CAT 2011 is over and the call getters are busy in preparing for the interviews and giving their best shot in the next round. For the rest, it's time to gear up for the next season. Quant enthusiast, Deepak Pandey elaborates on graph modification, a topic which is tricky for many students attempting CAT. This is the first article in the series of articles for CAT 2012 preparation.
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 simple 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) : To plot the graph of f(x+a), just 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.
Graph of F(x) = -Sinx (Obtained just by reflecting the graph of f(x) about x-axis)
6) y=|f(x)| : If we have the graph of f(x) and we want to find the graph of a function obtained by applying the modulus on the f(x), then just 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.
Let's do a couple of questions based on the above concept-
1) Prob: Given below are two graphs. Identify the number of relations that the pair of graphs satisfy from the given five relations.
Solution: The graph of F(x) can be obtained in the following ways-
- By reflecting the graph of f(x) about x-axis. Hence F(x) = -f(x)
- By reflecting the graph of f(x) about y-axis and then about x-axis. Hence F(x) = -f(-x)
- It is obvious that f(x) =|f(x)|, (as there is no part of the below x-axis). And the graph of F(x) can be obtaining by reflecting this graph about x-axis. Hence, F(x) = -|F(x)|
Hence the pair of graphs satisfy the three relations in total.
2) Prob: Given below is the graph of f(x) = e^x. With the help of this graph, obtain the graph of F(x)= -e^(-x).
Graph of f(x) = e^x
Solution: Let's first plot the graph of e^(-x) which can be obtained just by reflecting the graph of e^x about y-axis (as mentioned in point 1)
Graph of e^(-x)
Now the graph of F(x)= -e^(-x) can be obtained by reflecting the graph of e^(-x) about x-axis.
Graph of F(x)= -e^(-x)
I will wrap up this post here with some practice problems for you to have better understanding of the logic. Hope this will help you in solving problems related to Graphs Modification.
1) Question: Given below is the graph of f(x) = x^3. With the help of this graph, obtain the graph of F(x) = |x^3-3x^2+3x-1|+7.
2) Question:Given below is the graph of f(x)=logX. With the help of this graph, obtain the graph of
a) F(x) = -log(x-4),
b) F(x) = -log(|x|-4)
3) Question: Given below is the graph of the circle x^2+y^2=16. Obtain the graph of x^2+4x+y^2=12, with the help of this graph.
Image Source: Flickr
The Author, Deepak Pandey is a B. Tech (E & TC) with strong interpersonal skills and loves to connect with different people. Very inquisitive in nature, believes in continuous learning, compassionate from inside and an avid supporter of equal and ethical treatment to all the living beings. He can be reached at deepakpandey028[at]gmail.com.