1 §1.3 polynomial functions the student will learn about: polynomial function equations, graphs and...
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§1.3 Polynomial Functions
The student will learn about:
Polynomial function equations, graphs and roots.
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Introduction to PolynomialsA polynomial is an equation of the form
f (x) = an xn + an-1 x
n-1 + … + a1 x + a0
And graphs as a curve that wiggles back and forth across the x-axis.
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Definition Def: A polynomial function is a function of the form
f (x) = an xn + an-1 x
n-1 + … + a1 x + a0
For n a nonnegative integer called the degree of the polynomial.
The coefficients a0, a1, … , an are real
numbers with an ≠ 0.
The domain of a polynomial function is the set of real numbers.
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Graphs of a Polynomials
The shape of the graph of a polynomial function is connected to the degree and the sign of the leading coefficient an , and usually
wiggles back and forth across the x-axis.
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Graphs of a Polynomials Knowing the behavior of the ends of the graph helps us graph. What happens to y as x becomes very large or very small? That is, what happens at the tails?
Polynomial: Tail Behavior Chart
Leading coefficient positive
Leading coefficient negative
Degree even Both tails go up Both tails go down
Degree odd Left down, right up Left up, right down
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Graphs of a Polynomials
Polynomial: Tail Behavior Chart
Leading coefficient positive
Leading coefficient negative
Degree even Both tails go up Both tails go down
Degree odd Left down, right up Left up, right down
Remembering y = x 2 and y = x 3 may help!
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Graphs of a Polynomials The graph of a polynomial function of positive degree n can cross the x-axis at most n times.
An x-intercept is also called a zero or a root.
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Polynomial Root Approximation.
Theorem 1: If r is an x-intercept of the polynomial
P (x) = an xn + an - 1 x
n - 1 + an - 2 xn - 2 … + a1 x + a0
Let a n = the leading coefficient, and
Then |r| < (1 + | b | ) / | an | ) .
Let b = absolute value of the largest coefficient -
This gives us the maximum and minimum values for roots and helps in our search.
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Solving a Polynomial
Solutions may be found algebraically. Let f (x) = 0 and solve for x :
a. Factor – a bit of work that sometimes requires synthetic division (whatever that is).
b. Use the quadratic formula when you have second degree factors.
c. Use a graphing calculator and use the calc and zero buttons.
I love my calculator!
Remember you are responsible for both algebraic and calculator methods.
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Graphs of a Polynomials
The graph of a polynomial has origin (point, odd) symmetry, if all of the exponents are odd.
The graph of a polynomial has y-axis (even, vertical) symmetry if all of the exponents are even.
y = x 4 – 2x 2 - 1
y = x 3 – 2x
Note: both (x, y) and – x, y) are on the graph
Note: both (x, y) and (- x, - y) are on the graph
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Graphs of a Polynomials
A polynomial function is continuous with no holes or breaks.
The graph of a polynomial function of positive degree n can have at most (n – 1) turning points. These points are called relative maximum and relative minimum points.
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Exampley = x4 + 2 x3 – 4x2 - 8 x
I will factor using grouping
x4 + 2x3 – 4x2 - 8 x = x 3 (x + 2) – 4x (x + 2)
1. x-intercepts
Hence the x-intercepts are 0, 2, and – 2 as a root of multiplicity two.
I will factor using grouping
x4 + 2x3 – 4x2 - 8 x = x 3 (x + 2) – 4x (x + 2)
= (x 3 – 4x)(x + 2)
I will factor using grouping
x4 + 2x3 – 4x2 - 8 x = x 3 (x + 2) – 4x (x + 2)
= (x 3 – 4x)(x + 2) = x ( x 2 – 4)(x + 2)
= x (x – 2)(x + 2)(x + 2)
I will factor using grouping
x4 + 2x3 – 4x2 - 8 x = x 3 (x + 2) – 4x (x + 2)
= (x 3 – 4x)(x + 2) = x ( x 2 – 4)(x + 2)
Note that – 8 < r < 8. |r| < ( 1 + | b | ) / | an |
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Exampley = x4 + 2 x3 – 4x2 - 8 x
2. y-intercept
Allowing x = 0 in the original equation gives a y = 0 which means that the y-intercept is the origin,
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Exampley = x4 + 2 x3 – 4x2 - 8 x
3. Tail behavior.
The degree is even and the leading coefficient is positive so both ends go up.
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Exampley = x4 + 2 x3 – 4x2 - 8 x
3. Symmetry.
The function is neither even nor odd so it has no symmetry.
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Note: We will be finding the two relative minimum and the relative maximum with calculus although you already know how to do this with a graphing calculator!
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Polynomial Function Reviewf (x) = an x
n + an-1 xn-1 + … + a1 x + a0
The shape of the graph is connected to the degree and the leading coefficient.
Finding the x-intercepts is important.
The graph wiggles back and forth across the x-axis.
Finding the relative minimums and maximums will become important.
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Graphs of a Polynomials
Graphing polynomials is difficult and time consuming. Calculus will aide us greatly in determining the x-intercepts, (the y-intercept is always easy!), and the maximum and minimum points of a polynomial. Although this is helpful in graphing it is really more helpful in life as these characteristics have many applications.
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Summary.
• We had an introduction to polynomial functions and learned some of the properties of these functions.
• We had an introduction to rational functions.
• We learned about both the vertical and horizontal asymptotes associated with rational functions.
• We worked through an application that involved rational functions.
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ASSIGNMENT
§1.3; Page 13; 1 – 27 odd.