Algebra II Notes Unit Six: Polynomials Syllabus Objectives: 6.2 The student will simplify polynomial expressions.

Size: px
Start display at page:

Download "Algebra II Notes Unit Six: Polynomials Syllabus Objectives: 6.2 The student will simplify polynomial expressions."

Transcription

1 Algebra II Notes Unit Si: Polnomials Sllabus Objectives: 6. The student will simplif polnomial epressions. Review: Properties of Eponents (Allow students to come up with these on their own.) Let a and b be real numbers, and let m and n be integers. Product of Powers Propert Quotient of Powers Propert m n m n a a a m m a mn a 1 a or, a 0 n n nm a a a m Power of a Power Propert n a a mn ab a b Power of a Product Propert m m m m a a Power of a Quotient Propert, b 0 m b b m Negative Eponent Propert Zero Eponent Propert n m 1 a b a m or, a 0 a b a a 0 1, a 0 n Evaluating Numerical Epressions with Eponents E 1: Evaluate. Use the power of a power propert: Use the negative eponent propert: E : Evaluate. Use the power of a product propert: Use the negative eponent propert: Use the power of a quotient propert: 9 4 Page 1 of McDougal Littell

2 E : Evaluate 6 0. Algebra II Notes Unit Si: Polnomials Use the zero eponent propert: Use the quotient of a power propert: 6 Use the negative eponent propert: Note: We can use the quotient of a power propert to keep the eponent positive. Use the quotient of a power propert: Simplifing Algebraic Epressions E 1: Simplif the epression 8 1 and write with positive eponents. Use the power of a product propert: 1 Use the power of a power propert: Use the product of a power propert: Use the negative eponent propert: E : Simplif the epression 0 t v and write with positive eponents. Use the zero eponent propert: t 1 t 1 Use the negative eponent propert: t t t E : Simplif the epression 1 6 Use the product of a power propert: Use the power of a power propert: Use the quotient of a power propert: Simplif: 4 and write with positive eponents Page of McDougal Littell

3 Algebra II Notes Unit Si: Polnomials You Tr: Simplif the epression and write with positive eponents. Identif which properties of eponents ou used QOD: Which properties of eponents require ou to check that two or more bases are the same before appling the propert? Sample SAT Question(s): Taken from College Board online practice problems If m is a positive integer, which of the following is NOT equal to m? (A) (B) 4 m 4 m (C) m m (D) 4 m (E) Grid-In 16 m. For all positive integers a and b, let a b be defined b b a 1 ab a 1. What is the value of 4?. The positive integer n is not divisible b 7. The remainder when n is divided b are each equal to k. What is k? (A) 1 (B) (C) 4 (D) 6 (E) It cannot be determined from the information given. Page of McDougal Littell n is divided b 7 and the remainder when

4 Algebra II Notes Unit Si: Polnomials Sllabus Objective: 6.1 The student will graph a polnomial function with and without technolog. f a a... a a a, a 0 n n1 Polnomial Function: a function of the form n n1 1 0 Note: Eponents are whole numbers and coefficients are real numbers. n Leading Coefficient: a n Constant Term: a 0 Degree: n Note: Consider using the Fraer Model (Vocabular Concept Grid) Activit See resource pages. Standard Form of a Polnomial Function: The terms are written in descending order of the eponents Names of Polnomial Functions: This is kind of trick but a n is the name of the coefficient with the same degree. So, a n is the coefficient of the term that is the n th degree and a n 1 is the coefficient of the term that is degree n 1. Degree Tpe Standard Form 0 Constant f a0 1 Linear f a1 a0 Quadratic f a a a 1 0 Cubic f a a a a Quartic 4 f a a a a a Identifing Polnomial Functions f 8 a polnomial function? If es, write it in standard form. E 1: Is No. In order to be a polnomial function, all eponents must be whole numbers. f 8 a polnomial function? If es, write it in standard form. E : Is 4 Yes. All eponents are whole numbers and all coefficients are real numbers. f 8 Note: This is a quartic trinomial (degree = 4). Standard Form: 4 Evaluating Polnomial Functions Using Direct Substitution E 1: Find 4 f if f f So : f () 47 Page 4 of McDougal Littell

5 Algebra II Notes Unit Si: Polnomials Evaluating Polnomial Functions Using Snthetic Substitution E 1: Find 4 f if f 6 1 using snthetic substitution. Using the polnomial in standard form, write the coefficients in a row. Put the -value to the upper left Bring down the first coefficient, then multipl b the -value. multipl Add straight down the columns, and repeat The number in the bottom right is the value of So : f () 47 f. E : Find f if f 7 11 using snthetic substitution. This polnomial function is in standard form, however it is missing two terms. We can rewrite the f to fill in the missing terms. function as This also means that (, 17) is an ordered pair that would be a point on the graph. f 17 Graphing Polnomial Functions: To graph a polnomial function, make a table of values using snthetic substitution, plot the points, and determine the end behavior to draw the rest of the graph. End Behavior: the behavior of the graph as gets ver large (approaches positive infinit ) OR as gets ver small (or approaches negative infinit ). Notation: ( approaches positive infinit ) (The ver far right end of a graph). ( approaches negative infinit) (The ver far left end of a graph). Page of McDougal Littell

6 Algebra II Notes Unit Si: Polnomials Eploration Activit: Graph each function on the calculator. Determine the end behavior of f as approaches negative and positive infinit. Fill in the table and write our conclusion regarding the degree of the function and the end behavior. (Teacher Note: Answers are in red.) f Degree Sign of Leading Coefficient f f + + f + f + + f + 4 f f 4 + f + + f + 6 f f Page 6 of McDougal Littell

7 Conclusion: The graph of a polnomial function n n 1 f an an 1... a a1 a0 has the following end behavior. These patterns are ver predicatable. Algebra II Notes Unit Si: Polnomials Degree Lead Coefficient End Behavior Even Positive as, f as, f Even Negative as, f as, f Think of end behavior as what happens on either end of the graph. There can be a lot of curves, etc. in the middle, but polnomial functions either increase or decrease at the far ends (as, f( ) ). Odd Positive as, f as, f Odd Negative as, f as, f E 1: Graph the polnomial function 4 f 1 b hand. Check our graph on the graphing calculator. Step One: Make a table of values using snthetic substitution f Step Two: Determine end behavior using the degree and sign of the leading coefficient. The degree is even, and the leading coefficient is positive. So as, f as, f. Step Three: Graph the polnomial function Page 7 of McDougal Littell

8 Algebra II Notes Unit Si: Polnomials E : Graph the polnomial function f 4 b hand. Check our graph on the graphing calculator. Step One: Make a table of values using snthetic substitution f Wh didn t we use snthetic subsitution to find f(0)? Step Two: Determine end behavior using the degree and sign of the leading coefficient. The degree is odd and the leading coefficient is negative. So. Step Three: Graph the polnomial function. 1 as, f as, f You Tr: Graph the polnomial function f 1b hand. Check our graph on the graphing calculator. QOD: Which term of the polnomial function is most important when determining the end behavior of the function? Page 8 of McDougal Littell

9 Sample CCSD Common Eam Practice Question(s): Algebra II Notes Unit Si: Polnomials Which best represents the graph of the polnomial function? 4 Page 9 of McDougal Littell

10 Algebra II Notes Unit Si: Polnomials Sllabus Objective: 6. The student will simplif polnomial epressions. Adding Polnomials E 1: Add the polnomials Subtracting Polnomials. Vertical Method: Write each polnomial in standard form and line up like terms. Then add the like terms E 1: Subtract the polnomials To subtract, we will rewrite the problem as an addition problem b adding the opposite Horizontal Method: Combine each set of like terms Write the final answer in standard form Multipling Polnomials E 1: Find the product 4. Horizontal Method: Use the distributive propert b distributing each term of the first polnomial Combine like terms and write the answer in standard form Page 10 of McDougal Littell

