section 5.1 – polynomial functions students will be able to: graph polynomial functions,...
DESCRIPTION
Section 5.1 – Polynomial Functions A monomial is a real number, a variable, or a product of a real number and one or more variables with whole number exponents. The degree of a monomial in one variable is the exponent of the variable. A polynomial is a monomial or a sum of monomials The degree of a polynomial in one variable is the greatest degree among its monomial terms.TRANSCRIPT
![Page 1: Section 5.1 – Polynomial Functions Students will be able to: Graph polynomial functions, identifying zeros when suitable factorizations are available and](https://reader035.vdocument.in/reader035/viewer/2022062219/5a4d1b487f8b9ab0599a4662/html5/thumbnails/1.jpg)
Section 5.1 – Polynomial Functions Students will be able to:
•Graph polynomial functions, identifying zeros when suitable factorizations are available and
showing end behavior.
Lesson Vocabulary:Monomial Degree of a MonomialPolynomial Degree of a PolynomialPolynomial Function Standard FormTurning Point End Behavior
![Page 2: Section 5.1 – Polynomial Functions Students will be able to: Graph polynomial functions, identifying zeros when suitable factorizations are available and](https://reader035.vdocument.in/reader035/viewer/2022062219/5a4d1b487f8b9ab0599a4662/html5/thumbnails/2.jpg)
Section 5.1 – Polynomial Functions Essential Understanding:
A polynomial function has distinguishing “behaviors”.
You can look at its algebraic form and know something about it’s graph.
You can look at its graph and know something about its algebraic form.
![Page 3: Section 5.1 – Polynomial Functions Students will be able to: Graph polynomial functions, identifying zeros when suitable factorizations are available and](https://reader035.vdocument.in/reader035/viewer/2022062219/5a4d1b487f8b9ab0599a4662/html5/thumbnails/3.jpg)
Section 5.1 – Polynomial Functions A monomial is a real number, a variable, or a product
of a real number and one or more variables with whole number exponents.
The degree of a monomial in one variable is the exponent of the variable.
A polynomial is a monomial or a sum of monomials
The degree of a polynomial in one variable is the greatest degree among its monomial terms.
![Page 4: Section 5.1 – Polynomial Functions Students will be able to: Graph polynomial functions, identifying zeros when suitable factorizations are available and](https://reader035.vdocument.in/reader035/viewer/2022062219/5a4d1b487f8b9ab0599a4662/html5/thumbnails/4.jpg)
Section 5.1 – Polynomial Functions A polynomial with the variable x defines a polynomial function of x. The degree of the polynomial function
is the same as the degree of the polynomial.
![Page 5: Section 5.1 – Polynomial Functions Students will be able to: Graph polynomial functions, identifying zeros when suitable factorizations are available and](https://reader035.vdocument.in/reader035/viewer/2022062219/5a4d1b487f8b9ab0599a4662/html5/thumbnails/5.jpg)
Section 5.1 – Polynomial Functions You can classify a polynomial by its degree or by its
number of terms. Polynomials of degrees zero through five have specific names, as shown in this
table.
![Page 6: Section 5.1 – Polynomial Functions Students will be able to: Graph polynomial functions, identifying zeros when suitable factorizations are available and](https://reader035.vdocument.in/reader035/viewer/2022062219/5a4d1b487f8b9ab0599a4662/html5/thumbnails/6.jpg)
Section 5.1 – Polynomial Functions Problem 1:
Write each polynomial in standard form. What is the classification of each by degree? By number of
terms?
23 9 5x x 2 4 24 6 10 12x x x x
![Page 7: Section 5.1 – Polynomial Functions Students will be able to: Graph polynomial functions, identifying zeros when suitable factorizations are available and](https://reader035.vdocument.in/reader035/viewer/2022062219/5a4d1b487f8b9ab0599a4662/html5/thumbnails/7.jpg)
Section 5.1 – Polynomial Functions Problem 1:
Write each polynomial in standard form. What is the classification of each by degree? By number of
terms?
3 43 5x x x 5 23 4 2 10x x
![Page 8: Section 5.1 – Polynomial Functions Students will be able to: Graph polynomial functions, identifying zeros when suitable factorizations are available and](https://reader035.vdocument.in/reader035/viewer/2022062219/5a4d1b487f8b9ab0599a4662/html5/thumbnails/8.jpg)
Section 5.1 – Polynomial Functions The degree of a polynomial function affects the shape of its graph and determines the maximum number of turning points, or places where the graph changes
direction.
It also affects the end behavior, or the directions of the graph to the far left and to the far right.
![Page 9: Section 5.1 – Polynomial Functions Students will be able to: Graph polynomial functions, identifying zeros when suitable factorizations are available and](https://reader035.vdocument.in/reader035/viewer/2022062219/5a4d1b487f8b9ab0599a4662/html5/thumbnails/9.jpg)
Section 5.1 – Polynomial Functions The table on the next slide shows you examples of
polynomial functions and the four types of end behavior.
