Polynomial and Rational Functions

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Polynomial and Rational Functions"

Transcription

1 Polynomial and Rational Functions Quadratic Functions Overview of Objectives, students should be able to: 1. Recognize the characteristics of parabolas. 2. Find the intercepts a. x intercepts by solving a quadratic equation b. y intercept c. Recognize the connection between finding zeros, roots, x intercepts, and solutions to a quadratic function. 3. Find the vertex of a quadratic function Main Overarching Questions: 1. How can you find the x intercepts of a quadratic function on the graph and from the equation? 2. How can you find the y intercepts of a quadratic function from the graph and from the equation? 3. How can you find the vertex of a quadratic function from the graph and from the equation? 4. How can you determine whether the vertex of a quadratic function represents a minimum or a maximum value? 5. How can you tell whether the graph of a quadratic function should open up or down given its equation? 6. Explain how to find the zeros of a quadratic function. a. By observation in standard form b. Use the formula to find the vertex 4. Graph quadratic functions Objectives: Activities and Questions to ask students: Recognize the characteristics of parabolas. Give a few graphs of quadratic functions (that both open up and down) and the equations in 2 ax + bx + c form. Ask students to describe differences in the graphs. Why does a graph open up? Open down? When does the parabola have a maximum point? When does the parabola have a minimum point? How can we predict if a max/min exists by looking at the equation? Tell the students the max/min is always called the vertex. What kind of symmetry does the graph have? Find the intercepts Draw a parabola on the board (or provide a graphed parabola) and ask students to identify

2 i. x intercepts by solving a quadratic equation ii. y intercept iii. by looking at graph the y intercept. When the graph crosses the y axis, what is the value of x? Provide a quadratic function, and ask students to find the y intercept (plug in zero for x). Provide a few additional examples for independent or small group practice. After students have found solutions, ask them if there is a shortcut for determining the y intercept of a quadratic function (just use the c value). Using the original parabola, ask students to identify the x intercepts. When the graph crosses the x axis, what is the value of y? Provide a quadratic function, and ask students to find the x intercepts (plug in zero for y, or f(x) ). Ask students to recall methods for solving quadratic equations from previous lessons (factoring, completing the square, quadratic formula). Which method would be most efficient for this problem? Emphasize the idea that finding the x intercepts of a quadratic function is the same as solving a quadratic equation. This is also called finding the zeros, finding the roots, or finding the solutions of a quadratic function. Guide students through several additional examples, asking them to determine which method to use for each. Provide a problem set for independent practice. **Possible graphing calculator usage: After students have identified intercepts using the equations, allow them to graph the functions and check their answers using the calculate: value and calculate: zero features. Find the vertex of a quadratic function Introduce standard form: a ( x h) + k. Then, give several graphs of quadratic functions o By observation in standard form with their standard form equations and vertices plotted. o Use the formula to find the vertex Have students examine the differences in each situation and determine a pattern. Can the equation give us the vertex? How? 2 Now give the equation in ax + bx + c form. How would we find the vertex this time? Give the formula to find the x coordinate of the vertex. Have students practice finding the x coordinate. How would we find the y coordinate? Graph quadratic functions Have student students list the properties they would need to graph a quadratic function (vertex, x intercepts, y intercept, and domain and range). Have students describe in their own words how to find these properties. Give students a worksheet with a few quadratic equations to graph.

3 Polynomial Functions Overview of Objectives, students should be able to: 1. Identify polynomial functions Main Overarching Questions: 1. How do you identify properties of a polynomial function s graph and use them to graph? 2. Recognize characteristics of graphs of polynomial functions. 3. Understand the relationship between degree and turning points 4. Determine end behavior 5. Use factoring to find zeros of polynomial functions 6. Identify zeros and their multiplicities 7. Graph polynomial functions Objectives: Activities and Questions to ask students: Identify polynomial functions Ask students to define a polynomial. What do you think of when you hear the word? Some may say powers multiple terms or at least three terms Give students a few examples of polynomials and a few examples of non polynomials (rational functions, radical functions, and exponential functions). What is the difference between the polynomial and non polynomial functions? Recognize characteristics of graphs of polynomial functions To give students an idea of what polynomial graphs will look like, give a few polynomial functions graphs. Then, compare with non polynomial function graphs that have breaks and sharp turns. Have students compare. What are some traits of a polynomial function s

4 Understand the relationship between degree and turning points graph? (Smooth turns, peaks and valleys, and is infinite in both directions) Return to the several graphs of polynomials presented before. Include the function equations with each graph. Ask students to count the number of turning points in each graph. If further discussion of what a turning point is necessary explore the concept. If your examples are chosen wisely so that all have n 1 turning points, some students may see the connection between degree and the number of turning points. If not, ask students to determine the degree of each polynomial function and THEN determine the relationship. You will need to point out that functions can have at MOST n 1 turning points, but can certainly have less. Determine end behavior Have a discussion about what end behavior means to the students. Give an example of a polynomial function graph. Does the function end? Does it go up or down? On what side does it go up or down? Explain that end behavior refers to what the graph does far to the right (for large x) and far to the left (for small x). Refer to the earlier graphs. What is the end behavior? What two choices are there? Students should see that polynomial functions either increase without bound or decrease without bound on either side of the graph. Begin with two simple polynomials and their graphs. For example: f (x) = x 4 + x 2 and g(x) = x 4 + x 2. Ask students to compare the end behavior in each case. Can the equation determine the end behavior? Why did one graph go up on the left and up on the right but the other graph goes down on the left and down on the right? What change do you see in the equation? Have students draw the conclusion that the sign of the leading coefficient determines part of the end behavior. Students may think this is all they need to know. Now, give an example of an odd and even degree polynomial (keep the leading coefficient positive in both cases). f (x) = x 4 + x 2 and g(x) = x 3 + x 2. Again ask students to compare the end behavior of each graph? Why does one graph go up on the left and up on the right but the other graph goes down on the left and up on the right.