11 Algebra II Notes Unit Si: Polnomials E : Multipl the polnomials Vertical Method: Use long multiplication E : Multipl the polnomials 14. Multipl the polnomials two at a time. Because the are binomials, we can use FOIL to multipl the first two Use the distributive propert Combine like terms and write in standard form Review: Special Products (Allow students to come up with these on their own.) Memorize these! Sum and Difference Product a bab a b Square of a Binomial Cube of a Binomial a b a abb a b a abb a b a a bab b a b a a bab b E 1: Simplif the epression. Using the cube of a binomial: Page 11 of McDougal Littell

12 Application Problems Algebra II Notes Unit Si: Polnomials E 1: Find a polnomial epression for the volume of a rectangular prism with sides, 4, and. Volume of a Rectangular Prism = Length Width Height FOIL: 4 1 Vertical Method: E : From 198 through 1996, the number of flu shots given in one cit can be modeled b A t t t t for adults and b C t t t t Write a model for the total number F of flu shots given in these ears for children, where t is the number of ears since 198. To find the total flu shots, we need to add the polnomials. Vertical Method: Solution: 4 11.t 8.t 194t 4190t t 106t 1t 1t t 97.67t 194t 40t 81 F t t t t You Tr: Find the product: 7 1 QOD: What is the advantage of the vertical method when adding, subtracting, or multipling polnomials? Sample CCSD Common Eam Practice Question(s):? Which polnomial represents the product of 8 A. B. C. D Page 1 of McDougal Littell

13 Algebra II Notes Unit Si: Polnomials Sllabus Objectives: 6. The student will solve polnomial equations b factoring and graphing. Review: Factoring Patterns Factoring a General Trinomial E 1: Factor the trinomial 1. ac Method: ac 4 Split the middle term: Factor b grouping: Factoring a Perfect Square Trinomial E 1: Factor the trinomial 6 9. Difference of Two Squares Use a abb a b. E 1: Factor Common Monomial Factor Use a b aba b E 1: Factor the trinomial completel. 7 Factor the GCF and the binomial square Since this is not completel factored, use a b a ba b Sum and Difference of Two Cubes 4 a b ab a abb a b ab a abb. Page 1 of McDougal Littell

14 E 1: Factor the binomial Algebra II Notes Unit Si: Polnomials 8 1. Use a b a ba ab b E : Factor the binomial 7. 6 Use a b a ba ab b. 4 9 Factoring b Grouping E 1: Factor the polnomial Group each pair of terms and factor the GCF. 9 Factor the common binomial factor. 9 Factor the remaining terms if possible. Review: Zero-Product Propert If ab 0, then a 0 or b 0. Solving Polnomial Equations b Factoring E 1: Solve the equation Step One: Set the equation equal to zero. Step Two: Factor the polnomial Step Three: Set each factor equal to zero and solve. Solutions:,,0,, , 0, Page 14 of McDougal Littell

15 Algebra II Notes Unit Si: Polnomials E : Find the real-number solutions of the equation 0. Step One: Set the equation equal to zero. Step Two: Factor the polnomial Step Three: Set each factor equal to zero and solve. Real-Number Solution: 0 0, i i Application Problem E 1: An optical compan is going to make a glass prism that has a volume of 1 cm. The height will be h cm, and the base will be a right triangle with legs of length h cm and h cm. What will be the height? 1 h h h Volume of a Prism = Area of the Base Height 1 To solve this equation for h, we must set it equal to zero. 1 h h h 1 h h h 1 h h h Before factoring, we can multipl both sides of the equation b to eliminate (clear) the fractions. 1 0 h h 6h 1 0 h h 1h 0 Factor b grouping. 0 h h 6 h h h 0 6 Solve b setting each factor equal to 0. The height of the prism will be cm. h 0 h cm h h 60 6 No real solution Page 1 of McDougal Littell

16 Algebra II Notes Unit Si: Polnomials You Tr: Solve the equation QOD: Give an eample of a binomial that can be factored either as the difference of two squares or as the difference of two cubes. Show the complete factorization of our binomial. Sample CCSD Common Eam Practice Question(s): 1. Which of the following represents the solution set of the polnomial equation A.,, i, i B.,,, C., i, i i, i D., i, i, 7 1 0? 4. What is the factored form of the polnomial A. 9 B. 9 C. 9 D. 9 7?. Which lists the set of all real zeros of the following polnomial function? A. B., f 4 1 C.,, D.,,1, Page 16 of McDougal Littell

17 Algebra II Notes Unit Si: Polnomials Sllabus Objective: 6.6 The student will divide polnomials and relate the result to the remainder theorem and the factor theorem. Dividing Polnomials Using Long Division On Your Own: Find the quotient of 1,6 and 4 using long division. On Your Own: Find the quotient of and 4 1. For each step of long division, we will divide the term with the highest power in the dividend b the first term of the divisor Remember to put a place for the missing term. 4 (add the opposite) (bring down the net term) (remainder) Eploration: Use the polnomial function f. Use long division to divide f. Then use snthetic substitution to find f. What do ou notice? b Remainder Theorem: If a polnomial is f divided b k, then the remainder is r f k. Dividing Polnomials Using Snthetic Division (Note: This can onl be used when the divisor is in the form k.) E 1: Divide the polnomial 7 6 b. Use snthetic substitution for k. The coefficients of the quotient and remainder appear in snthetic substitution. Quotient: R Note for graphing: This means that (, 7) is an ordered pair that is on the graph of the function. Page 17 of McDougal Littell

18 Factor Theorem: A polnomial Algebra II Notes Unit Si: Polnomials f has a factor k if and onl if f k 0. E 1: Factor f given that f 6 0. Because f 6 0, we know that 6 is a factor of f snthetic division to find the other factors E : One zero of f 6 f b the Factor Theorem. We will use f 9 1 is 7. Find the other zeros of the function. f 7 0, we know that 7 Because snthetic division to find the other factors. is a factor of f Note for graphing: This means that ( 6,0) is an ordered pair that is on the graph of the function. 6 is called a zero. It is also an intercept. b the Factor Theorem. We will use f 7 1 f Set each factor equal to zero and 4 You Tr: Use long division to find the quotient of You Tr: Given that t is a zero of the function. f t 4t 9t t 1, find the other zeros. QOD: If f is a polnomial that has a as a factor, what do ou know about the value of f a? Sample CCSD Common Eam Practice Question(s): What is 9 divided b? A. B. C. D Page 18 of McDougal Littell

19 Algebra II Notes Unit Si: Polnomials Sllabus Objectives: 6.7 The student will identif all possible rational zeros of a polnomial function b using the rational root theorem. 6.4 The student will find rational zeros of a polnomial. Using the Rational Zero Theorem Review: Rational zero is a rational number that produces a function value of 0. It can be visualized as f( ) 0 where is a rational number. On the graph it is an -intercept. The Rational Zero Theorem If f ( ) a n n... a1a0has integer coefficients, then ever rational zero of p factor of constant term a0 f has the following form: q factor of leading coefficient an The first important step is to list the possible rational zeros. After the are listed we can test them to determine if the are rational zeros. If the value of the possible rational zeros =0, the are called zeros. List the possible rational zeros: E 1: Find the possible rational zeros of f ( ) 7 1 Step 1: The leading coefficient is 1. 1 is the onl factor of 1. Step : The constant is 1. All of the factors of 1 are 1,,, 4, 6, 1. Step : List the possible factors ,,,,,and *If we tested for actual zeros using snthetic substitution from previous lessons we would find that and 4 are zeros. This also means that could not be a zero, 7 could not be a zero, 1 could not be a zero. E : Find the possible rational zeros of f ( ) 7 1 Step 1: The leading coefficient is. The factors of are 1 and. Step : The constant is -. All of the factors of - are, Step : List the possible factors -,,, 1 1 *We will not test for actual zeros for this eample. This also means that could not be a zero, 4 could not be a zero, could not be a zero. When the leading coefficient is not 1, the list of possible zeros can increase dramaticall. There are man tools that are used to find the rational zeros. We know some of those tools now and others will be introduced later in the class. Eamples that follow will demonstrate some of them. Page 19 of McDougal Littell

