n The coefficients a i are real numbers, n is a whole number. The domain of any polynomial is R.

Size: px
Start display at page:

Download "n The coefficients a i are real numbers, n is a whole number. The domain of any polynomial is R."

Transcription

1 Section 4.1: Quadratic Functions Definition: A polynomial function has the form P ( x ) = a x n+ a x n a x 2+ a x + a (page 326) n n The coefficients a i are real numbers, n is a whole number. The domain of any polynomial is R. Reminder: 1. f ( x) = a is a constant function y= a. 2. f ( x) 0 = a x+ a is a linear function y= mx+ b Definition: The quadratic function f ( x) = a x2+ a x+ a can be written in the general form f ( x) = ax + bx+ c. Graphing Quadratic Functions Example: Compare y= x2, y= x2, y= 2x2, y= 1/2x2. a) Graphs with positive leading coefficients open up, and have a lowest point. b) Graphs with negative leading coefficients open down and have a highest point. c) The highest or lowest point on a parabola is the vertex (in these examples (0,0)). d) The vertical line passing through the vertex of a parabola is the axis of symmetry, in the examples x= 0. 1

2 Section 4.1: Quadratic Functions Example: Compare: y= x2, y= x2 2, y= x2+ 3 Example: Compare: y= x2, y= ( x 2) 2, y= ( x+ 3) 2 Example: Compare y= ( x+ 2) 2 1 and y= ( x 1) 2+ 3: The parabola y= ( x+ 2) 2 1 opens up, has vertex ( 2, 1) and axis of symmetryx= 2; the parabola y= ( x 1) 2+ 3 opens down, has vertex (1,3) and axis of symmetry x= 1. A quadratic function f ( x) = ax2+ bx+ ccan be written in graphing form (page 328) f ( x) = a( x h) 2+ k. The graph is a parabola that opens upward ifa> 0, downward ifa< 0, has vertex ( h, k ) and line of symmetry x= h. Follow your author's instructions for graphing on pages Page , 38 on WileyPlus 2

3 Section 4.1: Quadratic Functions If we use completing he square on the general parabola f ( x) = ax2+ bx+ c, f ( x) = ax2+ bx+ c f ( x) = a( x2+ b a x) + c f ( x) = a( x + a b x + ) 4 b c a2 + 4 b a 2 f ( x) = a( x+ b ) c b a 4a Easy method of finding the vertex of the parabola: The vertex of the parabola f ( x) = ax2+ bx+ c is given by ( h, k ) = b, f b. 2a 2a Page

4 Section 4.2: Polynomial Functions of Higher Degree Factoring easy quadratic polynomials reminder: Factor x2 49, x2 7, x2 48, x2+ 49, x2+ 5 Definition: A polynomial function has the form P ( x ) = a x n+ a x n a x 2+ a x + a (page 326) n n The coefficients a i are real numbers, n is a whole number. The domain of any polynomial is R. Polynomials Functions? 5 f ( x) = 3 x g( x) = 2 x( x 3) 2( x 4) 5 h( x) = 3 3x+ x 1 j( x) 5 = 3 f ( x) = x4 5x2+ 8x 2 6 x Use some simple observations to graph polynomials: I. Polynomial Graph Qualities (See page 344.) The graph of every polynomial function is (a) smooth, contains no sharp corners or cusps; (b) continuous, has no gaps or holes and can be drawn without lifting the pencil from the paper. 4

5 Section 4.2: Polynomial Functions of Higher Degree II. The Leading Term Test (End Behavior) Power functions, degree n, y= f ( x) = axn: Example: Graph y= x2, y= x4, y= x6 y= f ( x) = axn is u-shaped if n is even, and opens upward if a> 0. The domain is(, ); the range is [0, ). Example: Graph y= x, y= x3, and y= x5. f ( x) = axn is s-shaped for n odd, rising on the right, falling on the left when a> 0. Both the domain and range are (, ). The end behavior of a polynomial refers to what the graph does beyond the right- and leftmost x-intercepts. End Behavior Leading Coefficient Test (page 352) As x moves without bound to the left or right, the polynomial function P( x) = a x n a x+ a n 1 0 has the behavior as the power function y= a n nx : If n is odd and a n> 0 (positive), the graph of P( x ) rises right and falls left: As x, y ; and as x, y. If n is odd and a n< 0 is (negative), the graph of P( x ) falls right and rises left: As x, y ; and as x, y. 5

6 Section 4.2: Polynomial Functions of Higher Degree End Behavior Leading Coefficient Test (continued) If n is even and a n> 0 positive, P( x ) rises both right and left: As x, y ; and as x, y. If n is even and a n< 0 negative, P( x ) falls both right and left: As x, y ; and as x, y. III. Zeros (x-intercepts) At x-intercepts of a graph, the graph either crosses or is tangent to (touches) the x-axis. Between consecutive x-intercepts, the graph is entirely above or entirely below the x-axis. A factored polynomial P( x ) = 0 gives the x-intercepts ( a,0). Example: The polynomial P( x) = 2x4 2x3 6x2+ 2x+ 4 can be factored asp( x) = 2( x 2)( x 1)( x+ 1) 2. Setting each factor to 0, the intercepts are(2,0), (1,0), and( 1,0), having multiplicity 1, 1, and 2, respectively. Definition: If P( x ) is a polynomial function and c is a number for which P( c ) = 0, then c is called a zero of P. 6

7 Section 4.2: Polynomial Functions of Higher Degree Real Zeros of Polynomial Functions, page 259: The following are equivalent: 1) a is a zero of the function P( x ); 2) x= a is a solution(root) of equation P( x ) = 0; 3) ( x a) is a factor of P( x ); 4) ( a,0) is an x-intercept of P( x ). Even and Odd Multiplicity A factor of ( x a) k yields a repeated zero of multiplicity k. If a is a zero of odd multiplicity, the graph of the function crosses the x-axis at( a,0). If a is a zero of even multiplicity, the graph of the function is tangent to the x-axis at( a,0). Analyzing the Graph of a Polynomial (example 7 page 353) a. Graphing examples b. Find the polynomial of minimum degree with a graph that touches at x-intercept 2, crosses at x-intercept -1 and has y-intercept (0,-15). 7