5 Use factoring to find zeros of polynomial functions Have students draw the conclusion that the degree of the polynomial (odd or even) determines part of the end behavior. Summarize the 4 cases and have students practice determining end behavior. This is an extension of finding zeros of quadratic functions. Simply ask students how they found zeros of quadratic functions. Hopefully most will remember to set the function to zero and solve. If needed discuss what a zero is on the graph of a function. Why do we need it? What can the zeros tell us? How do we solve polynomial equations? What was the easiest way to solve quadratic equations? Factoring! Have students practice finding zeros. Identify zeros and their multiplicities This concept can be difficult for students to see on their own. Consider giving a factored polynomial function with zeros and their multiplicities given. You might need to give several examples. Consider using multiplicities 2 and greater and avoid a zero of 0. Ask students what they think multiplicity means. You can mention that the equation tells us the multiplicity of each zero. Hopefully students will see the relationship between the power on the factor and the multiplicity. If not, you can clue them in. Next, give the graph of a polynomial function with some zeros that cross the x axis and some that just touch the axis. Ask students to compare the zeros. How do they behave differently? Give a few graphs like this if necessary. Next, beside each zero include its multiplicity. Give several examples and let students consider if there is a relationship between the multiplicity and the behavior of the zero. Have students summarize the relationship between zeros multiplicities and the behavior of the zero. Graph polynomial functions This is an important point to stop and summarize all the properties studied thus far. Have students list all the properties studied and how we can use the equation of the function to describe these properties. With student involvement practice graphing a polynomial function. It s important to stress the work needed to describe the properties before even graphing the function.

6 Dividing Polynomials Overview of Objectives, students should be able to: 1. Use long division to divide polynomials Main Overarching Questions: 1. How do you divide polynomials? 2. Use synthetic division to divide polynomials Objectives: Activities and Questions to ask students: Use long division to divide polynomials Ask students to next consider what happens if we have more than one term in the 2 x 7x+ 12 denominator like. Can we split up the denominator? If students say yes, ask x 5 them to combine together 2 3 x + 5 Ask them once again to consider if they can split up the denominator. Once students are convinced that they cannot split the denominator, suggest using long division to divide. Give students all the terminology before beginning. As a good analogy, complete the steps to long division concurrently with an example from arithmetic. Have students compare/contrast each step of polynomial long division with the arithmetic version. Give students a few long division problems to try on their own. Use synthetic division to divide polynomials Before explaining all the steps, consider showing a long division problem fully worked out and by its side the same division completed using synthetic division. Have students compare the two solutions. Where did the number in the box come from? Where did the first row of numbers come from? The 2 nd row? The final row under the line? Where is the remainder? After students realize where the coefficients come from, illustrate the process of working

7 through the synthetic division. Have students summarize the process after completing an example. Are there any drawbacks to this method? What if the divisor isn t of the form x c? Have students practice this new method. Rational Functions Overview of Objectives, students should be able to: 1. Find the domains of rational functions Main Overarching Questions: 1. How do you identify properties of a rational function s graph and use them to graph? 2. Use arrow notation 3. Identify vertical asymptotes 4. Identify horizontal asymptotes 5. Use transformations to graph rational functions 6. Graph rational functions 7. Identify slant (oblique) asymptotes Objectives: Activities and Questions to ask students: Find the domains of rational functions Already covered in earlier lesson, but a review might be warranted. Use arrow notation By plotting points, have students graph f (x) = 1 x. What happens as x gets close to zero? Some students may say goes up while others may say go down. Discuss why both answers are correct. Then, ask what happens as x approaches zero on the left side. Once

8 Identify vertical asymptotes students answer, give the arrow notation that shows from the left As x 0, f (x). Repeat on the right side. Give them some examples to try. For the function f (x) = 1 x, what happens at x = 0? Does the function exist? What does the graph do at x = 0. Explain that at x = 0, the graph has a vertical asymptote defined as the vertical line x = 0. Give a few other graphs of rational functions along with their function equations that are of the form f (x) = 1 x c. Is there a relationship between the vertical asymptote and the equation of the function? What is the value of the denominator at the vertical asymptote value? Have students describe how they would find the vertical asymptote when given a function. If students insist on observing the value, give a more complicated denominator that requires solving or factoring then solving. How would we find the vertical asymptote for this function? Identify horizontal asymptotes Give three examples of the graphs and function equations that do not have a horizontal asymptote, three examples of a non zero horizontal asymptote rational function, and three examples of a zero horizontal asymptote rational function. Begin by asking students to determine the end behavior in each case. End behavior was discussed with polynomial functions, so students should have an understanding of what end behavior means. Is there a relationship between the differences in the degree of the numerator and denominator and the end behavior? What happens if the degree of the numerator is greater than the degree of the denominator? What happens if the degree of the numerator is equal to the degree of the denominator? What happens if the degree of the numerator is less than the degree of the denominator? Have students establish the relationship. Give a few examples to try. Use transformations to graph rational functions Transformation was covered earlier. Have students practice transforming the functions f (x) = 1 x and g(x) = 1 x 2. Graph rational functions Before graphing, ask students how they would find the zeros of a rational function. If we set the function equal to zero, what part of the fraction must be zero? Have students realize the numerator must be set to zero to find the zeros of a rational function. Now, have students list the properties and how to get them of rational functions. Stress the