20 Algebra II Notes Unit Si: Polnomials f 7 18 E : Find all the real zeros of f 18 7 Step 1: Put the function in standard order. Step : List possible rational zeros (1,,,4,6,8,9,1,18,4,6,7) Step : Tr the possible zeros until ou find one From previous lessons, the function can be reduced to: Then factored: 4 f f 6 ( 4) Zeros: ( 4) Note: Finding rational zeros is also referred to as finding real zeros. Rational numbers are also real numbers. There is a distinction between listing possible rational zeros and finding rational (real) zeros. f E 4: Find all of the real zeros of Step 1: Notice that each term contains a common factor of. The problem can be factored to f ( 4 6) and since 0onl ( 4 6) can be =0. Step : List possible rational zeros (1,,,6) (Since the leading coefficient is now 1) Step : Tr the possible zeros until ou find one From previous lessons, the function can be reduced to: Then factored: f f ( 1) 0 0 ( 1) 0 Zeros: 1 Page 0 of McDougal Littell

21 E : Find all the real zeros of 4 Algebra II Notes Unit Si: Polnomials f 4 81 Step 1: Mabe this could be graphed first. Step : Look at the graph for reasonable choices 1 It appears the might be,, and Step : Check the chosen values using snthetic division. Start with -. Wh not ½?? Wh not 7? Is a root (zero). f ( )( ) The factored form so far is 4 Step 4: Repeat the steps above using a different reasonable choice. Tr Is a root (zero) Step : Repeat the steps above using a different reasonable choice. Tr Is a root (zero). Step 6: The function 7 is left to be factored. ( 9) has no real factors. 1 Solution: There are real zeros:,, and 1 Yes, all three work! And each time, the function (polnomial) is reduced b one degree. f You Tr: Find all real zeros of QOD: If the leading coefficient of a polnomial with integer coefficients is 1, what tpe of numbers must an possible rational zeros be? Page 1 of McDougal Littell

22 Sample CCSD Common Eam Practice Question(s): Algebra II Notes Unit Si: Polnomials Which lists the set of all real zeros of the following polnomial function? A. B., C.,, D.,,1, f 4 1 Page of McDougal Littell

23 Algebra II Notes Unit Si: Polnomials Sllabus Objective: 6. The student will use the Fundamental Theorem of Algebra to determine the number of zeros of a polnomial function. The Fundamental Theorem of Algebra If f is a polnomial of degree n where n0, then the equation f( ) 0 has at least one root in the set of comple numbers. Finding the number of solutions or zeros Review: Find the solutions of the following eamples. State how man solutions each has and classif each zero as rational, irrational, or comple (imaginar). E 1: 1 0 E : 9 0 E : 1 0 for the quadratic factor.) (Hint: Use the factorization for the difference of cubes, then use the quadratic formula Do ou notice a pattern with the degree of the polnomial and the number of solutions each has? E 4: How man different solutions are there to How do ou eplain this number? ? E : How man different solutions are there to Solution: 4, i 4i 16 0? Note: On the graph, the imaginar roots do not cross the ais. Note: 4, i 4i are comple conjugate pairs. 1i,1 i are comple conjugate pairs. The comple roots of polnomial functions with real coefficients alwas occur in comple conjugate pairs. Is this also the case for irrational zeros? Page of McDougal Littell

24 Algebra II Notes Unit Si: Polnomials Finding the zeros of a polnomial function This activit involves finding the rational zeros as learned in the previous section, then using other tools, such as the quadratic formula or technolog, to find the irrational or comple roots. E 1: Find all zeros of f 4 ( ) Using the rational root theorem and snthetic division, it can be shown that is a repeated root and and -1 are roots. The factored form looks like this: ( ) ( )( 1). The graph is shown. When a factor k is raised to an odd power, the graph crosses through the -ais. When a factor k is raised to an even power, the graph is tangent to the -ais. Solution: There are four real zeros is a repeated root and and -1 are roots. E : Find all zeros of f 4 ( ) 0 Using the rational root theorem and snthetic division, it can be shown that and - are roots. 4 Using the pattern of E it can be shown that f( ) 0 factors to ( )( )( ) Using the quadratic formula ields zeros of = i Solution: There are four zeros, and - and i. Two are real and two are comple conjugates Using Zeros to Write Polnomial Functions E 1: Write a polnomial function f of least degree that has real coefficients, a leading coefficient of 1, and, and - as zeros. Step 1: Write f ( ) in factored form: f( ) ( )( )( ) Step : Review - Multipl the polnomials two at a time. Because the are binomials, we can use FOIL to multipl the first two. f( ) ( 6)( ) f( ) 190 Solution: f ( ) 19 0 Page 4 of McDougal Littell

25 Algebra II Notes Unit Si: Polnomials E : Write a polnomial function f of least degree that has real coefficients, a leading coefficient of 1, and 1, -, and 1- i as zeros. Step 1: Since 1- i is a zero, so is 1+ i Step : Write f ( ) in factored form: f ( ) ( 1)( )( (1- i) ( (1+ i)) Step : Regroup: f ( ) ( 1)( ) ( 1)- i ( 1)+ i 10 Step 4: Epand, multipl polnomials, and combine like terms. f( ) ( ) ( 1) - i f ( ) ( ) ( 1 1 f ( ) ( )( ) f 4 ( ) Note: This graph onl has two intercepts. Wh? Using Technolog to Approimate Zeros Specific instructions should be given based on the calculator used. This section will provide onl general direction. E 1: 4 Approimate the real zeros of f( ) Use a graphing calculator to graph and calculate the zeros. You Tr: State the number of zeros of f 1 and find what the are. QOD: What is the conjugate of a comple number, and wh is it important when finding all of the zeros of a polnomial function? Page of McDougal Littell

26 Sample CCSD Common Eam Practice Question(s): Algebra II Notes Unit Si: Polnomials According to the Fundamental Theorem of Algebra, how man comple zeros does the polnomial f 1 have? A. B. C. 4 D. 4 Page 6 of McDougal Littell

27 Algebra II Notes Unit Si: Polnomials Sllabus Objective: 6.8 The student will analze graphs of polnomial functions to determine its characteristics. Analzing polnomial graphs Concept Summar n n 1 Let f an an 1... a a1a0 be a polnomial function. The following statements are equivalent: Zero: k is a zero of the function f. Factor: Solution: k is a factor of polnomial f(). k is a solution of the polnomial function f()=0. - Intercept: k is an -intercept of the graph of the polnomial function f. Using -Intercepts to Graph a Polnomial Function f 1 E 1: Graph the function Step 1: Plot the -intercepts. Since + and 1 are factors, and 1 are zeros (-intercepts) Note: + is raised to an odd power so the graph crosses the -ais at =. 1 is raised to an even power so the graph is tangent to the -ais at = 1. When a factor k is raised to an odd power, the graph crosses through the -ais. When a factor k is raised to an even power, the graph is tangent to the -ais. Step : Plot a few points between the -intercepts. f(0) ; f( 1) 4 Step : Determine the end behavior of the graph. Cubic function (odd degree) with positive leading coefficient as, f as, f Step 4: Sketch the graph Page 7 of McDougal Littell