8 Section 4.2: Polynomial Functions of Higher Degree Example: Sketch P( x) = 3x4+ 12x3 12x2 x-intercepts: multiplicity: cross/touch: y-intercept End behavior Power function: As x, y ; as x, y. 8

9 Section 4.2: Polynomial Functions of Higher Degree Example: Sketch P( x) = 2( x 1) 3( x+ 2) 2 x-intercepts: multiplicity: cross/touch: y-intercept End behavior Power function: As x, y ; as x, y. 9

10 Section 4.3: Polynomial Division Example: Divide x 4 x 2 + 3x 10 using synthetic division. x+ 5 Synthetic division can only be used with x c. Division Algorithm for Polynomials (page 360) P( x) = Q( x) + r( x) or P( x) = d( x) Q( x) + r( x) d( x) d( x) P( x ) is the dividend, d( x) is the divisor, q( x) is the quotient, and r( x ) is the remainder. Notice that if d( x ) is of the formx c, then r( x ) is a number and P( x) = ( x c) Q( x) + R. 10

11 Section 4.4: Real Zeros of a Polynomial Factoring easy quadratic polynomials reminder: Factor: x2 49= x2 7= x2 48= x2+ 49= x2+ 7= x2+ 48= Remainder Theorem (page 367): If P( x ) is a polynomial divided byx a, then the remainder is P( a ), r= P( a). Example: Show the remainder theorem and synthetic division yield the same remainder when P( x) = x4 x2+ 3x 10 is divided byx+ 5. Example: Show the remainder theorem and synthetic division yield the same remainder when f ( x) = 4x3 3x2 8x 4 is divided byx+ 2. Factor Theorem (page 367): A polynomial P( x ) has a factor x a if and only if P( a ) = 0. Example: Doesx+ 2orx 2divide f ( x) = 4x3 3x2 8x 4evenly? On calculator, use TABLE. 11

12 Section 4.4: Real Zeros of a Polynomial The Remainder in Synthetic Division Summary: 1. The remainder r, gives the value of f at x= c, that is r= f ( c). 2. If r= 0, ( x c) is a factor of f ( x ). 3. If r= 0, ( c,0) is an x-intercept of the graph of f. Page Example: Factor f ( x) = 2x3 7x2 17x+ 10: f ( x) = 2x3 7x2 17x+ 10 = (2x 1)( x+ 2)( x 5) = 2( x 1/2)( x+ 2)( x 5) What is the leading coefficient? What is the constant term? What are the zeros? The real zeros of a polynomial are related to the leading coefficient and the constant term. The relationship is indicated in the Rational Zero Test. Rational Zeros Test (RZT) (page 369) If P( x) = a xn+ a xn a x2+ a x+ a has integer 12 n n coefficients, every rational zero of P has the form p/q where p divides a evenly and q divides a 0 nevenly. Rational zero = p factors of constant term q =± factors of leading coefficient

13 Section 4.4: Real Zeros of a Polynomial Example: List the possible rational zeros of f ( x) = 2x3 7x2 17x2+ 10 Page , How many zeros? RZT to find Candidate Zeros: Find Zeros: Zeros: Factorization: 56. How many zeros? RZT to find Candidate Zeros: Find Zeros: Zeros: Factorization: 13

14 Section 4.5: Complex Zeros, Fundamental Theorem of Algebra The Fundamental Theorem of Arithmetic: Any number can be factored into primes, not necessarily distinct. Example: Factor 24 into primes. The Fundamental Theorem of Algebra (page 382) Every P( x ) polynomial of degree n has at least one zero in the complex number system. n Zeros Theorem (page 383) Every P( x ) polynomial of degree n can be expressed as a product of n linear factors so has n linear factors, not necessarily distinct. Look at examples a. through e. on page 383. Example: Solve x2 6x= 10 Complex Zeros Occur in Conjugate Pairs Let f ( x ) be a polynomial function that has real coefficients. If r= a+ bi is a zero of f, the conjugate r= a bi is also a zero of f (page 383). Page 388 8, 14, 24, 34, 42 14

15 Section 4.6: Rational Functions Reminders: An equation of a vertical line has the form ; an equation of a horizontal line has the form. Definition: A rational function has the form f ( x) = n( x), n( x) and d( x ) are polynomials and d( x) 0. d( x) Example: Graph the rational function f ( x) = 1 1 x 2 + using transformations. The domain of f ( x) = 1 1 x 2 + is. Function f ( x ) is a transformation (shift right 2, up 1) of the graph of y= 1 x. Note: As x 2, f ( x) ; As x 2 +, f ( x). The line x= 2 is a vertical asymptote. Also note: As x, f ( x ) 1 + ; and as x, f ( x) 1. The line y= 1 is a horizontal asymptote. Asymptotes are lines that a function graph approaches but never touches (sometimes horizontal asymptotes are crossed). 15

16 Section 4.6: Rational Functions Diagrams of Vertical Asymptotes (page 392): x a+, f ( x) x a+, f ( x) x a, f ( x) x a, f ( x) Locating Vertical Asymptotes (page 393) Vertical Asymptotes occur at zeros of the denominator. See Note : If a rational function is not in lowest terms, the common factor x a results in a hole in the graph. 16

17 Section 4.6: Rational Functions Diagrams of Horizontal Asymptotes (page 392): x, f ( x) b + x, f ( x) b + x, f ( x) b x, f ( x) b Locating Horizontal Asymptotes (page 395) Determine the degrees of the numerator and denominator, n and m respectively. (a) If n< m, the x-axis, y= 0, is a horizontal asymptote. (b) If n= m, horizontal asymptote y= an. b (c) If n> m, there is no horizontal asymptote (there is another type of asymptote). Graphing Rational Functions, page 398 Page , 46, 56 m 17

Chapter 2 Formulas and Definitions:

Chapter 2 Formulas and Definitions: Chapter 2 Formulas and Definitions: (from 2.1) Definition of Polynomial Function: Let n be a nonnegative integer and let a n,a n 1,...,a 2,a 1,a 0 be real numbers with a n 0. The function given by f (x)

