Transcript 6-1 Polynomial Functions - Mr. Hale's Classes

6-1 Polynomial
Functions
Objectives
Exploring Polynomial Functions
Modeling Data with a Polynomial Function
Vocabulary
A polynomial is a monomial or the sum of monomials.
The exponent of the variable in a term determines the degree
of that term.
Ordering the terms by descending order by degree. This
order demonstrates the standard form of a polynomial.
P(x) = 2x³ - 5x² - 2x + 5
Leading
Coefficient
Cubic
Term
Quadratic
Term
Linear
Term
Constant
Term
Degrees of a Polynomial
Degree
Name Using
Degree
Polynomial Example
Number
of Terms
Name Using
Number of Terms
0
Constant
6
1
Monomial
1
Linear
x+3
2
Binomial
2
Quadratic
3x²
3
Cubic
2x³ - 5x² - 2x
3
Trinomial
4
Quartic
x 4  3x 3  5 x 2
5
Quintic
4
Polynomial of 4
Terms
 x  3x  5 x  1
5
3
2
Classifying Polynomials
Write each polynomial in standard form. Then classify it by
degree and by number of terms.
a. 9 + x3
x3 + 9
The term with the largest
degree is x3,so the polynomial
is degree 3. It has two terms.
The polynomial is a
cubic binomial.
b. x3 – 2x2 – 3x4
–3x4 + x3 – 2x2
The term with the largest
degree is –3x4, so the
polynomial is degree 4.
It has three terms.
The polynomial is
a quartic trinomial.
Comparing Models
Using a graphing calculator, determine whether a linear,
quadratic, or cubic model best fits the values in the table.
x
0
2
4
6
8
y
2.8
5
6
5.5
4
Linear model
Enter the data. Use the LinReg, QuadReg, and CubicReg
options of a graphing calculator to find the best-fitting model
for each polynomial classification.
Graph each model and compare.
Quadratic model
The quadratic model appears to best fit the given values.
Cubic model
Real World Connection
The table shows data on the number of employees that a small
company had from 1975 to 2000. Find a cubic function to model the data.
Use it to estimate the number of employees in 1998. Let 0 represent 1975.
Year
1975
1980
1985
1990
1995
2000
Number of
Employees
60
65
70
60
55
64
Enter the data. To find a cubic model, use the CubicReg
option of a graphing calculator. Graph the model.
The function ƒ(x) = 0.0096x3 – 0.375x2 + 3.541x + 58.96 is
an approximate model for the cubic function.
To estimate the number of employees for 1988, you can use the Table function
option of a graphing calculator to find that ƒ(13) 62.72. According to the model,
there were about 62 employees in 1988.