Graph theory

General tricks for sketching any curve:-
1. Suppose,we have a function y=f(x) and we replace x by (x-a),then the graph will shift horizontally by a units towards the right(assuming that a is positive here which is implicitly assumed everywhere) and this is called horizontal shift.
2. If we replace x by (x+a), then graph will shift horizontally a units towards the left.
3. If we have a function y=f(x) and we replace y by (y-a),then graph will shift vertically up by a units.This is called vertical shift.
4. If we replace y by (y+a),then graph will shift vertically down by a units.
5. If we replace x by (-x) in the given function,then the graph will rotate 180° about the y axis(or in simple terms mirror image about y axis)
6. Similarly,if we replace y by (-y) in the given function,then the curve will rotate 180° about the x axis.
7. If we interchange x and y(ie replace x by y and y by x),then the graph will be a mirror image about the line y=x. This is actually how the graphs of inverse of a function is drawn.
8. If we replace x by -y and y by -x,then graph is a mirror image about the line y=-x.

●These tricks are useful if there is slight variation in the given function from a standard function whose graphs are already known to us.For ex. we have to plot a function y=|x-2|.So,we already know the graph of y=|x|. It is just that x is replaced by (x-2) in the given function. So by using trick 1,the graph of y=|x| will shift 2 units toward the right and hence this gives the graph of the given function easily.
9. Checking if the curve passes through origin gives us an idea about our curve.So,this could be done by plugging x=0 and y=0 and checking if it is satisfied by the equation.
10. Finding the points of intersection of the curve with x and y axis also helps to trace the curve.
11. Find the value of the function as x approaches infinity. It tells us about the nature of the function(whether it tends to a point or tends to infinity as x approaches infinity).
12. Find dy/dx and check for what values of x the graph is increasing or decreasing and find points of maxima and minima.
13. We could even find double derivative of a f
13. Find the domain and range of a function to get the area of plane where the curve is confined.