More information

MAT116 Final Review Session Chapter 3: Polynomial and Rational Functions

MAT116 Final Review Session Chapter 3: Polynomial and Rational Functions MAT116 Final Review Session Chapter 3: Polynomial and Rational Functions Quadratic Function A quadratic function is defined by a quadratic or second-degree polynomial. Standard Form f x = ax 2 + bx + c,

More information

Polynomial and Rational Functions. Chapter 3

Polynomial and Rational Functions. Chapter 3 Polynomial and Rational Functions Chapter 3 Quadratic Functions and Models Section 3.1 Quadratic Functions Quadratic function: Function of the form f(x) = ax 2 + bx + c (a, b and c real numbers, a 0) -30

More information

Lesson 2.1: Quadratic Functions

Lesson 2.1: Quadratic Functions Quadratic Functions: Lesson 2.1: Quadratic Functions Standard form (vertex form) of a quadratic function: Vertex: (h, k) Algebraically: *Use completing the square to convert a quadratic equation into standard

More information

1) The line has a slope of ) The line passes through (2, 11) and. 6) r(x) = x + 4. From memory match each equation with its graph.

1) The line has a slope of ) The line passes through (2, 11) and. 6) r(x) = x + 4. From memory match each equation with its graph. Review Test 2 Math 1314 Name Write an equation of the line satisfying the given conditions. Write the answer in standard form. 1) The line has a slope of - 2 7 and contains the point (3, 1). Use the point-slope

More information

The Graph of a Quadratic Function. Quadratic Functions & Models. The Graph of a Quadratic Function. The Graph of a Quadratic Function

The Graph of a Quadratic Function. Quadratic Functions & Models. The Graph of a Quadratic Function. The Graph of a Quadratic Function 8/1/015 The Graph of a Quadratic Function Quadratic Functions & Models Precalculus.1 The Graph of a Quadratic Function The Graph of a Quadratic Function All parabolas are symmetric with respect to a line

More information

Chapter 2 Polynomial and Rational Functions

Chapter 2 Polynomial and Rational Functions Chapter 2 Polynomial and Rational Functions Section 1 Section 2 Section 3 Section 4 Section 5 Section 6 Section 7 Quadratic Functions Polynomial Functions of Higher Degree Real Zeros of Polynomial Functions

More information

Polynomial Functions. Linear Graphs and Linear Functions 1.3

Polynomial Functions. Linear Graphs and Linear Functions 1.3 Polynomial Functions Linear Graphs and Linear Functions 1.3 Forms for equations of lines (linear functions) Ax + By = C Standard Form y = mx +b Slope-Intercept (y y 1 ) = m(x x 1 ) Point-Slope x = a Vertical

More information

All quadratic functions have graphs that are U -shaped and are called parabolas. Let s look at some parabolas

All quadratic functions have graphs that are U -shaped and are called parabolas. Let s look at some parabolas Chapter Three: Polynomial and Rational Functions 3.1: Quadratic Functions Definition: Let a, b, and c be real numbers with a 0. The function f (x) = ax 2 + bx + c is called a quadratic function. All quadratic

More information

3.1 Power Functions & Polynomial Functions

3.1 Power Functions & Polynomial Functions 3.1 Power Functions & Polynomial Functions A power function is a function that can be represented in the form f() = p, where the base is a variable and the eponent, p, is a number. The Effect of the Power

More information

Chapter 2: Polynomial and Rational Functions

Chapter 2: Polynomial and Rational Functions Chapter 2: Polynomial and Rational Functions Section 2.1 Quadratic Functions Date: Example 1: Sketching the Graph of a Quadratic Function a) Graph f(x) = 3 1 x 2 and g(x) = x 2 on the same coordinate plane.

More information

Lesson 7.1 Polynomial Degree and Finite Differences

Lesson 7.1 Polynomial Degree and Finite Differences Lesson 7.1 Polynomial Degree and Finite Differences 1. Identify the degree of each polynomial. a. 3x 4 2x 3 3x 2 x 7 b. x 1 c. 0.2x 1.x 2 3.2x 3 d. 20 16x 2 20x e. x x 2 x 3 x 4 x f. x 2 6x 2x 6 3x 4 8

More information

Polynomial and Rational Functions. Copyright Cengage Learning. All rights reserved.

Polynomial and Rational Functions. Copyright Cengage Learning. All rights reserved. 2 Polynomial and Rational Functions Copyright Cengage Learning. All rights reserved. 2.2 Polynomial Functions of Higher Degree Copyright Cengage Learning. All rights reserved. What You Should Learn Use

More information

Polynomial Functions and Models

Polynomial Functions and Models 1 CA-Fall 2011-Jordan College Algebra, 4 th edition, Beecher/Penna/Bittinger, Pearson/Addison Wesley, 2012 Chapter 4: Polynomial Functions and Rational Functions Section 4.1 Polynomial Functions and Models

More information

H-Pre-Calculus Targets Chapter I can write quadratic functions in standard form and use the results to sketch graphs of the function.

H-Pre-Calculus Targets Chapter I can write quadratic functions in standard form and use the results to sketch graphs of the function. H-Pre-Calculus Targets Chapter Section. Sketch and analyze graphs of quadratic functions.. I can write quadratic functions in standard form and use the results to sketch graphs of the function. Identify

More information

Table of contents. Polynomials Quadratic Functions Polynomials Graphs of Polynomials Polynomial Division Finding Roots of Polynomials

Table of contents. Polynomials Quadratic Functions Polynomials Graphs of Polynomials Polynomial Division Finding Roots of Polynomials Table of contents Quadratic Functions Graphs of Polynomial Division Finding Roots of Jakayla Robbins & Beth Kelly (UK) Precalculus Notes Fall 2010 1 / 65 Concepts Quadratic Functions The Definition of

More information

Polynomial Functions. x n 2 a n. x n a 1. f x = a o. x n 1 a 2. x 0, , a 1

Polynomial Functions. x n 2 a n. x n a 1. f x = a o. x n 1 a 2. x 0, , a 1 Polynomial Functions A polynomial function is a sum of multiples of an independent variable raised to various integer powers. The general form of a polynomial function is f x = a o x n a 1 x n 1 a 2 x

More information

Chapter 3: Polynomial and Rational Functions

Chapter 3: Polynomial and Rational Functions Chapter 3: Polynomial and Rational Functions 3.1 Polynomial Functions and Their Graphs A polynomial function of degree n is a function of the form P (x) = a n x n + a n 1 x n 1 + + a 1 x + a 0 The numbers

More information

The coordinates of the vertex of the corresponding parabola are p, q. If a > 0, the parabola opens upward. If a < 0, the parabola opens downward.