28 Finding Turning Points Algebra II Notes Unit Si: Polnomials Turning points of polnomial functions: Another important characteristic of graphs of polnomial functions is that the have turning points corresponding to local maimum and minimum values. The coordinate of a turning point is a local maimum if the point is higher than all nearb points. The coordinate of a turning point is a local minimum if the point is lower than all nearb points. The graph of ever polnomial function of degree n has at most n 1 turning points. Moreover, if a polnomial has n distinct real zeros, then its graph has eactl n 1 turning points. E 1: Identif the zeros and turning points (estimate the zeros and turning points) Turning point (ma) zero zero Turning point (min) Turning point (ma) zero Turning point (min) Leading coefficient positive real zeros (including the double zero) {, 1, 1} turning points ( 1,4); (1,0) 1 local ma; 1 local min Leading coefficient positive 1 real zero, imaginar zeros {} turning points (0, ); (, 8) 1 local ma; 1 local min You Tr: Leading coefficient real zeros -10 Leading coefficient real zeros -10 Leading coefficient real zeros -10 Leading coefficient real zeros turning points turning points turning points turning points local ma; local min local ma; local min local ma; local min local ma; local min Page 8 of McDougal Littell

29 Algebra II Notes Unit Si: Polnomials E : Use a graphing calculator to graph and calculate the approimate local maimum(s) and local minimum(s) of f( ) ( )( )( ) Local maimum Coordinates:( 1, 6) Local ma is 6 at = 1 Local minimum You Tr: Graph the function use the calculator to find the turning points of the function. You Tr: Graph the function f 1 b hand. Check our graph on the graphing calculator and f 9 1 on the calculator and find the local etrema. QOD: What is the difference between local and absolute maima and minima? Sample CCSD Common Eam Practice Question(s): 1. Which describes the end behavior of the graph of 4 A. f B. f C. f 0 D. f f as? Page 9 of McDougal Littell

30 Algebra II Notes Unit Si: Polnomials. Use the graph of a polnomial function below. A. {} What are the zeros of the polnomial? B. { } C. {, 1, 4} D. {, 1, 4} Eploring Data and Statistics (Notes are not provided for this material) Modeling with Polnomial Functions Write a polnomial function whose intercepts are given. Finding and Using Finite Differences Properties of finite differences 1. If a polnomial function f() has degree n, then the n th -order differences of function values for equall spaced values are non-zero and constant.. Conversel, if the n th order differences of equall spaced data are non-zero and constant, then the data can be represented b a polnomial function of degree n. Polnomial Modeling with Technolog Graphing calculators make it eas to enter data, make a scatter plot, and calculate linear, quadratic, cubic, and quartic regressions. Page 0 of McDougal Littell

31 Unit Summar: Algebra II Notes Unit Si: Polnomials Polnomial equations provide some of the most classic problems in all of algebra. Finding zeros and etrema have man real-world applications. Real-life situations are modeled b writing equations based on data and using those equations to determine or estimate other data points (speed, volume, time, profits, patterns, etc). Page 1 of McDougal Littell

32 Resources Algebra II Notes Unit Si: Polnomials Vocabular (Concept Grid) This activit can be done as a whole class activit or as a smallgroup activit. It can be done quickl as a review, or in more detail as instruction with compare/contrast capabilities. The word or phrase is written in the center bo Eamples are written in the bottom left section (Quadrant III) Each of the quadrants should include eamples, illustrations, images as appropriate Teacher ma give some, but students should see the pattern and provide input Definition Eamples Characteristics Non Eamples Non eamples are written in the bottom right section (Quadrant IV). Teacher ma give some, but students should see the pattern and provide input Characteristics are written in the top right section (Quadrant I). Teacher ma provide some, but students should see the pattern and provide input A definition is written in the upper left section (Quadrant II). As students offer definitions, it ma be necessar to add more information in the quadrants or to point to information that will focus their definition. Variations: The order of placing information in the quadrants can be changed. For eample, on new material, the definition could be given first, then mabe an eample or two, then ask students for more eamples or non eamples, then characteristics.. f 4 f 8 f Definition Characteristics Polnomial function Eamples Onl whole number eponents No negative eponents No imaginar coefficients Can be written as f()= or = Continuous graph no missing domain End point behavior Non Eamples i 1 The word or phrase can be left blank and then determined b students based on the information in the quadrants. Go through the word categorization quickl, then use the back of the paper (or other space) to epand instruction or practice if needed. Page of McDougal Littell

Review: Properties of Exponents (Allow students to come up with these on their own.) m n m n. a a a. n n n m. a a a. a b a

Review: Properties of Exponents (Allow students to come up with these on their own.) m n m n. a a a. n n n m. a a a. a b a Algebra II Notes Unit Si: Polynomials Syllabus Objectives: 6. The student will simplify polynomial epressions. Review: Properties of Eponents (Allow students to come up with these on their own.) Let a

More information

Ready To Go On? Skills Intervention 6-1 Polynomials

Ready To Go On? Skills Intervention 6-1 Polynomials 6A Read To Go On? Skills Intervention 6- Polnomials Find these vocabular words in Lesson 6- and the Multilingual Glossar. Vocabular monomial polnomial degree of a monomial degree of a polnomial leading

More information

Algebra II Notes Unit Five: Quadratic Functions. Syllabus Objectives: 5.1 The student will graph quadratic functions with and without technology.

Algebra II Notes Unit Five: Quadratic Functions. Syllabus Objectives: 5.1 The student will graph quadratic functions with and without technology. Sllabus Objectives:.1 The student will graph quadratic functions with and without technolog. Quadratic Function: a function that can be written in the form are real numbers Parabola: the U-shaped graph

More information

Algebra II Notes Polynomial Functions Unit Introduction to Polynomials. Math Background

Algebra II Notes Polynomial Functions Unit Introduction to Polynomials. Math Background Introduction to Polynomials Math Background Previously, you Identified the components in an algebraic epression Factored quadratic epressions using special patterns, grouping method and the ac method Worked

More information

2.1 Evaluate and Graph Polynomial

2.1 Evaluate and Graph Polynomial 2. Evaluate and Graph Polnomial Functions Georgia Performance Standard(s) MM3Ab, MM3Ac, MM3Ad Your Notes Goal p Evaluate and graph polnomial functions. VOCABULARY Polnomial Polnomial function Degree of

More information

LESSON #28 - POWER FUNCTIONS COMMON CORE ALGEBRA II

LESSON #28 - POWER FUNCTIONS COMMON CORE ALGEBRA II 1 LESSON #8 - POWER FUNCTIONS COMMON CORE ALGEBRA II Before we start to analze polnomials of degree higher than two (quadratics), we first will look at ver simple functions known as power functions. The

More information

Polynomial and Rational Functions

Polynomial and Rational Functions Name Date Chapter Polnomial and Rational Functions Section.1 Quadratic Functions Objective: In this lesson ou learned how to sketch and analze graphs of quadratic functions. Important Vocabular Define

More information

LESSON #24 - POWER FUNCTIONS COMMON CORE ALGEBRA II

LESSON #24 - POWER FUNCTIONS COMMON CORE ALGEBRA II 1 LESSON #4 - POWER FUNCTIONS COMMON CORE ALGEBRA II Before we start to analze polnomials of degree higher than two (quadratics), we first will look at ver simple functions known as power functions. The

More information

Polynomials and Polynomial Functions

Polynomials and Polynomial Functions Unit 5: Polynomials and Polynomial Functions Evaluating Polynomial Functions Objectives: SWBAT identify polynomial functions SWBAT evaluate polynomial functions. SWBAT find the end behaviors of polynomial

More information

f(x) = 2x 2 + 2x - 4

f(x) = 2x 2 + 2x - 4 4-1 Graphing Quadratic Functions What You ll Learn Scan the tet under the Now heading. List two things ou will learn about in the lesson. 1. Active Vocabular 2. New Vocabular Label each bo with the terms

More information

Algebra I Notes Unit Eleven: Polynomials

Algebra I Notes Unit Eleven: Polynomials Syllabus Objective: 9.1 The student will add, subtract, multiply, and factor polynomials connecting the arithmetic and algebraic processes. Teacher Note: A nice way to illustrate operations with polynomials

More information

x Radical Sign: Radicand: the number beneath the radical sign

x Radical Sign: Radicand: the number beneath the radical sign Sllabus Objective: 9.4 The student will solve quadratic equations using graphic and algebraic techniques to include the quadratic formula, square roots, factoring, completing the square, and graphing.

