This question has been bothering me from many days. If I wanted to represent the graph of $xy =$ constant, why does the graph of $xy=$ constant come out to be a curve , something like this ?
$xy=$ constant means that $x$ is inversely proportional to $y$ and as $x$ increases $y$ decreases and vice versa.
But the same thing is also indicated in the following graph right ?
In this graph also , we see that as $x$ increases $y$ decreases. Then why is the graph of inverse proportionality always a curve ? I’m sorry if this question sound really elementary, but this has been bothering me from many days
Best Answer
If we give a closer look we will firstly get 2 different equations for the different curves,
[1] this curve gives us $$xy = 5 $$ and,
[2] this curve gives us $$x+y = 9$$ now,plotting a graph is basically seeing how the dependent variable (usually $y$) varies with the independent variable (usually $x$).
So, we can see that in [1] the equation loosely transforms to $y =\frac{5}{x}$ , and in [2] we have $y=9-x$.
Now, as we have 2 different functions for the 2 different equations [1] and [2] we can actually compare as @Abhishek pointed out, the slopes of these two functions at each point and see that they are not the same.
Which boils the argument down to the rate of change of $y$'s with the respective $x$'s which hence gives us a curve and a straight line.