9 importance of finding all this information and using it to find the graph. Have students practice finding all the properties and using them to graph. Identify slant (oblique) asymptotes Ask students about the end behavior when the degree of the numerator is greater than the degree of the denominator. Most will say goes up or goes down or that there is no horizontal asymptote. Explain that there are sometimes other types of end behavior asymptotes. To show this, give a graph of a rational function with a slant asymptote. Label the equation of the function and the equation of the slant asymptote. Ask students to use long division (or synthetic) to divide the numerator by the denominator. Is there a relationship between the quotient and the slant asymptote? Provide another example if necessary. To answer the question as to why the slant asymptote is the quotient (and not the remainder), ask students to give the remainder expression (not just the remainder, but the remainder divided by the divisor). What happens as x to the remainder expression? Have students try to plugin really large values of x. The remainder gets very close to zero, so the fraction is very closely approximated by the quotient, and the remainder has little to no effect. Polynomial and Rational Inequalities Overview of Objectives, students should be able to: 1. Solve polynomial inequalities 2. Solve rational inequalities Main Overarching Questions: 1. How do you find the solution to a polynomial inequality? 2. How do you find the solutions to a rational inequality? 3. Solve problems modeled by polynomial or rational inequalities. Objectives: Activities and Questions to ask students: Solve polynomial inequalities Give a simple example of a polynomial inequality like x 2 7x +10 < 0. How would we solve this? Students might have no idea, or might suggest isolating x. Ask how they would solve x 2 7x +10 = 0? Hopefully, they will suggest to factor and set each factor equal to 0.

10 Have students complete this and see that the solution is x = 2 and x = 5. Next, graph the function x 2 7x +10. For what values of x is the function below the x axis? If we are trying to solve x 2 7x +10 < 0, how can we use the graph to tell us the solution? The idea is to get students to observe that the values of x for points below the x axis are the solutions to this inequality. Repeat the discussion for x 2 7x +10 > 0. Next, discuss where the boundary points where the function moves from positive to negative, and negative to positive. Where did this happen? The zeros of the function. Give students a few more examples and have them verify that the only place where the functions change sign are at the zeros. Introduce the idea of a sign chart to the students and show them how we can use the chart to solve the inequality. Have students summarize the process of solving polynomial inequalities. Solve rational inequalities The above discussion can be repeated for rational inequalities. The students can discover that now the sign changes can occur at both zeros AND vertical asymptotes, so both types of points must be plotted on the number line. It will be important to have students work some examples with, to understand when to include the points (zeros) and to not include the points (the vertical asymptotes). Solve problems modeled by polynomial or rational inequalities. An interesting application from Blitzer s College Algebra introduces the quadratic equation: s(t) = 16t 2 + v 0 t + s 0. One exercise has students set up and solve the quadratic inequality in this situation: Divers in Acapulco, Mexico, dive headfirst at 8 feet per second from the top of a cliff 87 feet about the Pacific Ocean. During which time period will the diver s height exceed that of the cliff? Again, the key here is to be able to setup the problem. What values go in for what variables? Which variable are you solving for? The contents of this website were developed under Congressionally directed grants (P116Z090305) from the U.S. Department of Education. However, those contents do not necessarily represent the

Equations and Inequalities

Equations and Inequalities Rational Equations Overview of Objectives, students should be able to: 1. Solve rational equations with variables in the denominators.. Recognize identities, conditional equations, and inconsistent equations.

More information

FUNCTIONS. Introduction to Functions. Overview of Objectives, students should be able to:

FUNCTIONS. Introduction to Functions. Overview of Objectives, students should be able to: FUNCTIONS Introduction to Functions Overview of Objectives, students should be able to: 1. Find the domain and range of a relation 2. Determine whether a relation is a function 3. Evaluate a function 4.

More information

Exponential and Logarithmic Functions

Exponential and Logarithmic Functions Exponential and Logarithmic Functions Exponential Functions Overview of Objectives, students should be able to: 1. Evaluate exponential functions. Main Overarching Questions: 1. How do you graph exponential

More information

MA107 Precalculus Algebra Exam 2 Review Solutions

MA107 Precalculus Algebra Exam 2 Review Solutions MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write

More information

Math Rational Functions

Math Rational Functions Rational Functions Math 3 Rational Functions A rational function is the algebraic equivalent of a rational number. Recall that a rational number is one that can be epressed as a ratio of integers: p/q.

More information

Chapter 4. Polynomial and Rational Functions. 4.1 Polynomial Functions and Their Graphs

Chapter 4. Polynomial and Rational Functions. 4.1 Polynomial Functions and Their Graphs Chapter 4. Polynomial and Rational Functions 4.1 Polynomial Functions and Their Graphs A polynomial function of degree n is a function of the form P = a n n + a n 1 n 1 + + a 2 2 + a 1 + a 0 Where a s

More information

Algebra I Pacing Guide Days Units Notes 9 Chapter 1 ( , )

Algebra I Pacing Guide Days Units Notes 9 Chapter 1 ( , ) Algebra I Pacing Guide Days Units Notes 9 Chapter 1 (1.1-1.4, 1.6-1.7) Expressions, Equations and Functions Differentiate between and write expressions, equations and inequalities as well as applying order

More information

ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form

ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form Goal Graph quadratic functions. VOCABULARY Quadratic function A function that can be written in the standard form y = ax 2 + bx+ c where a 0 Parabola

More information

3.4 Limits at Infinity - Asymptotes

3.4 Limits at Infinity - Asymptotes 3.4 Limits at Infinity - Asymptotes Definition 3.3. If f is a function defined on some interval (a, ), then f(x) = L means that values of f(x) are very close to L (keep getting closer to L) as x. The line

More information

Polynomials Classwork

Polynomials Classwork Polynomials Classwork What Is a Polynomial Function? Numerical, Analytical and Graphical Approaches Anatomy of an n th -degree polynomial function Def.: A polynomial function of degree n in the vaiable

More information

with "a", "b" and "c" representing real numbers, and "a" is not equal to zero.

with a, b and c representing real numbers, and a is not equal to zero. 3.1 SOLVING QUADRATIC EQUATIONS: * A QUADRATIC is a polynomial whose highest exponent is. * The "standard form" of a quadratic equation is: ax + bx + c = 0 with "a", "b" and "c" representing real numbers,

