Algebra II
Unit #3: Polynomial Functions
5.1 Notes: Polynomial Functions
Name: ________________________
Block: __________
VOCABULARY: Fill in the blanks using your book pg. 280.
A _________________________ is a monomial (1-term function) or sum of monomials. (multiple-term
expression)
ex: 𝑦 = 4𝑥 3
𝑦 = 2𝑥 − 1
𝑦 = 7𝑥 9 + 𝑥 7 − 8𝑥 6 + 3𝑥 4 + 5𝑥 − 11
The degree of a polynomial is the greatest _____________________ among its terms.
ex: 𝑦 = 4𝑥 3 + 2𝑥 has a degree of _____
ex: 𝑦 = 8𝑥 5 − 3𝑥 9 + 8 has a degree of _____
The standard form of a polynomial arranges the terms by degree in _____________ numerical order.
(this means __________ to ___________)
We classify a polynomial by its degree or by its number of terms as shown below in the chart:
Part I: Classifying Polynomials: Write each polynomial in standard form (simplify first if needed).
Then classify the polynomial by degree and its number of terms.
1. 4x + x + 2
2. 3 + 3x  3x
5. (3  b5)
6. 3m + 5m2
9. a3(a2 + a + 1)
10. x(x + 5)  5(x + 5)
3. 4x5  5x2 + 3  2x2
7. 6(2x  1)
11. p(p  5) + 6
13. Open-Ended: Create your own polynomial with these characteristics.
a) quartic trinomial
b) cubic binomial
4. x2 + 3x  4x3
8. b(b  3)2
12. (3c2)2
c) quintic monomial
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Part II: Graphing Calculator Exploration – Investigating Shapes and End Behavior
VOCAB: End Behavior – the direction of the graph to the far left (as x approaches negative infinity)
and far right (as x approaches positive infinity)
For each function, sketch a graph and identify the left and right end behavior of the function (up or down).
**Use one color for ALL POSITIVE graphs, use a different color for ALL NEGATIVE graphs.**
a) 𝑦 = 𝑥1
𝑦 = −𝑥1
b) 𝑦 = 𝑥 2
𝑦 = −𝑥 2
(+) L: _____ R: _____
(-) L: _____ R: _____
(+) L: ____ R: ____
(-) L: _____ R: ____
e) 𝑦 = 𝑥 5
𝑦 = −𝑥 5
f) 𝑦 = 𝑥 6
𝑦 = −𝑥 6
(+) L: _____ R: _____
(-) L: _____ R: _____
(+) L: ____ R: ____
(-) L: _____ R: ____
c) 𝑦 = 𝑥 3
𝑦 = −𝑥 3
(+) L: ____ R: ____
(-) L: ____ R: _____
g) 𝑦 = 𝑥 7
𝑦 = −𝑥 7
(+) L: ____ R: ____
(-) L: ____ R: _____
d) 𝑦 = 𝑥 4
𝑦 = −𝑥 4
(+) L: _____ R: _____
(-) L: _____ R: _____
h) 𝑦 = 𝑥 8
𝑦 = −𝑥 8
(+) L: _____ R: _____
(-) L: _____ R: _____
Writing: Compare the graphs above. What patterns do you notice?
Which functions had matching end-behaviors (upup or down-down)?
Which functions had opposite end-behaviors (updown or down-up)?
What does the negative determine about the end
behavior of a function?
**Any other observations you made:
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Practice: Use your calculator to sketch a graph of each polynomial function (adjust the WINDOW to see
the key features). Determine the end behavior. (direction @ far left - far right).
1. 𝑦 = −4𝑥 2
2. 𝑦 = −5𝑥 3 + 9
3. 𝑦 = −4𝑥 4
4. 𝑦 = −8𝑥 5 − 1
End Behavior: ________________
End Behavior: ________________ End Behavior: ________________
5. y = 3x4 + 6x3  x2 + 12
6. y = 50  3x3 + 5x2
End Behavior: ________________
End Behavior: ________________ End Behavior: ________________
8. y = 12x4  x + 3x7  1
9. y = 5 + 2x + 7x2  5x3
End Behavior: ________________
End Behavior: ________________ End Behavior: ________________
7. y = x + x2 + 2
10. y = 20  5x6 + 3x  11x3
VOCABULARY: the lead coefficient is the coefficient on the term of highest degree (largest exponent
term) **Circle the lead term on each function above**
Using all graphs so far, complete the chart describing rules for the end behavior (left-right) of polynomials.
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Key Concept: The degree of a polynomial affects the shape of the graph, end behavior, and maximum
number of turning points!
VOCAB: Turning Points - places where the graph changes directions (increasing vs. decreasing) also
known as local maximums and/or local minimums or relative max/min
*For a polynomial with degree n, (𝑛 ≥ 1) there can be at most 𝑛 − 1 turning points**
If there are no turning points, the graph is either increasing or decreasing from _______ to _______.
How can you use your calculator
to determine a relative
maximum or minimum from a
polynomial graph?
**We use the maximum/minimum to identify intervals of x where the function is increasing or
decreasing. (The maximum or minimum tells us a turning point!)
Example #1: From left to right, the graph is decreasing when x is from ______ to _____. At the local
minimum (-1, -1) it changes directions and is increasing when x is between ____ and ____. Then, the
graph changes directions again at the local maximum (2, 2) and is decreasing when x is from ___ to ____.
Increasing Interval(s):
Decreasing Interval(s):
Practice: Sketch a graph of each function. Describe the shape including end behavior, turning points
(local max/min), and increasing/decreasing intervals (in terms of the x values).
Example #2: 𝑓(𝑥) = 2𝑥 3 − 5𝑥 2 + 1
End behavior: __________________
(turning point) Local Maximum:________________
(turning point) Local Minimum: _______________
Increasing Interval(s): - describe x using inequality statements
Decreasing Interval(s): - describe x using inequality statements
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1. 𝑓(𝑥) = 3𝑥 3 − 𝑥 − 3
End behavior: _______________
(turning point) Local Maximum:_____________
(turning point) Local Minimum: ____________
Increasing Interval(s): - describe x using inequality statements
Decreasing Interval(s): - describe x using inequality statements
2. 𝑓(𝑥) = −9𝑥 3 − 2𝑥 2 + 5𝑥 + 3
End behavior: _______________
(turning point) Local Maximum:_____________
(turning point) Local Minimum: ____________
Increasing Interval(s): - describe x using inequality statements
Decreasing Interval(s): - describe x using inequality statements
3. 𝑓(𝑥) = −0.5𝑥 4 + 𝑥 3 + 3𝑥 2
End behavior: _______________
(turning point) Local Maximum:_____________
(turning point) Local Minimum: ____________
Increasing Interval(s): - describe x using inequality statements
Decreasing Interval(s): - describe x using inequality statements
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Using what you know about end behavior, degree, and turning points:
Writing: Why is it easiest to determine the degree, end-behavior, and maximum number of turning
points from standard form? Use an example to support your answer.
Do MathXL 5.1
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