5.4 - Analyzing
Graphs of
Polynomial
Functions
Day 1
Example:
f(x) = –x3 – 4x2 + 5 by making a
table of values.
 Graph
Example:
 Which
graph is the graph of f(x) = x3 + 2x2 + 1?
A.
B.
C.
D.
Location Principle:
Example:
 Determine
consecutive values of x
between which each real zero of the
function f(x) = x4 – x3 – 4x2 + 1 is located.
Then draw the graph.
Maximum & Minimum Points
 Relative
Maximum – a point on the graph of a
function where no other nearby points have a
greater y-coordinate.
 Relative
Minimum - a point on the graph of a
function where no other nearby points have a
lesser y-coordinate.
Maximum & Minimum Points
 Extrema
 Turning
– max. and min. values of a function.
Point – when the graph turns. Another
name for relative max. and min.
- The graph of a polynomial function of
degree n has at most n – 1 turning points.
Example:
f(x) = x3 – 3x2 + 5. Find the
x-coordinates at which the relative maxima
and relative minima occur.
 Graph
Find Extrema on Calculator:
 Enter
equation into y =.
 2nd Calc
 Choose 3: minimum or 4: maximum.
 Curser on left of min/max, enter.
 Curser on right of min/max, enter.
 Enter.
Example:
the graph of f(x) = x3 + 3x2 + 2.
Estimate the x-coordinates at which the
relative maximum and relative minimum occur.
 Consider
A. relative minimum: x = 0
relative maximum: x = –2
B. relative minimum: x = –2
relative maximum: x = 0
C. relative minimum: x = –3
relative maximum: x = 1
D. relative minimum: x = 0
relative maximum: x = 2
5.4 - Analyzing
Graphs of
Polynomial
Functions
Day 2: Real-World Problems
Example:
a. The weight w, in pounds, of a patient during a
7-week illness is modeled by the function
w(n) = 0.1n3 – 0.6n2 + 110, where n is the number
of weeks since the patient became ill.
 Graph
the equation.
Example:
b. Describe the turning points of the graph and
its end behavior.
c. What trends in the patient’s weight does the
graph suggest?
d. Is it reasonable to assume the trend will
continue indefinitely?
Example:
The graph models the cross section of Mount
Rushmore.
What is the smallest degree possible for the
equation that corresponds with this graph?
Example:
The rainfall r, in inches per month, in a Midwestern
town during a 7-month period is modeled
by the function r(m) = 0.01m3 – 0.18m2 + 0.67m + 3.23,
where m is the number of months after March 1.
Graph the equation.
A.
B.
C.
D.
Example:
Describe the turning points of the graph and its end
behavior.
A. There is a relative minimum at
Month 2. r(m) decreases as m
increases.
B. There is a relative maximum at
Month 2. r(m) decreases as m
increases.
C. There is a relative maximum at
Month 2. r(m) increases as m
increases.
D. There is a relative minimum at
Month 2. r(m) decreases as m
decreases.
Example:
What trends in the amount of rainfall received by
the town does the graph suggest?
A. The rainfall decreased the first
two months, then increased.
B. The rainfall increased the first
two months, then decreased.
C. The rainfall continued to
increase throughout the entire
8 months.
D. The rainfall continued to
decrease throughout the entire
8 months.
Example:
Is it reasonable to assume the trend will continue
indefinitely?
A. yes
B. no