More information

Review for Calculus Rational Functions, Logarithms & Exponentials

Review for Calculus Rational Functions, Logarithms & Exponentials Definition and Domain of Rational Functions A rational function is defined as the quotient of two polynomial functions. F(x) = P(x) / Q(x) The domain of F is the set of all real numbers except those for

More information

Pre-Calculus Notes Vertical, Horizontal, and Slant Asymptotes of Rational Functions

Pre-Calculus Notes Vertical, Horizontal, and Slant Asymptotes of Rational Functions Pre-Calculus Notes Vertical, Horizontal, and Slant Asymptotes of Rational Functions A) Vertical Asymptotes A rational function, in lowest terms, will have vertical asymptotes at the real zeros of the denominator

More information

Precalculus A 2016 Graphs of Rational Functions

Precalculus A 2016 Graphs of Rational Functions 3-7 Precalculus A 2016 Graphs of Rational Functions Determine the equations of the vertical and horizontal asymptotes, if any, of each function. Graph each function with the asymptotes labeled. 1. ƒ(x)

More information

10.1 Notes-Graphing Quadratics

10.1 Notes-Graphing Quadratics Name: Period: 10.1 Notes-Graphing Quadratics Section 1: Identifying the vertex (minimum/maximum), the axis of symmetry, and the roots (zeros): State the maximum or minimum point (vertex), the axis of symmetry,

More information

Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.

Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P. MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called

More information

Analyzing Polynomial and Rational Functions

Analyzing Polynomial and Rational Functions Analyzing Polynomial and Rational Functions Raja Almukahhal, (RajaA) Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) To access a customizable version of this book, as

More information

Definition 2.1 The line x = a is a vertical asymptote of the function y = f(x) if y approaches ± as x approaches a from the right or left.

Definition 2.1 The line x = a is a vertical asymptote of the function y = f(x) if y approaches ± as x approaches a from the right or left. Vertical and Horizontal Asymptotes Definition 2.1 The line x = a is a vertical asymptote of the function y = f(x) if y approaches ± as x approaches a from the right or left. This graph has a vertical asymptote

More information

5.4 The Quadratic Formula

5.4 The Quadratic Formula Section 5.4 The Quadratic Formula 481 5.4 The Quadratic Formula Consider the general quadratic function f(x) = ax + bx + c. In the previous section, we learned that we can find the zeros of this function

More information

Title: Graphing Quadratic Equations in Standard Form Class: Math 100 or 107 Author: Sharareh Masooman Instructions to tutor: Read instructions under

Title: Graphing Quadratic Equations in Standard Form Class: Math 100 or 107 Author: Sharareh Masooman Instructions to tutor: Read instructions under Title: Graphing Quadratic Equations in Standard Form Class: Math 100 or 107 Author: Sharareh Masooman Instructions to tutor: Read instructions under Activity and follow all steps for each problem exactly

More information

Algebra 1 Course Title

Algebra 1 Course Title Algebra 1 Course Title Course- wide 1. What patterns and methods are being used? Course- wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept

More information

MINI LESSON. Lesson 5b Solving Quadratic Equations

MINI LESSON. Lesson 5b Solving Quadratic Equations MINI LESSON Lesson 5b Solving Quadratic Equations Lesson Objectives By the end of this lesson, you should be able to: 1. Determine the number and type of solutions to a QUADRATIC EQUATION by graphing 2.

More information

CHAPTER 2: POLYNOMIAL AND RATIONAL FUNCTIONS

CHAPTER 2: POLYNOMIAL AND RATIONAL FUNCTIONS CHAPTER 2: POLYNOMIAL AND RATIONAL FUNCTIONS 2.01 SECTION 2.1: QUADRATIC FUNCTIONS (AND PARABOLAS) PART A: BASICS If a, b, and c are real numbers, then the graph of f x = ax2 + bx + c is a parabola, provided

More information

Name: where Nx ( ) and Dx ( ) are the numerator and

Name: where Nx ( ) and Dx ( ) are the numerator and Oblique and Non-linear Asymptote Activity Name: Prior Learning Reminder: Rational Functions In the past we discussed vertical and horizontal asymptotes of the graph of a rational function of the form m

More information

The x-intercepts of the graph are the x-values for the points where the graph intersects the x-axis. A parabola may have one, two, or no x-intercepts.

The x-intercepts of the graph are the x-values for the points where the graph intersects the x-axis. A parabola may have one, two, or no x-intercepts. Chapter 10-1 Identify Quadratics and their graphs A parabola is the graph of a quadratic function. A quadratic function is a function that can be written in the form, f(x) = ax 2 + bx + c, a 0 or y = ax

More information

Solutions to Self-Test for Chapter 4 c4sts - p1

Solutions to Self-Test for Chapter 4 c4sts - p1 Solutions to Self-Test for Chapter 4 c4sts - p1 1. Graph a polynomial function. Label all intercepts and describe the end behavior. a. P(x) = x 4 2x 3 15x 2. (1) Domain = R, of course (since this is a

More information

a. all of the above b. none of the above c. B, C, D, and F d. C, D, F e. C only f. C and F

a. all of the above b. none of the above c. B, C, D, and F d. C, D, F e. C only f. C and F FINAL REVIEW WORKSHEET COLLEGE ALGEBRA Chapter 1. 1. Given the following equations, which are functions? (A) y 2 = 1 x 2 (B) y = 9 (C) y = x 3 5x (D) 5x + 2y = 10 (E) y = ± 1 2x (F) y = 3 x + 5 a. all

More information

MAT12X Intermediate Algebra

MAT12X Intermediate Algebra MAT1X Intermediate Algebra Workshop I Quadratic Functions LEARNING CENTER Overview Workshop I Quadratic Functions General Form Domain and Range Some of the effects of the leading coefficient a The vertex