The coordinates of the vertex of the corresponding parabola are p, q. If a > 0, the parabola opens upward. If a < 0, the parabola opens downward. Mathematics 10 Page 1 of 8 Quadratic Relations in Vertex Form The expression y ax p q defines a quadratic relation in form. The coordinates of the of the corresponding parabola are p, q. If a > 0, the

More information

Math Analysis Chapter 2 Notes: Polynomial and Rational Functions

Math Analysis Chapter 2 Notes: Polynomial and Rational Functions Math Analysis Chapter Notes: Polynomial and Rational Functions Day 13: Section -1 Comple Numbers; Sections - Quadratic Functions -1: Comple Numbers After completing section -1 you should be able to do

More information

MAT 129 Precalculus Chapter 5 Notes

MAT 129 Precalculus Chapter 5 Notes MAT 129 Precalculus Chapter 5 Notes Polynomial and Rational Functions David J. Gisch and Models Example: Determine which of the following are polynomial functions. For those that are, state the degree.

More information

Math 1310 Section 4.1: Polynomial Functions and Their Graphs. A polynomial function is a function of the form ...

Math 1310 Section 4.1: Polynomial Functions and Their Graphs. A polynomial function is a function of the form ... Math 1310 Section 4.1: Polynomial Functions and Their Graphs A polynomial function is a function of the form... where 0,,,, are real numbers and n is a whole number. The degree of the polynomial function

More information

Power and Polynomial Functions. College Algebra

Power and Polynomial Functions. College Algebra Power and Polynomial Functions College Algebra Power Function A power function is a function that can be represented in the form f x = kx % where k and p are real numbers, and k is known as the coefficient.

More information

1 Functions and Graphs

1 Functions and Graphs 1 Functions and Graphs 1.1 Functions Cartesian Coordinate System A Cartesian or rectangular coordinate system is formed by the intersection of a horizontal real number line, usually called the x axis,

More information

Chapter 3 Page 1 of 23. Lecture Guide. Math College Algebra Chapter 3. to accompany. College Algebra by Julie Miller

Chapter 3 Page 1 of 23. Lecture Guide. Math College Algebra Chapter 3. to accompany. College Algebra by Julie Miller Chapter 3 Page 1 of 23 Lecture Guide Math 105 - College Algebra Chapter 3 to accompany College Algebra by Julie Miller Corresponding Lecture Videos can be found at Prepared by Stephen Toner & Nichole DuBal

More information

Polynomials. Exponents. End Behavior. Writing. Solving Factoring. Graphing. End Behavior. Polynomial Notes. Synthetic Division.

Polynomials. Exponents. End Behavior. Writing. Solving Factoring. Graphing. End Behavior. Polynomial Notes. Synthetic Division. Polynomials Polynomials 1. P 1: Exponents 2. P 2: Factoring Polynomials 3. P 3: End Behavior 4. P 4: Fundamental Theorem of Algebra Writing real root x= 10 or (x+10) local maximum Exponents real root x=10

More information

Unit 2 Polynomial Expressions and Functions Note Package. Name:

Unit 2 Polynomial Expressions and Functions Note Package. Name: MAT40S Mr. Morris Unit 2 Polynomial Expressions and Functions Note Package Lesson Homework 1: Long and Synthetic p. 7 #3 9, 12 13 Division 2: Remainder and Factor p. 20 #3 12, 15 Theorem 3: Graphing Polynomials

More information

Let's look at some higher order equations (cubic and quartic) that can also be solved by factoring.

Let's look at some higher order equations (cubic and quartic) that can also be solved by factoring. GSE Advanced Algebra Polynomial Functions Polynomial Functions Zeros of Polynomial Function Let's look at some higher order equations (cubic and quartic) that can also be solved by factoring. In the video,

More information

f (x) = x 2 Chapter 2 Polynomial Functions Section 4 Polynomial and Rational Functions Shapes of Polynomials Graphs of Polynomials the form n

f (x) = x 2 Chapter 2 Polynomial Functions Section 4 Polynomial and Rational Functions Shapes of Polynomials Graphs of Polynomials the form n Chapter 2 Functions and Graphs Section 4 Polynomial and Rational Functions Polynomial Functions A polynomial function is a function that can be written in the form a n n 1 n x + an 1x + + a1x + a0 for

More information

Section 2: Polynomial and Rational Functions

Section 2: Polynomial and Rational Functions Section 2: Polynomial and Rational Functions The following maps the videos in this section to the Texas Essential Knowledge and Skills for Mathematics TAC 111.42(c). 2.01 Quadratic Functions Precalculus

More information

Introduction. A rational function is a quotient of polynomial functions. It can be written in the form

Introduction. A rational function is a quotient of polynomial functions. It can be written in the form RATIONAL FUNCTIONS Introduction A rational function is a quotient of polynomial functions. It can be written in the form where N(x) and D(x) are polynomials and D(x) is not the zero polynomial. 2 In general,

More information

3 Polynomial and Rational Functions

3 Polynomial and Rational Functions 3 Polnomial and Rational Functions 3.1 Quadratic Functions and Models 3.2 Polnomial Functions and Their Graphs 3.3 Dividing Polnomials 3.4 Real Zeros of Polnomials 3.5 Comple Zeros and the Fundamental

More information

Section Properties of Rational Expressions

Section Properties of Rational Expressions 88 Section. - Properties of Rational Expressions Recall that a rational number is any number that can be written as the ratio of two integers where the integer in the denominator cannot be. Rational Numbers:

More information

ZEROS OF POLYNOMIAL FUNCTIONS ALL I HAVE TO KNOW ABOUT POLYNOMIAL FUNCTIONS

ZEROS OF POLYNOMIAL FUNCTIONS ALL I HAVE TO KNOW ABOUT POLYNOMIAL FUNCTIONS ZEROS OF POLYNOMIAL FUNCTIONS ALL I HAVE TO KNOW ABOUT POLYNOMIAL FUNCTIONS TOOLS IN FINDING ZEROS OF POLYNOMIAL FUNCTIONS Synthetic Division and Remainder Theorem (Compressed Synthetic Division) Fundamental