The table also shows intervals where the functions are increasing and decreasing.
A function is increasing when the y-values increase as x-values increase.
A function is decreasing when the y-values decrease as the x-values increase.
![Page 10: Section 5.1 – Polynomial Functions Students will be able to: Graph polynomial functions, identifying zeros when suitable factorizations are available and](https://reader035.vdocument.in/reader035/viewer/2022062219/5a4d1b487f8b9ab0599a4662/html5/thumbnails/10.jpg)
Section 5.1 – Polynomial Functions
![Page 11: Section 5.1 – Polynomial Functions Students will be able to: Graph polynomial functions, identifying zeros when suitable factorizations are available and](https://reader035.vdocument.in/reader035/viewer/2022062219/5a4d1b487f8b9ab0599a4662/html5/thumbnails/11.jpg)
Section 5.1 – Polynomial Functions In general, the graph of a polynomial function of
degree n (n > 1) has at most n – 1 turning points.
The graph of a polynomial function of odd degree has an even number of turning points.
The graph of a polynomial function of even degree has an odd number of turning points.
![Page 12: Section 5.1 – Polynomial Functions Students will be able to: Graph polynomial functions, identifying zeros when suitable factorizations are available and](https://reader035.vdocument.in/reader035/viewer/2022062219/5a4d1b487f8b9ab0599a4662/html5/thumbnails/12.jpg)
Section 5.1 – Polynomial Functions Problem 2:
Consider the leading term of each polynomial function. What is the end behavior of the graph?
Check your answer with a graphing calculator.a. y = 4x3 – 3x
b. y = -2x4 + 8x3 – 8x2 + 2
![Page 13: Section 5.1 – Polynomial Functions Students will be able to: Graph polynomial functions, identifying zeros when suitable factorizations are available and](https://reader035.vdocument.in/reader035/viewer/2022062219/5a4d1b487f8b9ab0599a4662/html5/thumbnails/13.jpg)
Section 5.1 – Polynomial Functions Problem 3:
What is the graph of each function? Describe the graph, including end behavior, turning points, and
increasing/decreasing intervals.
a. y = ½x3
b. y = 3x - x3
![Page 14: Section 5.1 – Polynomial Functions Students will be able to: Graph polynomial functions, identifying zeros when suitable factorizations are available and](https://reader035.vdocument.in/reader035/viewer/2022062219/5a4d1b487f8b9ab0599a4662/html5/thumbnails/14.jpg)
Section 5.1 – Polynomial Functions Problem 3b:
What is the graph of each function? Describe the graph, including end behavior, turning points, and
increasing/decreasing intervals.
a. y = -x3 + 2x2 – x – 2
b. y = x3 - 1
![Page 15: Section 5.1 – Polynomial Functions Students will be able to: Graph polynomial functions, identifying zeros when suitable factorizations are available and](https://reader035.vdocument.in/reader035/viewer/2022062219/5a4d1b487f8b9ab0599a4662/html5/thumbnails/15.jpg)
Section 5.1 – Polynomial Functions Suppose you are given a set of polynomial function
outputs. You know that their inputs are an ordered set of x-values in which consecutive x-values differ by a constant. By analyzing the
differences of consecutive y-values, it is possible to determine the least-degree polynomial function
that could generate the data.
If the FIRST DIFFERENCES are constant, the function is linear. If the SECOND DIFFERENCES
are constant, the function is quadratic, If the THIRD DIFFERENCES are constant, the function
is cubic, and so on!!
![Page 16: Section 5.1 – Polynomial Functions Students will be able to: Graph polynomial functions, identifying zeros when suitable factorizations are available and](https://reader035.vdocument.in/reader035/viewer/2022062219/5a4d1b487f8b9ab0599a4662/html5/thumbnails/16.jpg)
Section 5.1 – Polynomial Functions Problem 4:
What is the degree of the polynomial function that generates the data shown at the left?
![Page 17: Section 5.1 – Polynomial Functions Students will be able to: Graph polynomial functions, identifying zeros when suitable factorizations are available and](https://reader035.vdocument.in/reader035/viewer/2022062219/5a4d1b487f8b9ab0599a4662/html5/thumbnails/17.jpg)
Section 5.1 – Polynomial Functions Problem 4b:
What is the degree of the polynomial function that generates the data shown at the left?
![Page 18: Section 5.1 – Polynomial Functions Students will be able to: Graph polynomial functions, identifying zeros when suitable factorizations are available and](https://reader035.vdocument.in/reader035/viewer/2022062219/5a4d1b487f8b9ab0599a4662/html5/thumbnails/18.jpg)
Section 5.1 – Polynomial Functions Problem 5:
Write an equation for a polynomial function that has three turning points and end behavior up and up.