Aim #36: How do we determine the properties of a function?
Homework: Handout
Do Now: Consider the function f(x) = -.5x + 2 on the domain 0 ≤ x ≤ 2
a. Write out in words the meaning of {(x,y) | 0 ≤ x ≤ 2 and y = f(x)}
b. Graph y = f(x)
Increasing/Decreasing
A function is increasing on a given interval when f(x1) < f(x2) then x1 < x2.
A function is decreasing on a given interval when f(x1) > f(x2) then x1 < x2.
y
Graph f(x) = x 3 - 3x on the domain -2 ≤ x ≤ 2
2
1
f(x)
-2
1
-1
-1
-2
On what domain interval(s) is the function increasing? Justify your answer.
On what domain interval(s) is the function decreasing? Justify your answer.
2
x
Relative Extrema
Relative Maximum: occurs when a function changes from increasing
to decreasing
Relative Minimum: occurs when a function changes from decreasing
to increasing
We call these points relative max/min because they may not be thehighest or
lowest points on the graph, but indicate where a functionchanges from either
increasing to decreasing or vice versa.
At what x value did f(x) = x3 - 3x have a relative max? rel. min?
Answer the following questions based on the given graph.
B
a. What point(s) are relative maxima?
b. What point(s) are relative minima?
c. On what intervals is the function increasing?
d. On what intervals is the function decreasing?
A
C
e. An absolute maximum or minimum is the highest or lowest output, respectively,
of a function. Does the graph above have an absolute maximum or minimum?
Positive/ Negative
A function f is called positive on an interval I if f(x) > 0 for all x in I.
A function f is called negative on an interval I if f(x) < 0 for all x in I.
Determine the following for the given function.
y
Positive: ______________________
Negative: _____________________
Rel. Max: _____________________
x
Rel. Min: ______________________
Increasing: ____________________
Decreasing: _____________________
Use your calculator to graph f(x) = x(x-2)(x-3), then answer the following. The
instructions for using your calculator to find the rel. max/min are on the
following page. Round the rel. max/min values to the nearest tenth.
y
Rel. Max: _______________________
Rel. Min: _______________________
Increasing: _____________________
x
Decreasing: _____________________
Positive: _______________________
Negative: ______________________