More information

9.1 Solving Quadratic Equations by Finding Square Roots Objectives 1. Evaluate and approximate square roots.

9.1 Solving Quadratic Equations by Finding Square Roots Objectives 1. Evaluate and approximate square roots. 9.1 Solving Quadratic Equations by Finding Square Roots 1. Evaluate and approximate square roots. 2. Solve a quadratic equation by finding square roots. Key Terms Square Root Radicand Perfect Squares Irrational

More information

Pre-AP Algebra 2 Unit 3 Lesson 1 Quadratic Functions

Pre-AP Algebra 2 Unit 3 Lesson 1 Quadratic Functions Unit 3 Lesson 1 Quadratic Functions Objectives: The students will be able to Identify and sketch the quadratic parent function Identify characteristics including vertex, axis of symmetry, x-intercept,

More information

Solving Quadratic Equations

Solving Quadratic Equations 9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation

More information

Situation: Dividing Linear Expressions

Situation: Dividing Linear Expressions Situation: Dividing Linear Expressions Date last revised: June 4, 203 Michael Ferra, Nicolina Scarpelli, Mary Ellen Graves, and Sydney Roberts Prompt: An Algebra II class has been examining the product

More information

a) x 2 8x = 25 x 2 8x + 16 = (x 4) 2 = 41 x = 4 ± 41 x + 1 = ± 6 e) x 2 = 5 c) 2x 2 + 2x 7 = 0 2x 2 + 2x = 7 x 2 + x = 7 2

a) x 2 8x = 25 x 2 8x + 16 = (x 4) 2 = 41 x = 4 ± 41 x + 1 = ± 6 e) x 2 = 5 c) 2x 2 + 2x 7 = 0 2x 2 + 2x = 7 x 2 + x = 7 2 Solving Quadratic Equations By Square Root Method Solving Quadratic Equations By Completing The Square Consider the equation x = a, which we now solve: x = a x a = 0 (x a)(x + a) = 0 x a = 0 x + a = 0

More information

Algebra Nation MAFS Videos and Standards Alignment Algebra 2

Algebra Nation MAFS Videos and Standards Alignment Algebra 2 Section 1, Video 1: Linear Equations in One Variable - Part 1 Section 1, Video 2: Linear Equations in One Variable - Part 2 Section 1, Video 3: Linear Equations and Inequalities in Two Variables Section

More information

Quadratic Equations and Inequalities

Quadratic Equations and Inequalities MA 134 Lecture Notes August 20, 2012 Introduction The purpose of this lecture is to... Introduction The purpose of this lecture is to... Learn about different types of equations Introduction The purpose

More information

1.1 Solving a Linear Equation ax + b = 0

1.1 Solving a Linear Equation ax + b = 0 1.1 Solving a Linear Equation ax + b = 0 To solve an equation ax + b = 0 : (i) move b to the other side (subtract b from both sides) (ii) divide both sides by a Example: Solve x = 0 (i) x- = 0 x = (ii)

More information

Portable Assisted Study Sequence ALGEBRA IIA

Portable Assisted Study Sequence ALGEBRA IIA SCOPE This course is divided into two semesters of study (A & B) comprised of five units each. Each unit teaches concepts and strategies recommended for intermediate algebra students. The first half of

More information

Rational Functions 5.2 & 5.3

Rational Functions 5.2 & 5.3 Math Precalculus Algebra Name Date Rational Function Rational Functions 5. & 5.3 g( ) A function is a rational function if f ( ), where g( ) and h( ) are polynomials. h( ) Vertical asymptotes occur at

More information

Course Name: Course Code: ALEKS Course: Instructor: Course Dates: Course Content: Textbook: Dates Objective Prerequisite Topics

Course Name: Course Code: ALEKS Course: Instructor: Course Dates: Course Content: Textbook: Dates Objective Prerequisite Topics Course Name: MATH 1204 Fall 2015 Course Code: N/A ALEKS Course: College Algebra Instructor: Master Templates Course Dates: Begin: 08/22/2015 End: 12/19/2015 Course Content: 271 Topics (261 goal + 10 prerequisite)

More information

Rational Polynomial Functions

Rational Polynomial Functions Rational Polynomial Functions Rational Polynomial Functions and Their Domains Today we discuss rational polynomial functions. A function f(x) is a rational polynomial function if it is the quotient of

More information

Rational Functions and Their Graphs

Rational Functions and Their Graphs Objectives Rational Functions and Their Graphs Section.6 Find domain of rational functions. Use transformations to graph rational functions. Use arrow notation. Identify vertical asymptotes. Identify horizontal

More information

Main page. Given f ( x, y) = c we differentiate with respect to x so that

Main page. Given f ( x, y) = c we differentiate with respect to x so that Further Calculus Implicit differentiation Parametric differentiation Related rates of change Small variations and linear approximations Stationary points Curve sketching - asymptotes Curve sketching the

More information

Graphing Quadratic Functions

Graphing Quadratic Functions Graphing Quadratic Functions In our consideration of polynomial functions, we first studied linear functions. Now we will consider polynomial functions of order or degree (i.e., the highest power of x

More information

BROCK UNIVERSITY MATHEMATICS MODULES

BROCK UNIVERSITY MATHEMATICS MODULES BROCK UNIVERSITY MATHEMATICS MODULES 11A.4: Maximum or Minimum Values for Quadratic Functions Author: Kristina Wamboldt WWW What it is: Maximum or minimum values for a quadratic function are the largest

More information

Algebra II End of Course Exam Answer Key Segment I. Scientific Calculator Only

Algebra II End of Course Exam Answer Key Segment I. Scientific Calculator Only Algebra II End of Course Exam Answer Key Segment I Scientific Calculator Only Question 1 Reporting Category: Algebraic Concepts & Procedures Common Core Standard: A-APR.3: Identify zeros of polynomials

