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
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1 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 and b be real numbers, and let m and n be integers. Product of Powers Property Quotient of Powers Property Power of a Power Property m n m n a a a m m a m n a 1 a or, a 0 n n n m a a a m n mn a a Power of a Product Property ab m m m a b m a a Power of a Quotient Property, b 0 m b b m Negative Eponent Property a m n 1 a b m or, a 0 a b a n Zero Eponent Property 0 a a 1, 0 Evaluating Numerical Epressions with Eponents E 1: Evaluate. Use the power of a power property: 1 1 Use the negative eponent property: E : Evaluate 5. Use the power of a product property: Use the negative eponent property: Use the power of a quotient property: Page 1 of 7 McDougal Littell
2 E : Evaluate Algebra II Notes Unit Si: Polynomials Use the zero eponent property: Use the quotient of a power property: Use the negative eponent property: 5 15 Note: We can use the quotient of a power property to keep the eponent positive Use the quotient of a power property: Simplifying Algebraic Epressions E 1: Simplify the epression 8 1 y y and write with positive eponents. Use the power of a product property: Use the power of a power property: Use the product of a power property: Use the negative eponent property: 8 1 y y y y y y y y 1 y y 7 7 E : Simplify the epression 0 t v and write with positive eponents. Use the zero eponent property: t 1 t 1 Use the negative eponent property: t t t E : Simplify the epression 1 y 6 5 4y Use the product of a power property: Use the power of a power property: Use the quotient of a power property: Simplify: 9 and write with positive eponents. 4 y 4 y y 1 y y y 1 y 1 y y 4y y Page of 7 McDougal Littell
3 Algebra II Notes Unit Si: Polynomials You Try: Simplify the epression and write with positive eponents. Identify which properties of eponents you used y y 6 y 1 4 y y 6 y 0 4 QOD: Which properties of eponents require you to check that two or more bases are the same before applying the property? Page of 7 McDougal Littell
4 Algebra II Notes Unit Si: Polynomials Syllabus Objective: 6.1 The student will graph a polynomial function with and without technology. Polynomial Function: a function of the form n f a a... a a a, a 0 n n 1 n n Note: Eponents are whole numbers and coefficients are real numbers. Leading Coefficient: a n Constant Term: a 0 Degree: n Note: Consider using the Frayer Model (Vocabulary Concept Grid) Activity See resource pages. Standard Form of a Polynomial Function: The terms are written in descending order of the eponents Names of Polynomial Functions: Degree Type Standard Form This is kind of tricky- 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. 0 Constant 1 Linear f a 0 f a1 a 0 Quadratic f a a1 a 0 Cubic f a a a1 a 0 4 Quartic 4 f a4 a a a1 a 0 Identifying Polynomial Functions E 1: Is f 5 8 a polynomial function? If yes, write it in standard form. No. In order to be a polynomial function, all eponents must be whole numbers. E : Is 5 4 f 8 a polynomial function? If yes, write it in standard form. Yes. All eponents are whole numbers and all coefficients are real numbers. Standard Form: f 8 Note: This is a quartic trinomial (degree = 4). 4 5 Evaluating Polynomial Functions Using Direct Substitution E 1: Find f if f 4 f So : f () 47 Page 4 of 7 McDougal Littell
5 Algebra II Notes Unit Si: Polynomials Evaluating Polynomial Functions Using Synthetic Substitution E 1: Find f if 4 f using synthetic substitution. Using the polynomial in standard form, write the coefficients in a row. Put the -value to the upper left Bring down the first coefficient, then multiply by the -value. multiply Add straight down the columns, and repeat The number in the bottom right is the value of f. So : f () 47 E : Find f if 5 f 7 11 using synthetic substitution. This polynomial function is in standard form, however it is missing two terms. We can rewrite the function as 5 4 f to fill in the missing terms f 17 This also means that (-,-17) is an ordered pair that would be a point on the graph. Graphing Polynomial Functions: To graph a polynomial function, make a table of values using synthetic 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 very large (approaches positive infinity ) OR as gets very small (or approaches negative infinity ). Notation: ( approaches positive infinity ) (The very far right end of a graph). ( approaches negative infinity) (The very far left end of a graph). Page 5 of 7 McDougal Littell
