GMAT Math : Functional shifts

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

Let’s use the simple function x² to explain the concepts.

x² 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, for any function f(x) :

Let’s do a GMAT data sufficiency question to reinforce our understanding:

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

Readers are encouraged to take some time to crack this question 🙂

…

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)

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