More information

Section 3.1 Quadratic Functions

Section 3.1 Quadratic Functions Chapter 3 Lecture Notes Page 1 of 72 Section 3.1 Quadratic Functions Objectives: Compare two different forms of writing a quadratic function Find the equation of a quadratic function (given points) Application

More information

Math 120, Sample Final Fall 2015

Math 120, Sample Final Fall 2015 Math 10, Sample Final Fall 015 Disclaimer: This sample final is intended to help students prepare for the final exam The final exam will be similar in structure and type of problems, however the actual

More information

3.1 Quadratic Functions and Their Models. Quadratic Functions. Graphing a Quadratic Function Using Transformations

3.1 Quadratic Functions and Their Models. Quadratic Functions. Graphing a Quadratic Function Using Transformations 3.1 Quadratic Functions and Their Models Quadratic Functions Graphing a Quadratic Function Using Transformations Graphing a Quadratic Function Using Transformations from Basic Parent Function, ) ( f Equations

More information

Topic 25: Quadratic Functions (Part 1) A quadratic function is a function which can be written as 2. Properties of Quadratic Functions

Topic 25: Quadratic Functions (Part 1) A quadratic function is a function which can be written as 2. Properties of Quadratic Functions Hartfield College Algebra (Version 015b - Thomas Hartfield) Unit FOUR Page 1 of 3 Topic 5: Quadratic Functions (Part 1) Definition: A quadratic function is a function which can be written as f x ax bx

More information

30 Wyner Math Academy I Fall 2015

30 Wyner Math Academy I Fall 2015 30 Wyner Math Academy I Fall 2015 CHAPTER FOUR: QUADRATICS AND FACTORING Review November 9 Test November 16 The most common functions in math at this level are quadratic functions, whose graphs are parabolas.

More information

Polynomial Functions and Their Graphs

Polynomial Functions and Their Graphs Polynomial Functions and Their Graphs Definition of a Polynomial Function Let n be a nonnegative integer and let a n, a n- 1,, a 2, a 1, a 0, be real numbers with a n 0. The function defined by f (x) a

More information

UMUC MATH-107 Final Exam Information

UMUC MATH-107 Final Exam Information UMUC MATH-07 Final Exam Information What should you know for the final exam? Here are some highlights of textbook material you should study in preparation for the final exam. Review this material from

More information

Cumulative Review. Name. 13) 2x = -4 13) SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Cumulative Review. Name. 13) 2x = -4 13) SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Cumulative Review Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Evaluate the algebraic expression for the given value or values of the variable(s).

More information

PreCalculus: Semester 1 Final Exam Review

PreCalculus: Semester 1 Final Exam Review Name: Class: Date: ID: A PreCalculus: Semester 1 Final Exam Review Short Answer 1. Determine whether the relation represents a function. If it is a function, state the domain and range. 9. Find the domain

More information

Chapter 3: Polynomial and Rational Functions

Chapter 3: Polynomial and Rational Functions Chapter 3: Polynomial and Rational Functions 3.1 Polynomial Functions A polynomial on degree n is a function of the form P(x) = a n x n + a n 1 x n 1 + + a 1 x 1 + a 0, where n is a nonnegative integer

More information

, a 1. , a 2. ,..., a n

, a 1. , a 2. ,..., a n CHAPTER Points to Remember :. Let x be a variable, n be a positive integer and a 0, a, a,..., a n be constants. Then n f ( x) a x a x... a x a, is called a polynomial in variable x. n n n 0 POLNOMIALS.

More information

Chapter 8 ~ Quadratic Functions and Equations In this chapter you will study... You can use these skills...

Chapter 8 ~ Quadratic Functions and Equations In this chapter you will study... You can use these skills... Chapter 8 ~ Quadratic Functions and Equations In this chapter you will study... identifying and graphing quadratic functions transforming quadratic equations solving quadratic equations using factoring

More information

Chapter 2 Polynomial and Rational Functions

Chapter 2 Polynomial and Rational Functions Chapter 2 Polynomial and Rational Functions Overview: 2.2 Polynomial Functions of Higher Degree 2.3 Real Zeros of Polynomial Functions 2.4 Complex Numbers 2.5 The Fundamental Theorem of Algebra 2.6 Rational

More information

A Partial List of Topics: Math Spring 2009

A Partial List of Topics: Math Spring 2009 A Partial List of Topics: Math 112 - Spring 2009 This is a partial compilation of a majority of the topics covered this semester and may not include everything which might appear on the exam. The purpose

More information

Chapter 8. Exploring Polynomial Functions. Jennifer Huss

Chapter 8. Exploring Polynomial Functions. Jennifer Huss Chapter 8 Exploring Polynomial Functions Jennifer Huss 8-1 Polynomial Functions The degree of a polynomial is determined by the greatest exponent when there is only one variable (x) in the polynomial Polynomial

More information

2 the maximum/minimum value is ( ).

2 the maximum/minimum value is ( ). Math 60 Ch3 practice Test The graph of f(x) = 3(x 5) + 3 is with its vertex at ( maximum/minimum value is ( ). ) and the The graph of a quadratic function f(x) = x + x 1 is with its vertex at ( the maximum/minimum

More information

King Fahd University of Petroleum and Minerals Prep-Year Math Program Math Term 161 Recitation (R1, R2)

King Fahd University of Petroleum and Minerals Prep-Year Math Program Math Term 161 Recitation (R1, R2) Math 001 - Term 161 Recitation (R1, R) Question 1: How many rational and irrational numbers are possible between 0 and 1? (a) 1 (b) Finite (c) 0 (d) Infinite (e) Question : A will contain how many elements

More information

More Polynomial Equations Section 6.4

More Polynomial Equations Section 6.4 MATH 11009: More Polynomial Equations Section 6.4 Dividend: The number or expression you are dividing into. Divisor: The number or expression you are dividing by. Synthetic division: Synthetic division