More information

SEE the Big Idea. Quonset Hut (p. 218) Zebra Mussels (p. 203) Ruins of Caesarea (p. 195) Basketball (p. 178) Electric Vehicles (p.

SEE the Big Idea. Quonset Hut (p. 218) Zebra Mussels (p. 203) Ruins of Caesarea (p. 195) Basketball (p. 178) Electric Vehicles (p. Polnomial Functions.1 Graphing Polnomial Functions. Adding, Subtracting, and Multipling Polnomials.3 Dividing Polnomials. Factoring Polnomials.5 Solving Polnomial Equations. The Fundamental Theorem of

More information

Factoring Polynomials

Factoring Polynomials 5. TEXAS ESSENTIAL KNOWLEDGE AND SKILLS 2A.7.D 2A.7.E Factoring Polnomials Essential Question How can ou factor a polnomial? Factoring Polnomials Work with a partner. Match each polnomial equation with

More information

Vocabulary. Term Page Definition Clarifying Example degree of a monomial. degree of a polynomial. end behavior. leading coefficient.

Vocabulary. Term Page Definition Clarifying Example degree of a monomial. degree of a polynomial. end behavior. leading coefficient. CHAPTER 6 Vocabular The table contains important vocabular terms from Chapter 6. As ou work through the chapter, fill in the page number, definition, and a clarifing eample. Term Page Definition Clarifing

More information

Course 15 Numbers and Their Properties

Course 15 Numbers and Their Properties Course Numbers and Their Properties KEY Module: Objective: Rules for Eponents and Radicals To practice appling rules for eponents when the eponents are rational numbers Name: Date: Fill in the blanks.

More information

Learning Goals. College of Charleston Department of Mathematics Math 101: College Algebra Final Exam Review Problems 1

Learning Goals. College of Charleston Department of Mathematics Math 101: College Algebra Final Exam Review Problems 1 College of Charleston Department of Mathematics Math 0: College Algebra Final Eam Review Problems Learning Goals (AL-) Arithmetic of Real and Comple Numbers: I can classif numbers as natural, integer,

More information

Lesson 7.1 Polynomial Degree and Finite Differences

Lesson 7.1 Polynomial Degree and Finite Differences Lesson 7.1 Polnomial Degree and Finite Differences 1. Identif the degree of each polnomial. a. 1 b. 0. 1. 3. 3 c. 0 16 0. Determine which of the epressions are polnomials. For each polnomial, state its

More information

BIG IDEAS MATH. Ron Larson Laurie Boswell. Sampler

BIG IDEAS MATH. Ron Larson Laurie Boswell. Sampler BIG IDEAS MATH Ron Larson Laurie Boswell Sampler 3 Polnomial Functions 3.1 Graphing Polnomial Functions 3. Adding, Subtracting, and Multipling Polnomials 3.3 Dividing Polnomials 3. Factoring Polnomials

More information

Algebra 1 Skills Needed for Success in Math

Algebra 1 Skills Needed for Success in Math Algebra 1 Skills Needed for Success in Math A. Simplifing Polnomial Epressions Objectives: The student will be able to: Appl the appropriate arithmetic operations and algebraic properties needed to simplif

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

Algebra 1 Skills Needed to be Successful in Algebra 2

Algebra 1 Skills Needed to be Successful in Algebra 2 Algebra 1 Skills Needed to be Successful in Algebra A. Simplifing Polnomial Epressions Objectives: The student will be able to: Appl the appropriate arithmetic operations and algebraic properties needed

More information

Summer Review For Students Entering Algebra 2

Summer Review For Students Entering Algebra 2 Summer Review For Students Entering Algebra Teachers and administrators at Tuscarora High School activel encourage parents and communit members to engage in children s learning. This Summer Review For

More information

Unit 2 Notes Packet on Quadratic Functions and Factoring

Unit 2 Notes Packet on Quadratic Functions and Factoring Name: Period: Unit Notes Packet on Quadratic Functions and Factoring Notes #: Graphing quadratic equations in standard form, verte form, and intercept form. A. Intro to Graphs of Quadratic Equations: a

More information

Math 154 :: Elementary Algebra

Math 154 :: Elementary Algebra Math :: Elementar Algebra Section. Section. Section. Section. Section. Section. Math :: Elementar Algebra Section. Eponents. When multipling like-bases, ou can add the eponents to simplif the epression..

More information

+ = + + = x = + = + = 36x

+ = + + = x = + = + = 36x Ch 5 Alg L Homework Worksheets Computation Worksheet #1: You should be able to do these without a calculator! A) Addition (Subtraction = add the opposite of) B) Multiplication (Division = multipl b the

More information

Review of Essential Skills and Knowledge

Review of Essential Skills and Knowledge Review of Essential Skills and Knowledge R Eponent Laws...50 R Epanding and Simplifing Polnomial Epressions...5 R 3 Factoring Polnomial Epressions...5 R Working with Rational Epressions...55 R 5 Slope

More information

Algebra/Pre-calc Review

Algebra/Pre-calc Review Algebra/Pre-calc Review The following pages contain various algebra and pre-calculus topics that are used in the stud of calculus. These pages were designed so that students can refresh their knowledge

More information

A. Simplifying Polynomial Expressions

A. Simplifying Polynomial Expressions A. Simplifing Polnomial Epressions I. Combining Like Terms - You can add or subtract terms that are considered "like", or terms that have the same variable(s) with the same eponent(s). E. 1: 5-7 + 10 +

More information

Glossary. Also available at BigIdeasMath.com: multi-language glossary vocabulary flash cards. An equation that contains an absolute value expression

Glossary. Also available at BigIdeasMath.com: multi-language glossary vocabulary flash cards. An equation that contains an absolute value expression Glossar This student friendl glossar is designed to be a reference for ke vocabular, properties, and mathematical terms. Several of the entries include a short eample to aid our understanding of important

More information

CHAPTER 2 Polynomial and Rational Functions

CHAPTER 2 Polynomial and Rational Functions CHAPTER Polnomial and Rational Functions Section. Quadratic Functions..................... 9 Section. Polnomial Functions of Higher Degree.......... Section. Real Zeros of Polnomial Functions............

More information

3.1 Graphing Quadratic Functions. Quadratic functions are of the form.

3.1 Graphing Quadratic Functions. Quadratic functions are of the form. 3.1 Graphing Quadratic Functions A. Quadratic Functions Completing the Square Quadratic functions are of the form. 3. It is easiest to graph quadratic functions when the are in the form using transformations.

More information

Chapter 5: Quadratic Equations and Functions 5.1 Modeling Data With Quadratic Functions Quadratic Functions and Their Graphs

Chapter 5: Quadratic Equations and Functions 5.1 Modeling Data With Quadratic Functions Quadratic Functions and Their Graphs Ch 5 Alg Note Sheet Ke Chapter 5: Quadratic Equations and Functions 5.1 Modeling Data With Quadratic Functions Quadratic Functions and Their Graphs Definition: Standard Form of a Quadratic Function The

More information

Ch 5 Alg 2 L2 Note Sheet Key Do Activity 1 on your Ch 5 Activity Sheet.