More information

Zeros of a Polynomial Function

Zeros of a Polynomial Function Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we

More information

ALGEBRA I A PLUS COURSE OUTLINE

ALGEBRA I A PLUS COURSE OUTLINE ALGEBRA I A PLUS COURSE OUTLINE OVERVIEW: 1. Operations with Real Numbers 2. Equation Solving 3. Word Problems 4. Inequalities 5. Graphs of Functions 6. Linear Functions 7. Scatterplots and Lines of Best

More information

Functions and Equations

Functions and Equations Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c

More information

Key Terms: Quadratic function. Parabola. Vertex (of a parabola) Minimum value. Maximum value. Axis of symmetry. Vertex form (of a quadratic function)

Key Terms: Quadratic function. Parabola. Vertex (of a parabola) Minimum value. Maximum value. Axis of symmetry. Vertex form (of a quadratic function) Outcome R3 Quadratic Functions McGraw-Hill 3.1, 3.2 Key Terms: Quadratic function Parabola Vertex (of a parabola) Minimum value Maximum value Axis of symmetry Vertex form (of a quadratic function) Standard

More information

55x 3 + 23, f(x) = x2 3. x x 2x + 3 = lim (1 x 4 )/x x (2x + 3)/x = lim

55x 3 + 23, f(x) = x2 3. x x 2x + 3 = lim (1 x 4 )/x x (2x + 3)/x = lim Slant Asymptotes If lim x [f(x) (ax + b)] = 0 or lim x [f(x) (ax + b)] = 0, then the line y = ax + b is a slant asymptote to the graph y = f(x). If lim x f(x) (ax + b) = 0, this means that the graph of

More information

Section P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities

Section P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities Section P.9 Notes Page P.9 Linear Inequalities and Absolute Value Inequalities Sometimes the answer to certain math problems is not just a single answer. Sometimes a range of answers might be the answer.

More information

POLYNOMIAL FUNCTIONS

POLYNOMIAL FUNCTIONS POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a

More information

Quadratic and Square Root Functions. Square Roots & Quadratics: What s the Connection?

Quadratic and Square Root Functions. Square Roots & Quadratics: What s the Connection? Activity: TEKS: Overview: Materials: Grouping: Time: Square Roots & Quadratics: What s the Connection? (2A.9) Quadratic and square root functions. The student formulates equations and inequalities based

More information

Algebra II Semester Exam Review Sheet

Algebra II Semester Exam Review Sheet Name: Class: Date: ID: A Algebra II Semester Exam Review Sheet 1. Translate the point (2, 3) left 2 units and up 3 units. Give the coordinates of the translated point. 2. Use a table to translate the graph

More information

Sec36NotesDone.notebook April 19, Sec. 3.6 Rational Functions and their Graphs. A rational function is of the form:

Sec36NotesDone.notebook April 19, Sec. 3.6 Rational Functions and their Graphs. A rational function is of the form: Sec. 3.6 Rational Functions and their Graphs A rational function is of the form: where P(x) and Q(x) are Polynomials The Domain of r(x) is all values of x where Q(x) is not equal to zero. The simplest

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Math 110 Review for Final Examination 2012 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Match the equation to the correct graph. 1) y = -

More information

Power functions: f(x) = x n, n is a natural number The graphs of some power functions are given below. n- even n- odd

Power functions: f(x) = x n, n is a natural number The graphs of some power functions are given below. n- even n- odd 5.1 Polynomial Functions A polynomial unctions is a unction o the orm = a n n + a n-1 n-1 + + a 1 + a 0 Eample: = 3 3 + 5 - The domain o a polynomial unction is the set o all real numbers. The -intercepts

More information

McMurry University Pre-test Practice Exam. 1. Simplify each expression, and eliminate any negative exponent(s).

McMurry University Pre-test Practice Exam. 1. Simplify each expression, and eliminate any negative exponent(s). 1. Simplify each expression, and eliminate any negative exponent(s). a. b. c. 2. Simplify the expression. Assume that a and b denote any real numbers. (Assume that a denotes a positive number.) 3. Find

More information

Zeros of Polynomial Functions

Zeros of Polynomial Functions Review: Synthetic Division Find (x 2-5x - 5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 3-5x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 3-5x 2 + x + 2. Zeros of Polynomial Functions Introduction

More information

Zeros of Polynomial Functions

Zeros of Polynomial Functions Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n-1 x n-1 + + a 1 x + a 0 has integer coefficients and p/q (where p/q is reduced) is a rational zero, then p is a factor of

More information

Section 2.3. Learning Objectives. Graphing Quadratic Functions

Section 2.3. Learning Objectives. Graphing Quadratic Functions Section 2.3 Quadratic Functions Learning Objectives Quadratic function, equations, and inequities Properties of quadratic function and their graphs Applications More general functions Graphing Quadratic

More information

Infinite Algebra 1 supports the teaching of the Common Core State Standards listed below.