More information

Final Exam C Name i D) 2. Solve the equation by factoring. 4) x2 = x + 72 A) {1, 72} B) {-8, 9} C) {-8, -9} D) {8, 9} 9 ± i

Final Exam C Name i D) 2. Solve the equation by factoring. 4) x2 = x + 72 A) {1, 72} B) {-8, 9} C) {-8, -9} D) {8, 9} 9 ± i Final Exam C Name First, write the value(s) that make the denominator(s) zero. Then solve the equation. 7 ) x + + 3 x - = 6 (x + )(x - ) ) A) No restrictions; {} B) x -, ; C) x -; {} D) x -, ; {2} Add

More information

Coach Stones Expanded Standard Pre-Calculus Algorithm Packet Page 1 Section: P.1 Algebraic Expressions, Mathematical Models and Real Numbers

Coach Stones Expanded Standard Pre-Calculus Algorithm Packet Page 1 Section: P.1 Algebraic Expressions, Mathematical Models and Real Numbers Coach Stones Expanded Standard Pre-Calculus Algorithm Packet Page 1 Section: P.1 Algebraic Expressions, Mathematical Models and Real Numbers CLASSIFICATIONS OF NUMBERS NATURAL NUMBERS = N = {1,2,3,4,...}

More information

2. Approximate the real zero of f(x) = x3 + x + 1 to the nearest tenth. Answer: Substitute all the values into f(x) and find which is closest to zero

2. Approximate the real zero of f(x) = x3 + x + 1 to the nearest tenth. Answer: Substitute all the values into f(x) and find which is closest to zero Unit 2 Examples(K) 1. Find all the real zeros of the function. Answer: Simply substitute the values given in all the functions and see which option when substituted, all the values go to zero. That is

More information

Section 5.1 Determine if a function is a polynomial function. State the degree of a polynomial function.

Section 5.1 Determine if a function is a polynomial function. State the degree of a polynomial function. Test Instructions Objectives Section 5.1 Section 5.1 Determine if a function is a polynomial function. State the degree of a polynomial function. Form a polynomial whose zeros and degree are given. Graph

More information

Chapter 2 notes from powerpoints

Chapter 2 notes from powerpoints Chapter 2 notes from powerpoints Synthetic division and basic definitions Sections 1 and 2 Definition of a Polynomial Function: Let n be a nonnegative integer and let a n, a n-1,, a 2, a 1, a 0 be real

More information

Test # 3 Review. È 3. Compare the graph of n 1 ÎÍ. Name: Class: Date: Short Answer. 1. Find the standard form of the quadratic function shown below:

Test # 3 Review. È 3. Compare the graph of n 1 ÎÍ. Name: Class: Date: Short Answer. 1. Find the standard form of the quadratic function shown below: Name: Class: Date: ID: A Test # 3 Review Short Answer 1. Find the standard form of the quadratic function shown below: 2. Compare the graph of m ( x) 9( x 7) 2 5 with m ( x) x 2. È 3. Compare the graph

More information

Unit 8 - Polynomial and Rational Functions Classwork

Unit 8 - Polynomial and Rational Functions Classwork Unit 8 - Polynomial and Rational Functions Classwork This unit begins with a study of polynomial functions. Polynomials are in the form: f ( x) = a n x n + a n 1 x n 1 + a n 2 x n 2 +... + a 2 x 2 + a

More information

Unit 2 Rational Functionals Exercises MHF 4UI Page 1

Unit 2 Rational Functionals Exercises MHF 4UI Page 1 Unit 2 Rational Functionals Exercises MHF 4UI Page Exercises 2.: Division of Polynomials. Divide, assuming the divisor is not equal to zero. a) x 3 + 2x 2 7x + 4 ) x + ) b) 3x 4 4x 2 2x + 3 ) x 4) 7. *)

More information

PreCalculus Notes. MAT 129 Chapter 5: Polynomial and Rational Functions. David J. Gisch. Department of Mathematics Des Moines Area Community College

PreCalculus Notes. MAT 129 Chapter 5: Polynomial and Rational Functions. David J. Gisch. Department of Mathematics Des Moines Area Community College PreCalculus Notes MAT 129 Chapter 5: Polynomial and Rational Functions David J. Gisch Department of Mathematics Des Moines Area Community College September 2, 2011 1 Chapter 5 Section 5.1: Polynomial Functions

More information

Lesson 7.1 Polynomial Degree and Finite Differences

Lesson 7.1 Polynomial Degree and Finite Differences Lesson 7.1 Polynomial Degree and Finite Differences 1. Identify the degree of each polynomial. a. 1 b. 0.2 1. 2 3.2 3 c. 20 16 2 20 2. Determine which of the epressions are polynomials. For each polynomial,

More information

10/22/16. 1 Math HL - Santowski SKILLS REVIEW. Lesson 15 Graphs of Rational Functions. Lesson Objectives. (A) Rational Functions

10/22/16. 1 Math HL - Santowski SKILLS REVIEW. Lesson 15 Graphs of Rational Functions. Lesson Objectives. (A) Rational Functions Lesson 15 Graphs of Rational Functions SKILLS REVIEW! Use function composition to prove that the following two funtions are inverses of each other. 2x 3 f(x) = g(x) = 5 2 x 1 1 2 Lesson Objectives! The

More information

Final Exam A Name. 20 i C) Solve the equation by factoring. 4) x2 = x + 30 A) {-5, 6} B) {5, 6} C) {1, 30} D) {-5, -6} -9 ± i 3 14

Final Exam A Name. 20 i C) Solve the equation by factoring. 4) x2 = x + 30 A) {-5, 6} B) {5, 6} C) {1, 30} D) {-5, -6} -9 ± i 3 14 Final Exam A Name First, write the value(s) that make the denominator(s) zero. Then solve the equation. 1 1) x + 3 + 5 x - 3 = 30 (x + 3)(x - 3) 1) A) x -3, 3; B) x -3, 3; {4} C) No restrictions; {3} D)

More information

Chapter 3 Polynomial Functions

Chapter 3 Polynomial Functions Trig / Coll. Alg. Name: Chapter 3 Polynomial Functions 3.1 Quadratic Functions (not on this test) For each parabola, give the vertex, intercepts (x- and y-), axis of symmetry, and sketch the graph. 1.