Ch 5 Alg 2 L2 Note Sheet Key Do Activity 1 on your Ch 5 Activity Sheet. Ch Alg L Note Sheet Ke Do Activit 1 on our Ch Activit Sheet. Chapter : Quadratic Equations and Functions.1 Modeling Data With Quadratic Functions You had three forms for linear equations, ou will have

More information

Secondary Mathematics 2 Table of Contents

Secondary Mathematics 2 Table of Contents Secondar Mathematics Table of Contents Unit 1: Etending the Number Sstem Cluster 1: Etending Properties of Eponents (N.RN.1 and N.RN.)... 3 Cluster : Using Properties of Rational and Irrational Numbers

More information

Algebra 2 CPA Summer Assignment 2018

Algebra 2 CPA Summer Assignment 2018 Algebra CPA Summer Assignment 018 This assignment is designed for ou to practice topics learned in Algebra 1 that will be relevant in the Algebra CPA curriculum. This review is especiall important as ou

More information

Algebra 2 Honors Summer Packet 2018

Algebra 2 Honors Summer Packet 2018 Algebra Honors Summer Packet 018 Solving Linear Equations with Fractional Coefficients For these problems, ou should be able to: A) determine the LCD when given two or more fractions B) solve a linear

More information

TEKS: 2A.10F. Terms. Functions Equations Inequalities Linear Domain Factor

TEKS: 2A.10F. Terms. Functions Equations Inequalities Linear Domain Factor POLYNOMIALS UNIT TEKS: A.10F Terms: Functions Equations Inequalities Linear Domain Factor Polynomials Monomial, Like Terms, binomials, leading coefficient, degree of polynomial, standard form, terms, Parent

More information

5.4 dividing POlynOmIAlS

5.4 dividing POlynOmIAlS SECTION 5.4 dividing PolNomiAls 3 9 3 learning ObjeCTIveS In this section, ou will: Use long division to divide polnomials. Use snthetic division to divide polnomials. 5.4 dividing POlnOmIAlS Figure 1

More information

Cubic and quartic functions

Cubic and quartic functions 3 Cubic and quartic functions 3A Epanding 3B Long division of polnomials 3C Polnomial values 3D The remainder and factor theorems 3E Factorising polnomials 3F Sum and difference of two cubes 3G Solving

More information

INTRODUCTION GOOD LUCK!

INTRODUCTION GOOD LUCK! INTRODUCTION The Summer Skills Assignment for has been developed to provide all learners of our St. Mar s Count Public Schools communit an opportunit to shore up their prerequisite mathematical skills

More information

Copyrighted by Gabriel Tang B.Ed., B.Sc. Page 111.

Copyrighted by Gabriel Tang B.Ed., B.Sc. Page 111. Algera Chapter : Polnomial and Rational Functions Chapter : Polnomial and Rational Functions - Polnomial Functions and Their Graphs Polnomial Functions: - a function that consists of a polnomial epression

More information

Algebra II Notes Unit Nine: Rational Equations and Functions

Algebra II Notes Unit Nine: Rational Equations and Functions Syllabus Objectives: 9. The student will solve a problem by applying inverse and joint variation. 9.6 The student will develop mathematical models involving rational epressions to solve realworld problems.

More information

Graphing Calculator Computations 2

Graphing Calculator Computations 2 Graphing Calculator Computations A) Write the graphing calculator notation and B) Evaluate each epression. 4 1) 15 43 8 e) 15 - -4 * 3^ + 8 ^ 4/ - 1) ) 5 ) 8 3 3) 3 4 1 8 3) 7 9 4) 1 3 5 4) 5) 5 5) 6)

More information

AP Calculus AB Summer Assignment Mrs. Berkson

AP Calculus AB Summer Assignment Mrs. Berkson AP Calculus AB Summer Assignment Mrs. Berkson The purpose of the summer assignment is to prepare ou with the necessar Pre- Calculus skills required for AP Calculus AB. Net ear we will be starting off the

More information

Which of the following expressions are monomials?

Which of the following expressions are monomials? 9 1 Stud Guide Pages 382 387 Polnomials The epressions, 6, 5a 2, and 10cd 3 are eamples of monomials. A monomial is a number, a variable, or a product of numbers and variables. An eponents in a monomial

More information

Polynomial and Rational Functions

Polynomial and Rational Functions Polnomial and Rational Functions 5 Figure 1 35-mm film, once the standard for capturing photographic images, has been made largel obsolete b digital photograph. (credit film : modification of work b Horia

More information

AP Calculus AB Summer Assignment Mrs. Berkson

AP Calculus AB Summer Assignment Mrs. Berkson AP Calculus AB Summer Assignment Mrs. Berkson The purpose of the summer assignment is to prepare ou with the necessar Pre- Calculus skills required for AP Calculus AB. Net ear we will be starting off the

More information

Ready To Go On? Skills Intervention 5-1 Using Transformations to Graph Quadratic Functions

Ready To Go On? Skills Intervention 5-1 Using Transformations to Graph Quadratic Functions Read To Go On? Skills Intervention 5-1 Using Transformations to Graph Quadratic Functions Find these vocabular words in Lesson 5-1 and the Multilingual Glossar. Vocabular quadratic function parabola verte

More information

FINAL EXAM REVIEW ITEMS Math 0312: Intermediate Algebra Name

FINAL EXAM REVIEW ITEMS Math 0312: Intermediate Algebra Name FINAL EXAM REVIEW ITEMS Math 0312: Intermediate Algebra Name 1) Find the SUM of the solutions of the equation. 82 + 0 = 16 Use the quadratic formula to solve the equation. (All solutions are real numbers.)

More information

ACTIVITY 14 Continued

ACTIVITY 14 Continued 015 College Board. All rights reserved. Postal Service Write your answers on notebook paper. Show your work. Lesson 1-1 1. The volume of a rectangular bo is given by the epression V = (10 6w)w, where w

More information

COUNCIL ROCK HIGH SCHOOL MATHEMATICS. A Note Guideline of Algebraic Concepts. Designed to assist students in A Summer Review of Algebra

COUNCIL ROCK HIGH SCHOOL MATHEMATICS. A Note Guideline of Algebraic Concepts. Designed to assist students in A Summer Review of Algebra COUNCIL ROCK HIGH SCHOOL MATHEMATICS A Note Guideline of Algebraic Concepts Designed to assist students in A Summer Review of Algebra [A teacher prepared compilation of the 7 Algebraic concepts deemed

More information

H.Algebra 2 Summer Review Packet

H.Algebra 2 Summer Review Packet H.Algebra Summer Review Packet 1 Correlation of Algebra Summer Packet with Algebra 1 Objectives A. Simplifing Polnomial Epressions Objectives: The student will be able to: Use the commutative, associative,

More information

Attributes of Polynomial Functions VOCABULARY

Attributes of Polynomial Functions VOCABULARY 8- Attributes of Polnomial Functions TEKS FCUS Etends TEKS ()(A) Graph the functions f () =, f () =, f () =, f () =, f () = b, f () =, and f () = log b () where b is,, and e, and, when applicable, analze

More information

math0320 FALL interactmath sections developmental mathematics sullivan 1e

math0320 FALL interactmath sections developmental mathematics sullivan 1e Eam final eam review 180 plus 234 TSI questions for intermediate algebra m032000 013014 NEW Name www.alvarezmathhelp.com math0320 FALL 201 1400 interactmath sections developmental mathematics sullivan

More information

Algebra 2 Semester Exam Review

Algebra 2 Semester Exam Review Algebra Semester Eam Review 7 Graph the numbers,,,, and 0 on a number line Identif the propert shown rs rs r when r and s Evaluate What is the value of k k when k? Simplif the epression 7 7 Solve the equation