Infinite Algebra 1 supports the teaching of the Common Core State Standards listed below. Infinite Algebra 1 Kuta Software LLC Common Core Alignment Software version 2.05 Last revised July 2015 Infinite Algebra 1 supports the teaching of the Common Core State Standards listed below. High School

More information

EQUATIONS and INEQUALITIES

EQUATIONS and INEQUALITIES EQUATIONS and INEQUALITIES Linear Equations and Slope 1. Slope a. Calculate the slope of a line given two points b. Calculate the slope of a line parallel to a given line. c. Calculate the slope of a line

More information

Graphing Quadratic Functions

Graphing Quadratic Functions Problem 1 The Parabola Examine the data in L 1 and L to the right. Let L 1 be the x- value and L be the y-values for a graph. 1. How are the x and y-values related? What pattern do you see? To enter the

More information

The Quadratic Formula

The Quadratic Formula Definition of the Quadratic Formula The Quadratic Formula uses the a, b and c from numbers; they are the "numerical coefficients"., where a, b and c are just The Quadratic Formula is: For ax 2 + bx + c

More information

Graphing Rational Functions

Graphing Rational Functions Graphing Rational Functions A rational function is defined here as a function that is equal to a ratio of two polynomials p(x)/q(x) such that the degree of q(x) is at least 1. Examples: is a rational function

More information

Review Session #5 Quadratics

Review Session #5 Quadratics Review Session #5 Quadratics Discriminant How can you determine the number and nature of the roots without solving the quadratic equation? 1. Prepare the quadratic equation for solving in other words,

More information

MSLC Workshop Series Math 1148 1150 Workshop: Polynomial & Rational Functions

MSLC Workshop Series Math 1148 1150 Workshop: Polynomial & Rational Functions MSLC Workshop Series Math 1148 1150 Workshop: Polynomial & Rational Functions The goal of this workshop is to familiarize you with similarities and differences in both the graphing and expression of polynomial

More information

Polynomial and Rational Functions

Polynomial and Rational Functions Chapter 5 Polnomial and Rational Functions Section 5.1 Polnomial Functions Section summaries The general form of a polnomial function is f() = a n n + a n 1 n 1 + +a 1 + a 0. The degree of f() is the largest

More information

x 2 + y 2 = 1 y 1 = x 2 + 2x y = x 2 + 2x + 1

x 2 + y 2 = 1 y 1 = x 2 + 2x y = x 2 + 2x + 1 Implicit Functions Defining Implicit Functions Up until now in this course, we have only talked about functions, which assign to every real number x in their domain exactly one real number f(x). The graphs

More information

Integration Unit 5 Quadratic Toolbox 1: Working with Square Roots. Using your examples above, answer the following:

Integration Unit 5 Quadratic Toolbox 1: Working with Square Roots. Using your examples above, answer the following: Integration Unit 5 Quadratic Toolbox 1: Working with Square Roots Name Period Objective 1: Understanding Square roots Defining a SQUARE ROOT: Square roots are like a division problem but both factors must

More information

College Algebra. Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGraw-Hill, 2008, ISBN: 978-0-07-286738-1

College Algebra. Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGraw-Hill, 2008, ISBN: 978-0-07-286738-1 College Algebra Course Text Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGraw-Hill, 2008, ISBN: 978-0-07-286738-1 Course Description This course provides

More information

CHAPTER 4. Test Bank Exercises in. Exercise Set 4.1

CHAPTER 4. Test Bank Exercises in. Exercise Set 4.1 Test Bank Exercises in CHAPTER 4 Exercise Set 4.1 1. Graph the quadratic function f(x) = x 2 2x 3. Indicate the vertex, axis of symmetry, minimum 2. Graph the quadratic function f(x) = x 2 2x. Indicate

More information

Mathematics. Accelerated GSE Analytic Geometry B/Advanced Algebra Unit 7: Rational and Radical Relationships

Mathematics. Accelerated GSE Analytic Geometry B/Advanced Algebra Unit 7: Rational and Radical Relationships Georgia Standards of Excellence Frameworks Mathematics Accelerated GSE Analytic Geometry B/Advanced Algebra Unit 7: Rational and Radical Relationships These materials are for nonprofit educational purposes

More information

7.1 Graphs of Quadratic Functions in Vertex Form

7.1 Graphs of Quadratic Functions in Vertex Form 7.1 Graphs of Quadratic Functions in Vertex Form Quadratic Function in Vertex Form A quadratic function in vertex form is a function that can be written in the form f (x) = a(x! h) 2 + k where a is called

More information

West Windsor-Plainsboro Regional School District Algebra I Part 2 Grades 9-12

West Windsor-Plainsboro Regional School District Algebra I Part 2 Grades 9-12 West Windsor-Plainsboro Regional School District Algebra I Part 2 Grades 9-12 Unit 1: Polynomials and Factoring Course & Grade Level: Algebra I Part 2, 9 12 This unit involves knowledge and skills relative

More information

High School Algebra 1 Common Core Standards & Learning Targets

High School Algebra 1 Common Core Standards & Learning Targets High School Algebra 1 Common Core Standards & Learning Targets Unit 1: Relationships between Quantities and Reasoning with Equations CCS Standards: Quantities N-Q.1. Use units as a way to understand problems

More information

Quadratic Functions. Copyright Cengage Learning. All rights reserved.

Quadratic Functions. Copyright Cengage Learning. All rights reserved. Quadratic Functions 4 Copyright Cengage Learning. All rights reserved. Solving by the Quadratic Formula 2 Example 1 Using the quadratic formula Solve the following quadratic equations. Round your answers

More information

CHAPTER 2: POLYNOMIAL AND RATIONAL FUNCTIONS

CHAPTER 2: POLYNOMIAL AND RATIONAL FUNCTIONS CHAPTER 2: POLYNOMIAL AND RATIONAL FUNCTIONS 2.01 SECTION 2.1: QUADRATIC FUNCTIONS (AND PARABOLAS) PART A: BASICS If a, b, and c are real numbers, then the graph of f x = ax2 + bx + c is a parabola, provided

More information

CCSS: N.CN.7: Solve quadratic equations with real coefficients that have complex solutions

CCSS: N.CN.7: Solve quadratic equations with real coefficients that have complex solutions 4.5 Completing The Square 1) Solve quadratic equations using the square root property 2) Generate perfect square trinomials by completing the square 3) Solve quadratic equations by completing the square

More information

Introduction to Modular Arithmetic, the rings Z 6 and Z 7

Introduction to Modular Arithmetic, the rings Z 6 and Z 7 Introduction to Modular Arithmetic, the rings Z 6 and Z 7 The main objective of this discussion is to learn modular arithmetic. We do this by building two systems using modular arithmetic and then by solving

