## Introduction

### Basics

Functional shifting is an important concept for advanced GMAT Math questions on the topic of functions.

Let’s use the simple function x^{2} to explain the concepts.

x^{2} appears this way on the xy axis.

Assuming a is a positive number :

(x-a)^{2} moves the graph forward by a units

(x+a)^{2} moves the graph backward by a units

(x)^{2} + a move the graph upwards by a units

(x)^{2} – a moves the graph downward by a units

(ax)^{2} compresses the graph horizontally (closer to y axis)

(x/a)^{2} expands the graph horizontally (closer to x axis)

(-x)^{2} gives the reflection of the graph against the y axis

**To generalize, f****or any function f(x) :**

**f(x-a) shifts the graph forward****f(x+a) shifts the graph backwards****f(x) + a shift the graph upwards****f(x) – a shifts the graph downward****f(ax) compresses the graph horizontally****f(x/a) expands the graph horizontally****f(-x) gives the reflection against the y axis****-f(x) gives the reflection against the x axis**

### When do we use this?

Any questions related to graphs and functions of the form described above – can be reframed into a motion and movement discussion – which is visual and much easier to process

## Problem statement

**f(x) and g(x) are represented below. Do f(x – k) – g and g(x + p) – r intersect only once?**

**Sum of k and p is equal to the sum of a and c****Average of r and g ≥ Average of b and d**

(A) Statement (**1**) ALONE is sufficient, but statement (2) alone is not sufficient.

(B) Statement (**2**) ALONE is sufficient, but statement (1) alone is not sufficient.

(C) BOTH statements **T**OGETHER are sufficient, but NEITHER statement ALONE is sufficient.

(D) **E**ACH statement ALONE is sufficient.

(E) Statements (1) and (2) TOGETHER are **n**ot sufficient

## Solution using the technique

From our understanding of functional shifts given in the question, we are expecting to see something like this :

**For f(x – k) – g and g(x + p) – r intersect to only once :**

**The two curves need to be horizontally aligned and their trough (maxima or minima) points should meet vertically (at a single point)**

Hence,

-a + k = c – p (horizontal alignment)

and

b – g = -d + r (vertical point touch)

Simplifying we have (two equality equations) :

k + p = a + c

r + g = b + d

Statement (1) from the question gives us our first equality above

Statement (2) tells us (r + g)/2 ≥ (b + d)/2 – simplifying we have r + g ≥ b + d

Clearly, we cannot conclusively say if r + g = b + d is always true.

So, we go with **option (E) **