*Saturday, February 16th, 2013*

**Basics of functions and modifications of graphs**

XAT’s Quant is always a little bit on the tougher side. It is said that the paper would be do-able and the level of difficulty will see a dip. That does not mean that the difficulty level would suddenly drop to the standard of elementary mathematics. XAT traditionally focuses more on topics like functions, probability, permutation & combination, etc. more than the CAT exam. In this post we will discuss some basic tips about functions and how graphs of functions change.

Let us see what the function y = f(x) = x^{3} + 7 looks like:

When we do y = -f (x) we will have to **reflect the graph in the x-axis**. Given below is what it is going to look like. As you can see, this image would be obtained if we had put a mirror on the X-axis.

When we do y = f(-x) = -x^{3} + 7 we will have to **reflect the graph in the y-axis**. Given below is what it is going to look like. As you can see, this image would be obtained if we had put a mirror on 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)

We also know that 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 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:

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) = x^{2}, 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) = x^{2}, 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 orignal 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**

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:

I guess we will wrap it up here and hope this would help you with your functions.