More information

Pre-Calculus 20 Chapter 3 Notes

Pre-Calculus 20 Chapter 3 Notes Section 3.1 Quadratic Functions in Vertex Form Pre-Calculus 20 Chapter 3 Notes Using a table of values, graph y = x 2 y x y=x 2-2 4-2 4 x Using a table of values, graph y = -1x 2 (or y = -x 2 ) y x y=-x

More information

8 Polynomials Worksheet

8 Polynomials Worksheet 8 Polynomials Worksheet Concepts: Quadratic Functions The Definition of a Quadratic Function Graphs of Quadratic Functions - Parabolas Vertex Absolute Maximum or Absolute Minimum Transforming the Graph

More information

Prentice Hall Mathematics: Algebra 1 2007 Correlated to: Michigan Merit Curriculum for Algebra 1

Prentice Hall Mathematics: Algebra 1 2007 Correlated to: Michigan Merit Curriculum for Algebra 1 STRAND 1: QUANTITATIVE LITERACY AND LOGIC STANDARD L1: REASONING ABOUT NUMBERS, SYSTEMS, AND QUANTITATIVE SITUATIONS Based on their knowledge of the properties of arithmetic, students understand and reason

More information

Section 3.7 Rational Functions

Section 3.7 Rational Functions Section 3.7 Rational Functions A rational function is a function of the form where P and Q are polynomials. r(x) = P(x) Q(x) Rational Functions and Asymptotes The domain of a rational function consists

More information

Definition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality.

Definition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality. 8 Inequalities Concepts: Equivalent Inequalities Linear and Nonlinear Inequalities Absolute Value Inequalities (Sections 4.6 and 1.1) 8.1 Equivalent Inequalities Definition 8.1 Two inequalities are equivalent

More information

Wentzville School District Formal Algebra II

Wentzville School District Formal Algebra II Wentzville School District Formal Algebra II Unit 6 - Rational Functions Unit Title: Rational Functions Course: Formal Algebra II Brief Summary of Unit: In this unit, students will graph and analyze the

More information

Math 120 Final Exam Practice Problems, Form: A

Math 120 Final Exam Practice Problems, Form: A Math 120 Final Exam Practice Problems, Form: A Name: While every attempt was made to be complete in the types of problems given below, we make no guarantees about the completeness of the problems. Specifically,

More information

Developmental Math Course Outcomes and Objectives

Developmental Math Course Outcomes and Objectives Developmental Math Course Outcomes and Objectives I. Math 0910 Basic Arithmetic/Pre-Algebra Upon satisfactory completion of this course, the student should be able to perform the following outcomes and

More information

Algebra 1 Chapter 3 Vocabulary. equivalent - Equations with the same solutions as the original equation are called.

Algebra 1 Chapter 3 Vocabulary. equivalent - Equations with the same solutions as the original equation are called. Chapter 3 Vocabulary equivalent - Equations with the same solutions as the original equation are called. formula - An algebraic equation that relates two or more real-life quantities. unit rate - A rate

More information

Examples of Tasks from CCSS Edition Course 3, Unit 5

Examples of Tasks from CCSS Edition Course 3, Unit 5 Examples of Tasks from CCSS Edition Course 3, Unit 5 Getting Started The tasks below are selected with the intent of presenting key ideas and skills. Not every answer is complete, so that teachers can

More information

3.4 Complex Zeros and the Fundamental Theorem of Algebra

3.4 Complex Zeros and the Fundamental Theorem of Algebra 86 Polynomial Functions.4 Complex Zeros and the Fundamental Theorem of Algebra In Section., we were focused on finding the real zeros of a polynomial function. In this section, we expand our horizons and

More information

Official Math 112 Catalog Description

Official Math 112 Catalog Description Official Math 112 Catalog Description Topics include properties of functions and graphs, linear and quadratic equations, polynomial functions, exponential and logarithmic functions with applications. A

More information

Introduction to Finite Systems: Z 6 and Z 7

Introduction to Finite Systems: Z 6 and Z 7 Introduction to : Z 6 and Z 7 The main objective of this discussion is to learn more about solving linear and quadratic equations. The reader is no doubt familiar with techniques for solving these equations

More information

3.3. GRAPHS OF RATIONAL FUNCTIONS. Some of those sketching aids include: New sketching aids include:

3.3. GRAPHS OF RATIONAL FUNCTIONS. Some of those sketching aids include: New sketching aids include: 3.3. GRAPHS OF RATIONAL FUNCTIONS In a previous lesson you learned to sketch graphs by understanding what controls their behavior. Some of those sketching aids include: y-intercept (if any) x-intercept(s)

More information

f is a parabola whose vertex is the point (h,k). The parabola is symmetric with

f is a parabola whose vertex is the point (h,k). The parabola is symmetric with Math 1014: Precalculus with Transcendentals Ch. 3: Polynomials and Rational Functions Sec. 3.1 Quadratic Functions I. Quadratic Functions A. Definition 1. A quadratic function is a function of the form

More information

Chapter 1 Notes: Quadratic Functions

Chapter 1 Notes: Quadratic Functions 1 Chapter 1 Notes: Quadratic Functions (Textbook Lessons 1.1 1.2) Graphing Quadratic Function A function defined by an equation of the form, The graph is a U-shape called a. Standard Form Vertex Form axis

More information

TOPIC 3: CONTINUITY OF FUNCTIONS

TOPIC 3: CONTINUITY OF FUNCTIONS TOPIC 3: CONTINUITY OF FUNCTIONS. Absolute value We work in the field of real numbers, R. For the study of the properties of functions we need the concept of absolute value of a number. Definition.. Let

More information

CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA

CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical

More information