6 Algebra II Notes Unit Si: Polynomials Eploration Activity: Graph each function on the calculator. Determine the end behavior of f as approaches negative and positive infinity. Fill in the table and write your conclusion regarding the degree of the function and the end behavior. (Teacher Note: Answers are in red.) f Degree Sign of Leading f Coefficient f + + f + f + + f + 4 f f f f f f Page 6 of 7 McDougal Littell
7 Conclusion: The graph of a polynomial function n f a a... a a a n Algebra II Notes Unit Si: Polynomials n 1 n has the following end behavior. These patterns are very predicatable. 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 polynomial 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 polynomial function calculator. 4 f 5 1 by hand. Check your graph on the graphing Step One: Make a table of values using synthetic substitution f Step Two: Determine end behavior using the degree and sign of the leading coefficient. as, f The degree is even, and the leading coefficient is positive. So as, f. Step Three: Graph the polynomial function. Page 7 of 7 McDougal Littell
8 E : Graph the polynomial function calculator. Algebra II Notes Unit Si: Polynomials f 4 by hand. Check your graph on the graphing Step One: Make a table of values using synthetic substitution f Why didn t we use synthetic division 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 polynomial function. as, f as, f You Try: Graph the polynomial function f 1by hand. Check your graph on the graphing calculator. QOD: Which term of the polynomial function is most important when determining the end behavior of the function? Page 8 of 7 McDougal Littell
9 Algebra II Notes Unit Si: Polynomials Syllabus Objective: 6. The student will simplify polynomial epressions. Adding Polynomials E 1: Add the polynomials Subtracting Polynomials Vertical Method: Write each polynomial in standard form and line up like terms. Then add the like terms E 1: Subtract the polynomials To subtract, we will rewrite the problem as an addition problem by adding the opposite Horizontal Method: Combine each set of like terms. Write the final answer in standard form Multiplying Polynomials E 1: Find the product 4. Horizontal Method: Use the distributive property by distributing each term of the first polynomial Combine like terms and write the answer in standard form Page 9 of 7 McDougal Littell
10 Algebra II Notes Unit Si: Polynomials E : Multiply the polynomials Vertical Method: Use long multiplication E : Multiply the polynomials Multiply the polynomials two at a time. Because they are binomials, we can use FOIL to multiply the first two Use the distributive property 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 b a b a b Square of a Binomial Cube of a Binomial a b a ab b a b a ab b a b a a b ab b a b a a b ab b E 1: Simplify the epression y. Using the cube of a binomial: y y y 7 y 54 y 6y 8 Page 10 of 7 McDougal Littell
11 Application Problems Algebra II Notes Unit Si: Polynomials E 1: Find a polynomial 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 1985 through 1996, the number of flu shots given in one city can be modeled by 4 A 11.t 8.5t 194t 4190t 759 for adults and by 4 C 6.87t 106t 51t 15t 540 for children, where t is the number of years since Write a model for the total number F of flu shots given in these years. To find the total flu shots, we need to add the polynomials. Vertical Method: Solution: 4 11.t 8.5t 194t 4190t t 106t 51t 15t t t 194t 4055t 81 4 F 18.t t 194t 4055t 81 You Try: Find the product: 7 1 QOD: What is the advantage of the vertical method when adding, subtracting, or multiplying polynomials? Page 11 of 7 McDougal Littell
12 Algebra II Notes Unit Si: Polynomials Syllabus Objectives: 6. The student will solve polynomial equations by factoring and graphing. Review: Factoring Patterns Factoring a General Trinomial E 1: Factor the trinomial 5 1. ac Method: ac 4 Split the middle term: Factor by grouping: Factoring a Perfect Square Trinomial E 1: Factor the trinomial 6 9. Use a ab b a b. Difference of Two Squares E 1: Factor 6 49y. Use a b a b a b. 6 7y 6 7y 6 7y Common Monomial Factor E 1: Factor the trinomial completely. 7 Factor the GCF and the binomial square Since this is not completely factored, use a b a b a b. 4 Sum and Difference of Two Cubes a b a b a ab b a b a b a ab b Page 1 of 7 McDougal Littell
13 E 1: Factor the binomial Algebra II Notes Unit Si: Polynomials 8 1. Use a b a b a ab b E : Factor the binomial 6 y 7. Use a b a b a ab b. y y y y y y y 9 4 Factoring by Grouping E 1: Factor the polynomial Group each pair of terms and factor the GCF. Factor the common binomial factor. 9 9 Factor the remaining terms if possible. Review: Zero-Product Property If ab 0, then a 0 or b 0. Solving Polynomial Equations by Factoring E 1: Solve the equation Step One: Set the equation equal to zero. Step Two: Factor the polynomial Step Three: Set each factor equal to zero and solve. Solutions:,,0,, , 0, Page 1 of 7 McDougal Littell
