plot these on the xy-plane and connect them, I get this picture

of the graph

and this is good enough for us FINITE CREATURES.!!

IMPORTANT REMARK.:-

It should be kept in mind that the curve in above is only an approximation to the graph of

When a graph is obtained by plotting points, whether by hand, calculator, or computer, there is no guarantee that the resulting curve has the correct shape. For example, the curve in the Figure here pass through the points tabulated in above table.

1.3.1Definition:-
The GRAPH of an equation in two variables x and y is the set of all points in the xy-plane whose coordinates are members of the solution set of the equation So the Graph of an equation is just all those points in the xy-plane which when plugged into the x and y satisfy the equation. So with this in mind, we will now freely use the words SKETCH or PLOT the GRAPH of an equation,. This will mean that you are trying to give a picture of the set that contains the solutions of a given equation. This is where algebra is being converted to geometry!! Here is an example: Example#2 Sketch the graph of Solution:- Now here we are looking for all those ordered pairs (a,b) in the xy-plane which when substituted for the x and the y in the equation satisfy it. Well an easy way to find such numbers is by picking values for x, plugging them into the equation and noting the resulting y value. This pair of x and y will DEFINITELY satisfy the equation because it came out of the equation. Well, this can be done forever, because there are infinitely many real numbers that I can square!!!! So the solution set of has infinitely many elements, and we cant write them all down. And if we cant write them all, we certainly can sketch them on the xy-plane!! What to do?? Well, we don't have to get them all. Let's get a few and sketch them and see if we can get somewhere. Here are the values I will choose for x, the corresponding y values, and the resulting pair (x,y) that satisfies the equation When I plot these on the xy-plane and connect them, I get this picture of the graph and this is good enough for us FINITE CREATURES.!! IMPORTANT REMARK.:- It should be kept in mind that the curve in above is only an approximation to the graph of When a graph is obtained by plotting points, whether by hand, calculator, or computer, there is no guarantee that the resulting curve has the correct shape. For example, the curve in the Figure here pass through the points tabulated in above table.
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