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) :
- 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
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 TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are not 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)
-a + k = c – p (horizontal alignment)
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)