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© 2014 Pacific Crest
111
M
odel
2: A
nalysis of
S
ample
F
unctions
Scenario A:
Analyze the function:
f
(
x
) =
x
2/3
Step
Model
1.
Convert to function notation
Already in function notation
2.
Determine the function’s family
Power function
3.
Determine the domain of the function All real numbers
4.
Graph the function
5.
Asymptotes
No asymptotes
6.
Determine the range of the function Non-negative real numbers
7.
Determine the function’s intercepts
x
- and
y
-intercepts at (0,0)
8.
Determine if the function is increasing
or decreasing
Decreasing from –∞ to 0.
Increasing from 0 to ∞.
9.
Determine function’s end behavior
The graph is always positive, decreasing from –∞ to 0
and increasing from 0 to ∞. Unbounded.
10. Validate your analysis
Should behave like
x
1/3
for
x
> 0. Check against the
function card for the family of power functions.
Scenario B:
Analyze the function:
x
2
+
y
2
= 4
Step
Model
1.
Convert to function notation
2
2
2
2
2
4
4
4
x y
y
x
y
x
+ =
= −
= ± −
This equation does not represent a
function since for x values of –2 and
2, the value of y is 0 (it does not
pass the vertical line test).
2.5 Analyzing a Function