14 Algebra II Notes Unit Si: Polynomials E : Find the real-number solutions of the equation Step One: Set the equation equal to zero. Step Two: Factor the polynomial Step Three: Set each factor equal to zero and solve i,5i Real-Number Solution: 5 Application Problem E 1: An optical company is going to make a glass prism that has a volume of 15 cm. The height will be h cm, and the base will be a right triangle with legs of length h cm and h the height? cm. What will be Volume of a Prism = Area of the Base Height To solve this equation for h, we must set it equal to zero h h h 1 h h h 1 5 h h h h h 6h 15 Before factoring, we can multiply both sides of the equation by to eliminate (clear) the fractions h h 6h 15 0 h 5h 1h 0 Factor by grouping. 0 h h 5 6 h 5 0 h 5 h 6 Solve by setting each factor equal to 0. h h cm h h No real solution The height of the prism will be 5 cm. Page 14 of 7 McDougal Littell
15 Algebra II Notes Unit Si: Polynomials You Try: 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 your binomial. Page 15 of 7 McDougal Littell
16 Algebra II Notes Unit Si: Polynomials Syllabus Objective: 6.6 The student will divide polynomials and relate the result to the remainder theorem and the factor theorem. Dividing Polynomials Using Long Division On Your Own: Find the quotient of 1,6 and 4 using long division. On Your Own: Find the quotient of 4 y y y 5 and y y 1. For each step of long division, we will divide the term with the highest power in the dividend by the first term of the divisor. y y 4 y y y y y y y y Remember to put a place for the missing term. y y y 4 y y y (add the opposite) y y y y y 5 y y (bring down the net term) (remainder) Eploration: Use the polynomial function. Then use synthetic substitution to find f f 5. Use long division to divide f by. What do you notice? Remainder Theorem: If a polynomial is f divided by k, then the remainder is r f k. Dividing Polynomials Using Synthetic Division (Note: This can only be used when the divisor is in the form k.) E 1: Divide the polynomial 7 6 by. Use synthetic substitution for k. The coefficients of the quotient and remainder appear in synthetic substitution. Quotient: R Note for graphing: This means that (-,-7) is an ordered pair that is on the graph of the function. Page 16 of 7 McDougal Littell
17 Algebra II Notes Unit Si: Polynomials Factor Theorem: A polynomial f has a factor k if and only 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 by the Factor Theorem. We will use synthetic division to find the other factors f f 6 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. E : One zero of f 9 1 is 7. Find the other zeros of the function. Because f 7 0, we know that 7 is a factor of f by the Factor Theorem. We will use synthetic division to find the other factors f 7 5 f 7 1 Set each factor equal to zero You Try: Use long division to find the quotient of and. You Try: Given that t 5 is a zero of the function f t 4t 9t 5t 15, find the other zeros. QOD: If f is a polynomial that has a as a factor, what do you know about the value of f a? Page 17 of 7 McDougal Littell
18 Algebra II Notes Unit Si: Polynomials Syllabus Objectives: 6.7 The student will identify all possible rational zeros of a polynomial function by using the rational root theorem. 6.4 The student will find rational zeros of a polynomial. 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... a1 a0has integer coefficients, then every rational zero of p factor of constant term a0 f has the following form: q factor of leading coefficient a n The first important step is to list the possible rational zeros. After they are listed we can test them to determine if they are rational zeros. If the value of the possible rational zeros =0, they 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 only 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 synthetic substitution from previous lessons we would find that and 4 are zeros. This also means that 5 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 ( ) Step 1: The leading coefficient is. The factors of are 1 and. Step : The constant is -5. All of the factors of -5 are 5, 1. 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 dramatically. There are many 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 18 of 7 McDougal Littell