More information

d. 2x 3 7x 2 5x 2 2x 2 3x 1 x 2x 3 3x 2 1x 2 4x 2 6x 2 3. a. x 5 x x 2 5x 5 5x 25 b. x 4 2x 2x 2 8x 3 3x 12 c. x 6 x x 2 6x 6 6x 36

d. 2x 3 7x 2 5x 2 2x 2 3x 1 x 2x 3 3x 2 1x 2 4x 2 6x 2 3. a. x 5 x x 2 5x 5 5x 25 b. x 4 2x 2x 2 8x 3 3x 12 c. x 6 x x 2 6x 6 6x 36 Vertices: (.8, 5.), (.37, 3.563), (.6, 0.980), (5.373, 6.66), (.8, 7.88), (.95,.) Graph the equation for an value of P (the second graph shows the circle with P 5) and imagine increasing the value of P,

More information

MHF 4U Unit 1 Polynomial Functions Outline

MHF 4U Unit 1 Polynomial Functions Outline MHF 4U Unit 1 Polnomial Functions Outline Da Lesson Title Specific Epectations 1 Average Rate of Change and Secants D1., 1.6, both D1.1A s - Instantaneous Rate of Change and Tangents D1.6, 1.4, 1.7, 1.5,

More information

The American School of Marrakesh. Algebra 2 Algebra 2 Summer Preparation Packet

The American School of Marrakesh. Algebra 2 Algebra 2 Summer Preparation Packet The American School of Marrakesh Algebra Algebra Summer Preparation Packet Summer 016 Algebra Summer Preparation Packet This summer packet contains eciting math problems designed to ensure our readiness

More information

Higher. Polynomials and Quadratics. Polynomials and Quadratics 1

Higher. Polynomials and Quadratics. Polynomials and Quadratics 1 Higher Mathematics Contents 1 1 Quadratics EF 1 The Discriminant EF 3 3 Completing the Square EF 4 4 Sketching Parabolas EF 7 5 Determining the Equation of a Parabola RC 9 6 Solving Quadratic Inequalities

More information

1.2 Functions and Their Properties PreCalculus

1.2 Functions and Their Properties PreCalculus 1. Functions and Their Properties PreCalculus 1. FUNCTIONS AND THEIR PROPERTIES Learning Targets for 1. 1. Determine whether a set of numbers or a graph is a function. Find the domain of a function given

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

4Cubic. polynomials UNCORRECTED PAGE PROOFS

4Cubic. polynomials UNCORRECTED PAGE PROOFS 4Cubic polnomials 4.1 Kick off with CAS 4. Polnomials 4.3 The remainder and factor theorems 4.4 Graphs of cubic polnomials 4.5 Equations of cubic polnomials 4.6 Cubic models and applications 4.7 Review

More information

Ready To Go On? Skills Intervention 12-1 Inverse Variation

Ready To Go On? Skills Intervention 12-1 Inverse Variation 12A Find this vocabular word in Lesson 12-1 and the Multilingual Glossar. Identifing Inverse Variation Tell whether the relationship is an inverse variation. Eplain. A. Read To Go On? Skills Intervention

More information

review math0410 (1-174) and math 0320 ( ) aafinm mg

review math0410 (1-174) and math 0320 ( ) aafinm mg Eam Name review math04 (1-174) and math 0320 (17-243) 03201700aafinm0424300 mg MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Simplif. 1) 7 2-3 A)

More information

College Algebra Final, 7/2/10

College Algebra Final, 7/2/10 NAME College Algebra Final, 7//10 1. Factor the polnomial p() = 3 5 13 4 + 13 3 + 9 16 + 4 completel, then sketch a graph of it. Make sure to plot the - and -intercepts. (10 points) Solution: B the rational

More information

CHAPTER 1-5 CREDIT INTERVENTION ASSIGNMENT

CHAPTER 1-5 CREDIT INTERVENTION ASSIGNMENT MHFU NAME: DATE: CHAPTER - CREDIT INTERVENTION ASSIGNMENT Circle the best option and write our choice in the space provided Show our work clearl and in the correct order on separate sheets Which one of

More information

LESSON #42 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART 2 COMMON CORE ALGEBRA II

LESSON #42 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART 2 COMMON CORE ALGEBRA II LESSON #4 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART COMMON CORE ALGEBRA II You will recall from unit 1 that in order to find the inverse of a function, ou must switch and and solve for. Also,

More information

ab is shifted horizontally by h units. ab is shifted vertically by k units.

ab is shifted horizontally by h units. ab is shifted vertically by k units. Algera II Notes Unit Eight: Eponential and Logarithmic Functions Sllaus Ojective: 8. The student will graph logarithmic and eponential functions including ase e. Eponential Function: a, 0, Graph of an

More information

Name Class Date. Finding Real Roots of Polynomial Equations Extension: Graphing Factorable Polynomial Functions

Name Class Date. Finding Real Roots of Polynomial Equations Extension: Graphing Factorable Polynomial Functions Name Class Date -1 Finding Real Roots of Polnomial Equations Etension: Graphing Factorable Polnomial Functions Essential question: How do ou use zeros to graph polnomial functions? Video Tutor prep for

More information

Mathematics. Polynomials and Quadratics. hsn.uk.net. Higher. Contents. Polynomials and Quadratics 1. CfE Edition

Mathematics. Polynomials and Quadratics. hsn.uk.net. Higher. Contents. Polynomials and Quadratics 1. CfE Edition Higher Mathematics Contents 1 1 Quadratics EF 1 The Discriminant EF 3 3 Completing the Square EF 4 4 Sketching Parabolas EF 7 5 Determining the Equation of a Parabola RC 9 6 Solving Quadratic Inequalities

More information

Additional Factoring Examples:

Additional Factoring Examples: Honors Algebra -3 Solving Quadratic Equations by Graphing and Factoring Learning Targets 1. I can solve quadratic equations by graphing. I can solve quadratic equations by factoring 3. I can write a quadratic

More information

3.1 Exponential Functions and Their Graphs

3.1 Exponential Functions and Their Graphs .1 Eponential Functions and Their Graphs Sllabus Objective: 9.1 The student will sketch the graph of a eponential, logistic, or logarithmic function. 9. The student will evaluate eponential or logarithmic

More information

Syllabus Objective: 2.9 The student will sketch the graph of a polynomial, radical, or rational function.

Syllabus Objective: 2.9 The student will sketch the graph of a polynomial, radical, or rational function. Precalculus Notes: Unit Polynomial Functions Syllabus Objective:.9 The student will sketch the graph o a polynomial, radical, or rational unction. Polynomial Function: a unction that can be written in

More information

Chapter 8 Notes SN AA U2C8

Chapter 8 Notes SN AA U2C8 Chapter 8 Notes SN AA U2C8 Name Period Section 8-: Eploring Eponential Models Section 8-2: Properties of Eponential Functions In Chapter 7, we used properties of eponents to determine roots and some of

More information

k y = where k is the constant of variation and

k y = where k is the constant of variation and Syllabus Objectives: 9. The student will solve a problem by applying inverse and joint variation. 9.6 The student will develop mathematical models involving rational epressions to solve realworld problems.

More information

Name Period Date. Practice FINAL EXAM Intro to Calculus (50 points) Show all work on separate sheet of paper for full credit!

Name Period Date. Practice FINAL EXAM Intro to Calculus (50 points) Show all work on separate sheet of paper for full credit! Name Period Date Practice FINAL EXAM Intro to Calculus (0 points) Show all work on separate sheet of paper for full credit! ) Evaluate the algebraic epression for the given value or values of the variable(s).