More information

Practice Test - Chapter 2

Practice Test - Chapter 2 Graph and analyze each function. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. 1. f (x) = 0.25x 3 Evaluate the function for several

More information

Calculus First Semester Review Name: Section: Evaluate the function: (g o f )( 2) f (x + h) f (x) h. m(x + h) m(x)

Calculus First Semester Review Name: Section: Evaluate the function: (g o f )( 2) f (x + h) f (x) h. m(x + h) m(x) Evaluate the function: c. (g o f )(x + 2) d. ( f ( f (x)) 1. f x = 4x! 2 a. f( 2) b. f(x 1) c. f (x + h) f (x) h 4. g x = 3x! + 1 Find g!! (x) 5. p x = 4x! + 2 Find p!! (x) 2. m x = 3x! + 2x 1 m(x + h)

More information

MAC1105-College Algebra

MAC1105-College Algebra MAC1105-College Algebra Chapter -Polynomial Division & Rational Functions. Polynomial Division;The Remainder and Factor Theorems I. Long Division of Polynomials A. For f ( ) 6 19 16, a zero of f ( ) occurs

More information

Pre-Calculus Midterm Practice Test (Units 1 through 3)

Pre-Calculus Midterm Practice Test (Units 1 through 3) Name: Date: Period: Pre-Calculus Midterm Practice Test (Units 1 through 3) Learning Target 1A I can describe a set of numbers in a variety of ways. 1. Write the following inequalities in interval notation.

More information

MATH 121: EXTRA PRACTICE FOR TEST 2. Disclaimer: Any material covered in class and/or assigned for homework is a fair game for the exam.

MATH 121: EXTRA PRACTICE FOR TEST 2. Disclaimer: Any material covered in class and/or assigned for homework is a fair game for the exam. MATH 121: EXTRA PRACTICE FOR TEST 2 Disclaimer: Any material covered in class and/or assigned for homework is a fair game for the exam. 1 Linear Functions 1. Consider the functions f(x) = 3x + 5 and g(x)

More information

MATH 1314 College Algebra Scott Travis Fall 2014 Review for Exam #2

MATH 1314 College Algebra Scott Travis Fall 2014 Review for Exam #2 MATH 1314 College Algebra Scott Travis Fall 2014 Review for Exam #2 There are eight sections from Chapters 4 and 5 included in the exam: 4.1, 4.3, 5.1 to 5.6. This review should help you prepare. For each

More information

A repeated root is a root that occurs more than once in a polynomial function.

A repeated root is a root that occurs more than once in a polynomial function. Unit 2A, Lesson 3.3 Finding Zeros Synthetic division, along with your knowledge of end behavior and turning points, can be used to identify the x-intercepts of a polynomial function. This information allows

More information

Reading Mathematical Expressions & Arithmetic Operations Expression Reads Note

Reading Mathematical Expressions & Arithmetic Operations Expression Reads Note Math 001 - Term 171 Reading Mathematical Expressions & Arithmetic Operations Expression Reads Note x A x belongs to A,x is in A Between an element and a set. A B A is a subset of B Between two sets. φ

More information

Student: Date: Instructor: kumnit nong Course: MATH 105 by Nong https://xlitemprodpearsoncmgcom/api/v1/print/math Assignment: CH test review 1 Find the transformation form of the quadratic function graphed

More information

Rational Functions 4.5

Rational Functions 4.5 Math 4 Pre-Calculus Name Date Rational Function Rational Functions 4.5 g ( ) A function is a rational function if f ( ), where g ( ) and ( ) h ( ) h are polynomials. Vertical asymptotes occur at -values

More information

Solutions to the Worksheet on Polynomials and Rational Functions

Solutions to the Worksheet on Polynomials and Rational Functions Solutions to the Worksheet on Polynomials and Rational Functions Math 141 1 Roots of Polynomials A Indicate the multiplicity of the roots of the polynomialh(x) = (x 1) ( x) 3( x +x+1 ) B Check the remainder

More information

9.5. Polynomial and Rational Inequalities. Objectives. Solve quadratic inequalities. Solve polynomial inequalities of degree 3 or greater.

9.5. Polynomial and Rational Inequalities. Objectives. Solve quadratic inequalities. Solve polynomial inequalities of degree 3 or greater. Chapter 9 Section 5 9.5 Polynomial and Rational Inequalities Objectives 1 3 Solve quadratic inequalities. Solve polynomial inequalities of degree 3 or greater. Solve rational inequalities. Objective 1

More information

Question 1. Find the coordinates of the y-intercept for. f) None of the above. Question 2. Find the slope of the line:

Question 1. Find the coordinates of the y-intercept for. f) None of the above. Question 2. Find the slope of the line: of 4 4/4/017 8:44 AM Question 1 Find the coordinates of the y-intercept for. Question Find the slope of the line: of 4 4/4/017 8:44 AM Question 3 Solve the following equation for x : Question 4 Paul has

More information

College Algebra Notes

College Algebra Notes Metropolitan Community College Contents Introduction 2 Unit 1 3 Rational Expressions........................................... 3 Quadratic Equations........................................... 9 Polynomial,

More information

( ) ( ) ( ) ( ) Given that and its derivative are continuous when, find th values of and. ( ) ( )

( ) ( ) ( ) ( ) Given that and its derivative are continuous when, find th values of and. ( ) ( ) 1. The piecewise function is defined by where and are constants. Given that and its derivative are continuous when, find th values of and. When When of of Substitute into ; 2. Using the substitution, evaluate

More information

3 What is the degree of the polynomial function that generates the data shown below?

3 What is the degree of the polynomial function that generates the data shown below? hapter 04 Test Name: ate: 1 For the polynomial function, describe the end behavior of its graph. The leading term is down. The leading term is and down.. Since n is 1 and a is positive, the end behavior

More information

( 3) ( ) ( ) ( ) ( ) ( )

( 3) ( ) ( ) ( ) ( ) ( ) 81 Instruction: Determining the Possible Rational Roots using the Rational Root Theorem Consider the theorem stated below. Rational Root Theorem: If the rational number b / c, in lowest terms, is a root

