Five-Minute Check (over Lesson 5–3)
CCSS
Then/Now
New Vocabulary
Example 1: Graph of a Polynomial Function
Key Concept: Location Principle
Example 2: Locate Zeros of a Function
Example 3: Maximum and Minimum Points
Example 4: Real-World Example: Graph a Polynomial Model
Over Lesson 5–3
State the degree and leading coefficient of
–4x5 + 2x3 + 6.
A. 5; –4
B. 5; 4
C. 4; 5
D. 8; 6
Over Lesson 5–3
Find p(3) and p(–5) for p(x) = 12 – x2.
A. 1; –10
B. 2; –12
C. 3; –13
D. 4; –14
Over Lesson 5–3
Find p(3) and p(–5) for p(x) = x3 – 10x + 40.
A. 37; 35
B. 23; –20
C. –15; 20
D. 37; –35
Over Lesson 5–3
If p(x) = x2 – 3x + 4, find p(x + 2).
A. x2 + 3x + 6
B. x2 + x + 2
C. x – 6
D. x + 4
Over Lesson 5–3
Determine whether the statement is sometimes,
always, or never true.
The graph of a polynomial of degree three will
intersect the x-axis three times.
A. sometimes
B. always
C. never
Over Lesson 5–3
Describe the end behavior of the graph of function
f(x) = –x2 + 4.
A.
as x → –∞, f(x) → –∞ and
as x → +∞, f(x) → –∞
B.
as x → –∞, f(x) → –∞ and
as x → +∞, f(x) → +∞
C.
as x → –∞, f(x) → +∞ and
as x → +∞, f(x) → +∞
D.
as x → –∞, f(x) → +∞ and
as x → +∞, f(x) → –∞
Content Standards
F.IF.4 For a function that models a relationship
between two quantities, interpret key features of
graphs and tables in terms of the quantities, and
sketch graphs showing key features given a verbal
description of the relationship.
F.IF.7.c Graph polynomial functions, identifying zeros
when suitable factorizations are available, and
showing end behavior.
Mathematical Practices
3 Construct viable arguments and critique the
reasoning of others.
You used maxima and minima and graphs of
polynomials.
• Graph polynomial functions and locate their
zeros.
• Find the relative maxima and minima of
polynomial functions.
• Location Principle
• relative maximum
• relative minimum
• extrema
• turning points
Graph of a Polynomial Function
Graph f(x) = –x3 – 4x2 + 5 by making a
table of values.
Answer:
Graph of a Polynomial Function
Graph f(x) = –x3 – 4x2 + 5 by making a
table of values.
This is an odd degree
polynomial with a
Answer:
negative leading
coefficient, so
f(x)  + as x  –
and
f(x)  – as x  +.
Notice that the graph
intersects the
x-axis at 3 points
indicating that there are
3 real zeros.
Which graph is the graph of f(x) = x3 + 2x2 + 1?
A.
B.
C.
D.
Locate Zeros of a Function
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.
Make a table of values. Since f(x) is a 4th degree
polynomial function, it will have 0, 2, or 4 real zeros.
Locate Zeros of a Function
Look at the values of f(x) to locate the zeros. Then use
the points to sketch the graph of the function.
Answer:
There are zeros
between x = –2 and
–1, x = –1 and 0,
x = 0 and 1, and x = 2
and 3.
Determine consecutive values of x between which
each real zero of the function f(x) = x3 – 4x2 + 2 is
located.
A. x = –1 and x = 0
B. x = –1 and x = 0, x = 0 and
x = 1, x = 3 and x = 4
C. x = –1 and x = 0, x = 3 and
x=4
D. x = –1 and x = 0, x = 0 and
x = 1, x = 2 and x = 3
Maximum and Minimum Points
Graph f(x) = x3 – 3x2 + 5. Estimate the x-coordinates
at which the relative maxima and relative minima
occur.
Make a table of values and graph the function.
Maximum and Minimum Points
Answer: The value of f(x) at x = 0 is greater than the
surrounding points, so there must be a relative
maximum near x = 0. The value of f(x) at x = 2 is less
than the surrounding points, so there must be a
relative minimum near x = 2.
Maximum and Minimum Points
Check: You can use a graphing calculator to find the
relative maximum and relative minimum of a
function and confirm your estimate.
Enter y = x3 – 3x2 + 5 in the Y= list and graph
the function.
Use the CALC menu to find each maximum
and minimum.
When selecting the left bound, move the
cursor to the left of the maximum or
minimum. When selecting the right bound,
move the cursor to the right of the maximum
or minimum.
Maximum and Minimum Points
Press ENTER twice.
The estimates for a relative maximum near
x = 0 and a relative minimum near x = 2 are
accurate.
Consider the graph of f(x) = x3 + 3x2 + 2. Estimate
the x-coordinates at which the relative maximum
and relative minimum occur.
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
Graph a Polynomial Model
A. HEALTH 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.
Make a table of values for weeks 0 through 7. Plot the
points and connect with a smooth curve.
Graph a Polynomial Model
Answer:
Graph a Polynomial Model
B. Describe the turning points of the graph and its
end behavior.
Answer: There is a
relative minimum at
week 4.
w(n) → ∞ as
n → ∞.
Graph a Polynomial Model
C. What trends in the patient’s weight does the
graph suggest?
Answer: The patient lost
weight for each of 4 weeks
after becoming ill. After
4 weeks, the patient
gained weight and
continues to gain weight.
Graph a Polynomial Model
D. Is it reasonable to assume the trend will continue
indefinitely?
Answer: The trend may
continue for a few weeks,
but it is unlikely that the
patient’s weight will rise
indefinitely.
A. WEATHER 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.
B. WEATHER 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.
C. WEATHER 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.
D. Is it reasonable to assume the trend will
continue indefinitely?
A. yes
B. no