The Xavier Admission Tests (XAT) Quantitative Aptitude is always a bit on the tougher side. Sometimes, the difficulty level may take a dip, but that does not mean that it would suddenly drop to the standards of elementary mathematics. Unlike the Common Admission Test (CAT), XAT traditionally focuses more on topics like functions, probability, permutation & combination, etc.
Here we will discuss some basic tips about functions and how graphs of functions change.
Let us see what the function y= f(x) = x3 + 7 looks like:
When we do y = -f (x), the graph will change to its mirror image across the X-axis.
When we do y = f(-x) = -x3 + 7, the graph will change to its mirror image across the Y-axis.
We also know that for even functions, f(x) = f(-x), so their graphs would be identical in nature. We can also say that a function is even if its mirror image in the Y-axis is identical to the original.
Given below are the graphs of even functions cos(x) and cos(-x)
For odd functions, f(-x) = – f (x) , the graph of an odd function has rotational symmetry with respect to the origin, meaning that its graph remains unchanged after a rotation of 180 degrees about the origin. This means that if you reflect the graph of an odd function first in the X-axis and then in the Y-axis, the resultant graph would be same as the identical.
Let us check this out:
y = f(x) = x + 5 sin(x)
To find out its reflection in the X-axis, we will need to y = – f(x) = – . As you can see, if you reflect the above graph in Y-axis, you will get back the original.
In the modifications discussed above, we talked about reflection about the X-axis or the Y-axis. However, there can be other modifications as well, in which the graph shifts up, down, left or right. Let us look at those.
If y = f(x), the graph of y = f(x) + c (where c is a constant) will be the graph of y = f(x) shifted c units upwards (in the direction of the y-axis). If y = f(x), the graph of y = f(x) – c (where c is a constant) will be the graph of y = f(x) shifted c units downwards (in the direction of the y-axis).
If we consider f(x) = x2, given below are the graphs of f(x), f(x) + 20 and f(x) 10. As you can see, the red graph is shifted 20 units upwards and the orange graph is shifted 10 units downwards from the original blue graph.
If y = f(x), the graph of y = f(x + c) will be the graph of y = f(x) shifted c units to the left.
If y = f(x), the graph of y = f(x c) will be the graph of y = f(x) shifted c units to the right.
If we consider f(x) = x2, given below are the graphs of f(x), f(x+5) and f(x-3). As you can see, the red graph is shifted 5 units left and the orange graph is shifted 3 units right from the original blue graph.
The graph of y = af(x) is a stretch scale factor a in the y-axis. This is because all the y-values become a times bigger.
The graph of f(ax) is also a stretch. This time the multiple affects the x-values. (Everything happens a times quicker.) Therefore, the graph of f(ax) is a stretch scale factor 1/a in the x-axis.
Another modification which happens is in the case of y = |f(x)| In this case, whatever portion is below the X-axis gets reflected in the x-axis.
Check the examples below:
Given below is the graph of f(x) = |Sin(x)| to clarify it further.
Hope this would help you with your functions problems.
Ravi Handa, an alumnus of IIT Kharagpur, has been teaching for CAT and various other competitive exams for around a decade. He currently runs an online CAT coaching and CAT Preparation course on his website http://www.handakafunda.com