Graphing Polynomial Functions
Goal:
Evaluate and graph
polynomial functions.
CCSS: F.IF.4
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Given a function, identify key features in
graphs and tables including: intercepts;
intervals where the function is increasing,
decreasing, positive, or negative; relative
maximums and minimums; symmetries; end
behavior; and periodicity.
Standards for Mathematical
Practice
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1. Make sense of problems and persevere in solving
them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the
reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Essential Question:
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How do I graph a polynomial function?
Polynomial Function
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Function of the form:
f ( x)  an x  an1x
n
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n1
 .........  a1x  a0
an is the leading coefficient
a0 is the constant term
n is the degree
Polynomial is in standard form if its terms are written in
descending order of exponents from left to right.
EVALUATING POLYNOMIAL FUNCTIONS
A polynomial function is a function of the form
f(x) = an x nn + an – 1 x nn – 11 +· · ·+ a 1 x + aa00
Where ann  00 and the exponents are all whole numbers.
For this polynomial function, aan is the leading coefficient,
coefficient
n
aa00 is the constant
constant term,
term and n is the degree.
degree
A polynomial function is in standard form if its terms are
descending order
order of
of exponents
exponents from
from left
left to
to right.
right.
written in descending
Common Types of Polynomials
Degree
Type
Standard Form
0
Constant
f(x) = a0
1
Linear
f(x) = a1x + a0
2
Quadratic
f(x) = a2x2 + a1x + a0
3
Cubic
f(x) = a3x3 + a2x2 + a1x + a0
4
Quartic
f(x) = a4x4 + a3x3 + a2x2 + a1x + a0
Identifying Polynomial Functions
Decide whether the function is a polynomial function. If it is, write the function in
Standard form and state its degree, type, and leading coefficient.
1 2
a. f ( x)  x  3 x 4  7
2
b. f ( x)  x 3  3x
Identifying Polynomial Functions
Decide whether the function is a polynomial function. If it is, write the function in
Standard form and state its degree, type, and leading coefficient.
c. f ( x )  6 x 2  2 x 1  x
d . f ( x)  0.5 x   x 2  2
Evaluate the Polynomial Function
Using Synthetic Substitution
f ( x)  5x  4x  8x  1 when x = 2
3
2
Evaluating a Polynomial Function in
Real Life
The time t (in seconds) it takes a camera battery to recharge after
flashing n times can be modeled by:
f ( x)  0.000015n  0.0034n  0.25n  5.3
3
2
Find the recharge time after 100 flashes.
6.2 Continued:
Graphing Polynomial Functions
Will use end behavior to analyze the graphs of
polynomial functions.
End Behavior
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Behavior of the graph as x approaches
positive infinity (+∞) or negative infinity (-∞)
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The expression x→+∞ : as x approaches
positive infinity
The expression x→-∞ : as x approaches
negative infinity
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End Behavior of Graphs of Linear
Equations
f(x) = x
f(x)→+∞ as x→+∞
f(x)→-∞ as x→-∞
f(x) = -x
f(x)→-∞ as x→+∞
f(x)→+∞ as x→-∞
End Behavior of Graphs of Quadratic
Equations
f(x) = x²
f(x) = -x²
f(x)→+∞ as x→+∞
f(x)→+∞ as x→-∞
f(x)→-∞ as x→+∞
f(x)→-∞ as x→-∞
Investigating Graphs of Polynomial
Functions
1.
Use a Graphing Calculator to grph each function then
analyze the functions end behavior by filling in this
statement: f(x)→__∞ as x→+∞ and f(x)→__∞ as x→-∞
a. f(x) = x³
c. f(x) = x4
e. f(x) = x5
g. f(x) = x6
b. f(x) = -x³
d. f(x) = -x4
f. f(x) = -x5
h. f(x) = -x6
Investigating Graphs of Polynomial
Functions
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How does the sign of the leading coefficient
affect the behavior of the polynomial function
graph as x→+∞?
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How is the behavior of a polynomial functions
graph as x→+∞ related to its behavior as
x→-∞ when the functions degree is odd?
When it is even?
End Behavior for Polynomial
Functions
For the graph of
f ( x)  an xn  an1xn1  .........  a1x  a0
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If an>0 and n even, then f(x)→+∞ as x→+∞ and f(x)→+∞ as x→-∞
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If an>0 and n odd, then f(x)→+∞ as x→+∞ and f(x)→-∞ as x→-∞
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If an<0 and n even, then f(x)→-∞ as x→+∞ and f(x)→-∞ as x→-∞
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If an<0 and n odd, then f(x)→-∞ as x→+∞ and f(x)→+∞ as x→-∞
Graphing Polynomial Functions
f(x)= x³ + x² – 4x – 1
x
f(x)
-3
-2
-1
0
1
2
3
Graphing Polynomial Functions
f(x)= -x4 – 2x³ + 2x² + 4x
x
f(x)
-3
-2
-1
0
1
2
3