19 E : Find all the real zeros of Algebra II Notes Unit Si: Polynomials f Step 1: Put the function in standard order. f Step : List possible rational zeros (1,,,4,6,8,9,1,18,4,6,7) Step : Try the possible zeros until you find one From previous lessons, the function can be reduced to: f 4 Then factored: 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. E 4: Find all of the real zeros of f Step 1: Notice that each term contains a common factor of. The problem can be factored to f ( 4 6) and since 0only ( 4 6) can be =0. Step : List possible rational zeros (1,,,6) (Since the leading coefficient is now 1) Step : Try the possible zeros until you find one From previous lessons, the function can be reduced to: f Then factored: f ( 1) 0 0 ( 1) 0 Zeros: 1 Page 19 of 7 McDougal Littell
20 E 5: Find all the real zeros of Algebra II Notes Unit Si: Polynomials 5 4 f 4 81 Step 1: Maybe this could be graphed first. Step : Look at the graph for reasonable choices 1 It appears they might be,, and Step : Check the chosen values using synthetic division. Start with -. Why not -½? -5? Why not 7? Is a root (zero). The factored form so far is 4 f ( )( ) Step 4: Repeat the steps above using a different reasonable choice. Try Step 5: Repeat the steps above using a different reasonable choice. Try Step 6: The function Is a root (zero). 7 is left to be factored. ( 9) has no real factors. 1 Solution: There are real zeros:,, and Is a root (zero). 1 Yes, all three work! And each time, the function (polynomial) is reduced by one degree. You Try: Find all real zeros of f QOD: If the leading coefficient of a polynomial with integer coefficients is 1, what type of numbers must any possible rational zeros be? Page 0 of 7 McDougal Littell
21 Algebra II Notes Unit Si: Polynomials Syllabus Objective: 6.5 The student will use the Fundamental Theorem of Algebra to determine the number of zeros of a polynomial function. The Fundamental Theorem of Algebra If f is a polynomial of degree n where n 0, 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 many solutions each has and classify each zero as rational, irrational, or comple (imaginary). E 1: 1 0 E : 9 0 E : 1 0 (Hint: Use the factorization for the difference of cubes, then use the quadratic formula for the quadratic factor.) Do you notice a pattern with the degree of the polynomial and the number of solutions each has? E 4: How many different solutions are there to How do you eplain this number? ? E 5: How many different solutions are there to Solution: 4 i, 4i 16 0? Note: On the graph, the imaginary roots do not cross the -ais. Note: 4 i, 4i are comple conjugate pairs. 1 i, 1 i are comple conjugate pairs. The comple roots of polynomial functions with real coefficients always occur in comple conjugate pairs. Is this also the case for irrational zeros? Page 1 of 7 McDougal Littell
22 Algebra II Notes Unit Si: Polynomials Finding the zeros of a polynomial function This activity involves finding the rational zeros as learned in the previous section, then using other tools, such as the quadratic formula or technology, to find the irrational or comple roots. E 1: Find all zeros of 4 f ( ) Using the rational root theorem and synthetic 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 4 f ( ) 5 0 Using the rational root theorem and synthetic division, it can be shown that and - are roots. Using the pattern of E 5 it can be shown that Using the quadratic formula 5 yields zeros of = i 5 f 4 ( ) 5 0 factors to ( )( )( 5) Solution: There are four zeros, and - and i 5. Two are real and two are comple conjugates Using Zeros to Write Polynomial Functions E 1: Write a polynomial function f of least degree that has real coefficients, a leading coefficient of 1, and, and -5 as zeros. Step 1: Write f( ) in factored form: f ( ) ( )( )( 5) Step : Review - Multiply the polynomials two at a time. Because they are binomials, we can use FOIL to multiply the first two. f ( ) ( 5 6)( 5) f ( ) 19 0 Solution: f ( ) 19 0 Page of 7 McDougal Littell
23 Algebra II Notes Unit Si: Polynomials E : Write a polynomial 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 Step 4: Epand, multiply polynomials, and combine like terms. f ( ) ( ) ( 1) - i f ( ) ( ) ( 1 1 f ( ) ( )( ) f 4 ( ) 6 4 Note: This graph only has two -intercepts. Why? Using Technology to Approimate Zeros Specific instructions should be given based on the calculator used. This section will provide only general direction. E 1: 4 Approimate the real zeros of f ( ) Use a graphing calculator to graph and calculate the zeros. You Try: State the number of zeros of f 15 and find what they are. QOD: What is the conjugate of a comple number, and why is it important when finding all of the zeros of a polynomial function? Page of 7 McDougal Littell