More information

Operations w/polynomials 4.0 Class:

Operations w/polynomials 4.0 Class: Exponential LAWS Review NO CALCULATORS Name: Operations w/polynomials 4.0 Class: Topic: Operations with Polynomials Date: Main Ideas: Assignment: Given: f(x) = x 2 6x 9 a) Find the y-intercept, the equation

More information

Department of Mathematics, University of Wisconsin-Madison Math 114 Worksheet Sections (4.1),

Department of Mathematics, University of Wisconsin-Madison Math 114 Worksheet Sections (4.1), Department of Mathematics, University of Wisconsin-Madison Math 114 Worksheet Sections (4.1), 4.-4.6 1. Find the polynomial function with zeros: -1 (multiplicity ) and 1 (multiplicity ) whose graph passes

More information

3.4. ZEROS OF POLYNOMIAL FUNCTIONS

3.4. ZEROS OF POLYNOMIAL FUNCTIONS 3.4. ZEROS OF POLYNOMIAL FUNCTIONS What You Should Learn Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions. Find rational zeros of polynomial functions. Find

More information

Chapter 2. Polynomial and Rational Functions. 2.3 Polynomial Functions and Their Graphs. Copyright 2014, 2010, 2007 Pearson Education, Inc.

Chapter 2. Polynomial and Rational Functions. 2.3 Polynomial Functions and Their Graphs. Copyright 2014, 2010, 2007 Pearson Education, Inc. Chapter Polynomial and Rational Functions.3 Polynomial Functions and Their Graphs Copyright 014, 010, 007 Pearson Education, Inc. 1 Objectives: Identify polynomial functions. Recognize characteristics

More information

2.1 Quadratic Functions

2.1 Quadratic Functions Date:.1 Quadratic Functions Precalculus Notes: Unit Polynomial Functions Objective: The student will sketch the graph of a quadratic equation. The student will write the equation of a quadratic function.

More information

Using the Laws of Exponents to Simplify Rational Exponents

Using the Laws of Exponents to Simplify Rational Exponents 6. Explain Radicals and Rational Exponents - Notes Main Ideas/ Questions Essential Question: How do you simplify expressions with rational exponents? Notes/Examples What You Will Learn Evaluate and simplify

More information

MATH 115: Review for Chapter 5

MATH 115: Review for Chapter 5 MATH 5: Review for Chapter 5 Can you find the real zeros of a polynomial function and identify the behavior of the graph of the function at its zeros? For each polynomial function, identify the zeros of

More information

Algebra III Chapter 2 Note Packet. Section 2.1: Polynomial Functions

Algebra III Chapter 2 Note Packet. Section 2.1: Polynomial Functions Algebra III Chapter 2 Note Packet Name Essential Question: Section 2.1: Polynomial Functions Polynomials -Have nonnegative exponents -Variables ONLY in -General Form n ax + a x +... + ax + ax+ a n n 1

More information

R1: Sets A set is a collection of objects sets are written using set brackets each object in onset is called an element or member

R1: Sets A set is a collection of objects sets are written using set brackets each object in onset is called an element or member Chapter R Review of basic concepts * R1: Sets A set is a collection of objects sets are written using set brackets each object in onset is called an element or member Ex: Write the set of counting numbers

More information

6.1 Polynomial Functions

6.1 Polynomial Functions 6.1 Polynomial Functions Definition. A polynomial function is any function p(x) of the form p(x) = p n x n + p n 1 x n 1 + + p 2 x 2 + p 1 x + p 0 where all of the exponents are non-negative integers and

More information

Section 4.1: Polynomial Functions and Models

Section 4.1: Polynomial Functions and Models Section 4.1: Polynomial Functions and Models Learning Objectives: 1. Identify Polynomial Functions and Their Degree 2. Graph Polynomial Functions Using Transformations 3. Identify the Real Zeros of a Polynomial

More information

PART 1: USING SCIENTIFIC CALCULATORS (41 PTS.) 1) The Vertex Form for the equation of a parabola in the usual xy-plane is given by y = 3 x + 4

PART 1: USING SCIENTIFIC CALCULATORS (41 PTS.) 1) The Vertex Form for the equation of a parabola in the usual xy-plane is given by y = 3 x + 4 MIDTERM SOLUTIONS (CHAPTERS AND 3: POLYNOMIAL, RATIONAL, EXP L, LOG FUNCTIONS) MATH 141 FALL 018 KUNIYUKI 150 POINTS TOTAL: 41 FOR PART 1, AND 109 FOR PART PART 1: USING SCIENTIFIC CALCULATORS (41 PTS.)

More information

Polynomial and Synthetic Division

Polynomial and Synthetic Division Chapter Polynomial and Rational Functions y. f. f Common function: y Horizontal shift of three units to the left, vertical shrink Transformation: Vertical each y-value is multiplied stretch each y-value

More information

( ) = 1 x. g( x) = x3 +2

( ) = 1 x. g( x) = x3 +2 Rational Functions are ratios (quotients) of polynomials, written in the form f x N ( x ) and D x ( ) are polynomials, and D x ( ) does not equal zero. The parent function for rational functions is f x

More information

3 Polynomial and Rational Functions

3 Polynomial and Rational Functions 3 Polynomial and Rational Functions 3.1 Polynomial Functions and their Graphs So far, we have learned how to graph polynomials of degree 0, 1, and. Degree 0 polynomial functions are things like f(x) =,

More information

The degree of the polynomial function is n. We call the term the leading term, and is called the leading coefficient. 0 =

The degree of the polynomial function is n. We call the term the leading term, and is called the leading coefficient. 0 = Math 1310 A polynomial function is a function of the form = + + +...+ + where 0,,,, are real numbers and n is a whole number. The degree of the polynomial function is n. We call the term the leading term,

More information

Final Exam Review for DMAT 0310

Final Exam Review for DMAT 0310 Final Exam Review for DMAT 010 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Factor the polynomial completely. What is one of the factors? 1) x

More information

Functions and Equations

Functions and Equations Canadian Mathematics Competition An activity of the Centre for Education in Mathematics and Computing, University of Waterloo, Waterloo, Ontario Euclid eworkshop # Functions and Equations c 006 CANADIAN

More information