Basic HTML Version
Q
uantitative
R
easoning &
P
roblem
S
olving
108
© 2014 Pacific Crest
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Not describing the function from left to right
Example
: Analyze
the function
f
(
x
) =
x
2
+ 2
x
+ 2
The function is increasing throughout
with a vertex at (–1, 1).
Why?
The analysis starts at the vertex goes to the left and right from that point rather than
going from left to right starting far from 0 on the left. The function is decreasing
from –∞ to –1 and increasing from –1 to ∞.
●
Missing important subtleties of the behavior
Example
: Analyze
the function
2
2
( )
2
x x
f x
x
− −
=
−
The function is a linear function that is
always increasing with an
x
-intercept at
(-1, 1) and a
y
-intercept at (0, 1).
Why?
The function is undefined at
x
= 2 which would have been discovered if the student
had correctly determined the domain to be all real numbers except
x
= 2. This is a
subtle behavior that often cannot be seen on computer generated graphs that are
using a finite number of points to graph a function.
The function is increasing from ∞ to 2 and increasing from 2 to ∞ with an
x
-intercept
at (–1, 0) and a
y
-intercept at (0, 1). The function is not increasing for all real numbers
because is not increasing from 1 to 3 since
f
(2)
≯
f
(1) as
f
(2) is not defined.