Algebra II Pre AP
NOTES – Graphing Polynomial Functions
End Behavior
The end behavior of a polynomial function (how the graph begins and ends) depends on the leading
coefficient and the degree of the polynomial.
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If the degree of the polynomial is odd, the end behavior of the function will be the same as a line.
o
Positive lead coefficient, Odd degree polynomial
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Negative lead coefficient, Odd degree polynomial
If the degree of the polynomial is even, the end behavior of the function will be the same as a parabola.
o
Positive lead coefficient, Even degree polynomial
o
Negative lead coefficient, Even degree polynomial
Real Roots
The real roots or zeroes of a polynomial function are the x intercepts of the graphed function.
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If a real number occurs as a root an odd number of times, the graph will cross the x -axis at that point.
The multiplicity of the root will determine how the graph appears as it passes through the x -axis.
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If a real number occurs as a root an even number of times, the graph will “bounce” on the x -axis. The
multiplicity of the root will determine how the graph appears as it bounces off the x -axis.
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If x is a common factor of the polynomial, the zero is a root.
Imaginary Roots
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Imaginary roots cannot be graphed in the real number plane.
Imaginary roots occur in conjugate pairs.
Y-intercept – Let x = 0 and solve for y .
Examples:
1. f ( x)  3x 2  x  5 2 x  5
4
roots:________________________
y-intercept: _______________
lead term: _______________
sign of lead coefficient _____
degree of polynomial _____
2. f ( x)  2
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2  x ( x  2)  x  1
2
roots:________________________
y-intercept: _______________
lead term: _______________
sign of lead coefficient _____
degree of polynomial _____
3. f  x   x  3x  3x  3x  2
4
3
2
a) possible rational roots
____________________________
b) pni chart
c) roots ___________________
d) y intercept _________
Additional Facts regarding the Graphing of Polynomial Functions
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The total number of roots (real and imaginary) is the same as the degree of the polynomial.
The graph of a polynomial function will have one less change of direction than the degree of the
polynomial (or less than that by an even number).
A sketch of the graph of a polynomial function can be made using the roots and knowledge of the end
behavior. If the function is not in factored form, the Rational Root Theorem, Descartes’ Rule of Signs,
the Remainder and Factor Theorems, and the Upper and Lower Bound Theorems (LUB & GLB) are all
used to help find all of the roots of the function in order to sketch the graph.