2.1 Evaluate and Graph Polynomial Functions...Decide whether the function is a polynomial function....
Transcript of 2.1 Evaluate and Graph Polynomial Functions...Decide whether the function is a polynomial function....
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2.1 Evaluate and Graph Polynomial Functions(NG pg. 75)
Vocabulary
Polynomial:
A monomial or a sum of monomials.
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Polynomial function:
A function of the form f(x) = anxn + ... where an ≠0, the exponents are whole numbers, and the coefficients are all real numbers.
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Degree of a polynomial function:
The exponent in the term of a polynomial function where the variable is raised to the greatest power.
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Leading coefficient:
The coefficient in the term of a polynomial function that has the greatest exponent.
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Standard form of a polynomial function:
When the terms are written in descending order of exponents from left to right.
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Synthetic substitution:
An alternate method to evaluate a polynomial function using fewer operations than direct substitution.
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Example 1:
Decide whether the function is a polynomial function. If so, write it in standard form and state its degree and leading coefficient.
a. f(x) = 3x3 + 4x2.5 6x2
The function __________ a polynomial function because the term __________ has an exponent that is ___________________________.
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b. f(x) = x2 + 3.7x + 9x4
The function __________ a polynomial function written as __________________in its standard form. It has degree _____ and a leading coefficient of _____.
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Checkpoint #1:
State the degree and leading coefficient of f(x) = 2x3 + 2x2 3x4 + 5.
degree: _____leading coefficient: _____
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Example 2 ½:
Use direct substitution to evaluate g(x) = 4x3 + 3x2 7 when x = 2.
Substitute 2 for x, then simplify.
g(2) =
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Checkpoint #1 ½ (pg. 78 put in margin):
Use direct substitution to evaluate f(x) = 2x3 + 8x2 + 3x 1 when x = 4.
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Example 2:
Use synthetic substitution to evaluate f(x) = 2x4 + 3x3 6x2 + 3 when x = 2.
Write the coefficients of f(x) in order of _______________ exponents. Write the value of x to the left. Bring down the leading coefficient. Multiply the leading coefficient by ___ and write the product under the second coefficient. __________. Multiply the previous sum by ___ and write the product under the third coefficient. Add. Repeat for all of the remaining coefficients.
___ 2 3 6 0 3 coefficients
f(2) = _____
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Checkpoint #2 on pg. 78.
Complete the following exercise using the function f(x) = x4 + 3x3 + x2 4x 1.
Evaluate f(x) for x = 2 using synthetic substitution.
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Classwork: NG pg. 79 #114 all
and
Homework: Textbook pg. 69 #1 7 all
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Example 3:
(a) Graph the function f(x) = x3 + 2x2 + 2x 1, (b) find the domain and the range of the
function,(c) describe the degree and leading coefficient of the function, and(d) decide any symmetries in the graph.
a. Make a table of values and plot the corresponding points. Connect the points with a smooth curve.
x 1 0 1 2 3f(x) __ __ __ __ __
b. The domain is __________________ and the range is __________________.
c. The degree is _____ and the leading coefficient is _________________.
d. The function is _____________________ because f(x) = (x)3 + 2(x)2 + 2(x) 1 = ____________________ which is not equal _____ or _____. The graph has _________________.
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Even, Odd, or Neither?
Even:• must have even exponents (on variables)• graph is symmetric to yaxis
Odd:• must have odd exponents (on variables)• graph is symmetric to origin
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Checkpoint #3 on pg. 78.
Graph f(x) = x4 + 3x3 + x2 4x 1.
a. Table of values
b. Domain = Range =
c. Degree= Leading coefficient =
d. Even, odd, or neither?
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End Behavior of Polynomial Functions
For the graph of f(x) = anxn + .............. :• If n is odd and an > 0, then f(x) ___ as x +∞ and f(x) ___ as x ∞.• If n is odd and an < 0, then f(x) ___ as x +∞ and f(x) ___ as x ∞.• If n is even and an > 0, then f(x) ___ as x +∞ and f(x) ___ as x ∞.• If n is even and an < 0, then f(x) ___ as x +∞ and f(x) ___ as x ∞.
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Example 4:
Domain=
Range =
Degree =
Leading Coefficient =
Even, odd, or neither?
Symmetry?
End behavior: f(x) ___ as x ∞ and f(x) ___ as x +∞
Intervals of increase/decrease:
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Domain=
Range =
Degree =
Leading Coefficient =
Even, odd, or neither?
Symmetry?
End behavior: f(x) ___ as x ∞ and f(x) ___ as x +∞
Intervals of increase/decrease:
Example 5:
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Example 6:
Domain=
Range =
Degree =
Leading Coefficient =
Even, odd, or neither?
Symmetry?
End behavior: f(x) ___ as x ∞ and f(x) ___ as x +∞
Intervals of increase/decrease:
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Example 7:
Domain=
Range =
Degree =
Leading Coefficient =
Even, odd, or neither?
Symmetry?
End behavior: f(x) ___ as x ∞ and f(x) ___ as x +∞
Intervals of increase/decrease:
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Example 8:
Domain=
Range =
Degree =
Leading Coefficient =
Even, odd, or neither?
Symmetry?
End behavior: f(x) ___ as x ∞ and f(x) ___ as x +∞
Intervals of increase/decrease:
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Classwork/Homework:
Even & Odd Functions Practice
and
NG pg. 80 #15 21 all
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Textbook pg. 69 #8 20 all
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