More information

f 0 ab a b: base f

f 0 ab a b: base f Precalculus Notes: Unit Eponential and Logarithmic Functions Sllabus Objective: 9. The student will sketch the graph of a eponential, logistic, or logarithmic function. 9. The student will evaluate eponential

More information

Polynomial and Rational Functions

Polynomial and Rational Functions Polnomial and Rational Functions Figure -mm film, once the standard for capturing photographic images, has been made largel obsolete b digital photograph. (credit film : modification of ork b Horia Varlan;

More information

Unit 3 NOTES Honors Common Core Math 2 1. Day 1: Properties of Exponents

Unit 3 NOTES Honors Common Core Math 2 1. Day 1: Properties of Exponents Unit NOTES Honors Common Core Math Da : Properties of Eponents Warm-Up: Before we begin toda s lesson, how much do ou remember about eponents? Use epanded form to write the rules for the eponents. OBJECTIVE

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

Lesson #33 Solving Incomplete Quadratics

Lesson #33 Solving Incomplete Quadratics Lesson # Solving Incomplete Quadratics A.A.4 Know and apply the technique of completing the square ~ 1 ~ We can also set up any quadratic to solve it in this way by completing the square, the technique

More information

REVIEW KEY VOCABULARY REVIEW EXAMPLES AND EXERCISES

REVIEW KEY VOCABULARY REVIEW EXAMPLES AND EXERCISES Etra Eample. Graph.. 6. 7. (, ) (, ) REVIEW KEY VOCABULARY quadratic function, p. 6 standard form of a quadratic function, p. 6 parabola, p. 6 verte, p. 6 ais of smmetr, p. 6 minimum, maimum value, p.

More information

Northwest High School s Algebra 2/Honors Algebra 2

Northwest High School s Algebra 2/Honors Algebra 2 Northwest High School s Algebra /Honors Algebra Summer Review Packet 0 DUE Frida, September, 0 Student Name This packet has been designed to help ou review various mathematical topics that will be necessar

More information

Precalculus Honors - AP Calculus A Information and Summer Assignment

Precalculus Honors - AP Calculus A Information and Summer Assignment Precalculus Honors - AP Calculus A Information and Summer Assignment General Information: Competenc in Algebra and Trigonometr is absolutel essential. The calculator will not alwas be available for ou

More information

MATH 0312 FINAL EXAM REVIEW ITEMS

MATH 0312 FINAL EXAM REVIEW ITEMS MATH 012 FINAL EXAM REVIEW ITEMS Name The items on this review are representative of the items that ou might see on our course final eam. No formul sheets are allowed and calculators are not allowed on

More information

Fair Game Review. Chapter 9. Find the square root(s) ± Find the side length of the square. 7. Simplify Simplify 63.

Fair Game Review. Chapter 9. Find the square root(s) ± Find the side length of the square. 7. Simplify Simplify 63. Name Date Chapter 9 Find the square root(s). Fair Game Review... 9. ±. Find the side length of the square.. s. s s Area = 9 ft s Area = 0. m 7. Simplif 0. 8. Simplif. 9. Simplif 08. 0. Simplif 88. Copright

More information

CHAPTER 3 Polynomial Functions

CHAPTER 3 Polynomial Functions CHAPTER Polnomial Functions Section. Quadratic Functions and Models............. 7 Section. Polnomial Functions of Higher Degree......... 7 Section. Polnomial and Snthetic Division............ Section.

More information

LESSON #1 - BASIC ALGEBRAIC PROPERTIES COMMON CORE ALGEBRA II

LESSON #1 - BASIC ALGEBRAIC PROPERTIES COMMON CORE ALGEBRA II 1 LESSON #1 - BASIC ALGEBRAIC PROPERTIES COMMON CORE ALGEBRA II Mathematics has developed a language all to itself in order to clarif concepts and remove ambiguit from the analsis of problems. To achieve

More information

review for math TSI 55 practice aafm m

review for math TSI 55 practice aafm m Eam TSI Name review for math TSI practice 01704041700aafm042430m www.alvarezmathhelp.com MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the

More information

LESSON #48 - INTEGER EXPONENTS COMMON CORE ALGEBRA II

LESSON #48 - INTEGER EXPONENTS COMMON CORE ALGEBRA II LESSON #8 - INTEGER EXPONENTS COMMON CORE ALGEBRA II We just finished our review of linear functions. Linear functions are those that grow b equal differences for equal intervals. In this unit we will

More information

5.3 Polynomials and Polynomial Functions

5.3 Polynomials and Polynomial Functions 70 CHAPTER 5 Eponents, Polnomials, and Polnomial Functions 5. Polnomials and Polnomial Functions S Identif Term, Constant, Polnomial, Monomial, Binomial, Trinomial, and the Degree of a Term and of a Polnomial.

More information

KEY IDEAS. Chapter 1 Function Transformations. 1.1 Horizontal and Vertical Translations Pre-Calculus 12 Student Workbook MHR 1

KEY IDEAS. Chapter 1 Function Transformations. 1.1 Horizontal and Vertical Translations Pre-Calculus 12 Student Workbook MHR 1 Chapter Function Transformations. Horizontal and Vertical Translations A translation can move the graph of a function up or down (vertical translation) and right or left (horizontal translation). A translation

More information

Review Topics for MATH 1400 Elements of Calculus Table of Contents

Review Topics for MATH 1400 Elements of Calculus Table of Contents Math 1400 - Mano Table of Contents - Review - page 1 of 2 Review Topics for MATH 1400 Elements of Calculus Table of Contents MATH 1400 Elements of Calculus is one of the Marquette Core Courses for Mathematical

More information

4.1 Practice A. Name Date. as x +. Describe the degree and leading coefficient of the function. as x and f( x)

4.1 Practice A. Name Date. as x +. Describe the degree and leading coefficient of the function. as x and f( x) Name Date. Practice A In Exercises, decide whether the function is a polnomial function. If so, write it in standard form and state its degree, tpe, and leading coefficient.. f( x) = x x + 5x 7. ( ). g(

More information

UNCORRECTED SAMPLE PAGES. 3Quadratics. Chapter 3. Objectives

UNCORRECTED SAMPLE PAGES. 3Quadratics. Chapter 3. Objectives Chapter 3 3Quadratics Objectives To recognise and sketch the graphs of quadratic polnomials. To find the ke features of the graph of a quadratic polnomial: ais intercepts, turning point and ais of smmetr.

More information

Polynomial and Rational Functions

Polynomial and Rational Functions Polnomial and Rational Functions Figure -mm film, once the standard for capturing photographic images, has been made largel obsolete b digital photograph. (credit film : modification of ork b Horia Varlan;

More information

a. plotting points in Cartesian coordinates (Grade 9 and 10), b. using a graphing calculator such as the TI-83 Graphing Calculator,

a. plotting points in Cartesian coordinates (Grade 9 and 10), b. using a graphing calculator such as the TI-83 Graphing Calculator, GRADE PRE-CALCULUS UNIT C: QUADRATIC FUNCTIONS CLASS NOTES FRAME. After linear functions, = m + b, and their graph the Quadratic Functions are the net most important equation or function. The Quadratic

More information

3.1 Graph Quadratic Functions

3.1 Graph Quadratic Functions 3. Graph Quadratic Functions in Standard Form Georgia Performance Standard(s) MMA3b, MMA3c Goal p Use intervals of increase and decrease to understand average rates of change of quadratic functions. Your

More information

Algebra Final Exam Review Packet

Algebra Final Exam Review Packet Algebra 1 00 Final Eam Review Packet UNIT 1 EXPONENTS / RADICALS Eponents Degree of a monomial: Add the degrees of all the in the monomial together. o Eample - Find the degree of 5 7 yz Degree of a polynomial:

More information