24 Algebra II Notes Unit Si: Polynomials Syllabus Objective: 6.8 The student will analyze graphs of polynomial functions to determine its characteristics. Analyzing polynomial graphs Concept Summary n n 1 Let f a a... a a a be a polynomial function. n n The following statements are equivalent: Zero: k is a zero of the function f. Factor: Solution: -k is a factor of polynomial f(). k is a solution of the polynomial function f()=0. X- Intercept: k is an -intercept of the graph of the polynomial function f. Using -Intercepts to Graph a Polynomial Function E 1: Graph the function f 1 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 When a factor k is raised to an odd power, the graph crosses through the -ais. 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 4 of 7 McDougal Littell
25 Finding Turning Points Algebra II Notes Unit Si: Polynomials Turning points of polynomial functions: Another important characteristic of graphs of polynomial functions is that they have turning points corresponding to local maimum and minimum values. The y coordinate of a turning point is a local maimum if the point is higher than all nearby points. The y coordinate of a turning point is a local minimum if the point is lower than all nearby points. The graph of every polynomial function of degree n has at most n 1 turning points. Moreover, if a polynomial has n distinct real zeros, then its graph has eactly n 1 turning points. E 1: Identify the zeros and turning points (estimate the zeros and turning points) Turning point (ma) zero zero Turning point (min) Local ma Leading coefficient positive real zeros (including the double zero) {-, 1, 1} turning points (-1,4); (1,0) 1 local ma; 1 local min Turning point (ma) Leading coefficient positive 1 real zero, imaginary zeros {5} turning points (0,-); (,-8) 1 local ma; 1 local min zero Turning point (min) You Try: Leading coefficient real zeros Leading coefficient real zeros Leading coefficient real zeros 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 5 of 7 McDougal Littell
26 Algebra II Notes Unit Si: Polynomials E : Use a graphing calculator to graph and calculate the approimate local maimum(s) and local minimum(s) of f ( ) ( )( )( 5) Local maimum Coordinates:( - 1, 6) Local ma is 6 at = - 1 Local minimum You Try: Graph the function use the calculator to find the turning points of the function. f 1 by hand. Check your graph on the graphing calculator and You Try: Graph the function f 9 1 on the calculator and find the local etrema. QOD: What is the difference between local and absolute maima and minima? Eploring Data and Statistics (Notes are not provided for this material) Modeling with Polynomial Functions Write a polynomial function whose intercepts are given. Finding and Using Finite Differences Properties of finite differences 1. If a polynomial function f() has degree n, then the n th -order differences of function values for equally spaced values are non-zero and constant.. Conversely, if the n th order differences of equally spaced data are non-zero and constant, then the data can be represented by a polynomial function of degree n. Polynomial Modeling with Technology Graphing calculators make it easy to enter data, make a scatter plot, and calculate linear, quadratic, cubic, and quartic regressions. Unit Summary: Polynomial equations provide some of the most classic problems in all of algebra. Finding zeros and etrema have many real-world applications. Real-life situations are modeled by writing equations based on data and using those equations to determine or estimate other data points (speed, volume, time, profits, patterns, etc). Page 6 of 7 McDougal Littell
27 Resources Algebra II Notes Unit Si: Polynomials Vocabulary (Concept Grid) This activity can be done as a whole-class activity or as a smallgroup activity. It can be done quickly 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 may 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 may give some, but students should see the pattern and provide input Characteristics are written in the top right section (Quadrant I). Teacher may 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 may be necessary 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 maybe an eample or two, then ask students for more eamples or non-eamples, then characteristics.. f 4 f f 8 4 y 4 16 Definition Characteristics Polynomial function Eamples Only whole number eponents No negative eponents No imaginary coefficients Can be written as f()= or y= Continous graph no missing domain End point behavior Non-Eamples y 4 y y i 1 The word or phrase can be left blank and then determined by students based on the information in the quadrants. Go through the word categorization quickly, then use the back of the paper (or other space) to epand instruction or practice if needed. Page 7 of 7 McDougal Littell
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