Graph Modification

Sometimes we come across questions related to graphs modification. These questions may look difficult in the first sight but are really easy to crack if we know some of the fundas. The questions may be like you are given a graph f(x) and you are asked to find the graph of say |f(-x+a)|+b or you may be given two or three graphs and you are asked to find the relationship or the number of relationships between the graphs among some given relationships.

Now, let us take a function y=f(x) and see the various methods in which graph of f(x) can be modified:-

1) y= -f(x) : Reflect the graph of y=f(x) about X-axis.

For example, if we have a graph of f(x)= x^3+7, then the graph of g(x)= -x^3-7 or –(x^3+7) can be obtained just by reflecting the graph along x-axis.

2) y=f(-x) : Reflect the graph of y=f(x) about Y-axis.

For example, if we are given the graph of f(x) = x^2+2x+8, then the graph of g(x)=x^2-2x+8 or (-x)^2-2x+8 can be obtained just by reflecting the graph along y-axis.

(Remember for even functions, graphs for y=f(x) and y=f(-x) are same. For example,  f(x)=cosx, f(x)= x^2, f(x)=|x| are even functions)

3) y=-f(-x): Reflect the graph of y=f(x) about X-axis and then Y-axis.

(For odd functions, graphs for y=f(x) and y=- f(-x) are same. For example, f(x)= sinx, f(x)=x^3, f(x)=ax+b are some of the odd functions. We can also say for odd functions f(x)+f(-x)=0)

4) y=f(x)+a : Shift the graph of y=f(x) along Y-axis by |a| units in the direction same as the sign of ‘a’. For example, if the graph of f(x) = x^4+5x+7 is given, then the graph of g(x)=x^4+5x+12 or (x^4+5x+7)+5 can be obtained just by shifting the graph in positive y-axis direction.

5)  y=f(x+a) : Translate the graph of y=f(x) along X-axis by |a| units in  the direction opposite to the sign of ‘a’ .

For example, if f(x)= 5x+7, then the graph of f(x+6) i.e. 5x +37 or 5(x+6)+7 can be obtained just by shifting the graph by 6 units in the negative x-axis direction.
6) y=|f(x)| : Reflect that part of the graph y=f(x) which is below the X-axis about X-axis.

For example, if f(x)=logx then graph of g(x)=|logx| can be obtained just by reflecting the graph below x-axis about the x axis.

(Remember, in this case there won’t be any graph below the X-axis (i.e. for -ve values of ‘y’) because |f(x)| is always +ve and hence ‘y’ can’t be –ve)
7) y=f(|x|) : Just consider the part of the graph y=f(x) for x>0  (i.e. the part of the graph to the right of Y-axis) and omit the rest. Now reflect this new graph about Y-axis to obtain the final graph.

For example, if you are given the graph of, say f(x)=ax+b, then the graph of a|x|+b can be obtained just by considering the part of the graph to the right of y-axis, omitting the rest of the part and reflecting this new graph about y-axis.”

Contributed by :

Deepak Pandey,