Polynomial and Rational Functions


 Gervase Black
 11 months ago
 Views:
Transcription
1 Chapter Section.1 Quadratic Functions Polnomial and Rational Functions Objective: In this lesson ou learned how to sketch and analze graphs of quadratic functions. Course Number Instructor Date Important Vocabular Define each term or concept. Constant function A polnomial function with degree 0. That is, f() = a. Linear function A polnomial function with degree 1. That is, f() = m + b, m 0. Quadratic function Let a, b, and c be real numbers with a 0. The function f() = a + b + c is called a quadratic function. Ais of smmetr A line about which a parabola is smmetric. Also called simpl the ais of the parabola. Verte The point where the ais intersects the parabola. I. The Graph of a Quadratic Function (Pages 9 94) Let n be a nonnegative integer and let a n, a n 1,..., a, a 1, a 0 be real numbers with a n 0. A polnomial function of with degree n is... the function f() = a n n + a n 1 n a + a 1 + a 0. How to analze graphs of quadratic functions A quadratic function is a polnomial function of second degree. The graph of a quadratic function is a special U shaped curve called a(n) parabola. If the leading coefficient of a quadratic function is positive, the graph of the function opens upward and the verte of the parabola is the minimum point on the graph. If the leading coefficient of a quadratic function is negative, the graph of the function opens downward and the verte of the parabola is the maimum point on the graph. Copright Houghton Mifflin Compan. All rights reserved. 3
2 4 Chapter Polnomial and Rational Functions II. The Standard Form of a Quadratic Function (Pages 95 96) The standard form of a quadratic function is f() = a( h) + k, a 0. How to write quadratic functions in standard form and use the results to sketch graphs of functions For a quadratic function in standard form, the ais of the associated parabola is = h and the verte is (h, k). To write a quadratic function in standard form,... process of completing the square on the variable. use the To find the intercepts of the graph of solve the equation a + b + c = 0. f ( ) = a + b + c,... Eample 1: Sketch the graph of f ( ) = + 8 and identif the verte, ais, and intercepts of the parabola. ( 1, 9); = 1; ( 4, 0) and (, 0) III. Finding Minimum and Maimum Values (Pages 97 98) For a quadratic function in the form f ( ) = a + b + c, when a > 0, f has a minimum that occurs at b/(a). When a < 0, f has a maimum that occurs at b/(a). To find the minimum or maimum value, evaluate the function at b/(a). How to find minimum and maimum values of quadratic functions in reallife applications Eample : Homework Assignment Page(s) Eercises Find the minimum value of the function f ( ) = At what value of does this minimum occur? Minimum function value is 71/1 when = 11/6 Copright Houghton Mifflin Compan. All rights reserved.
3 Section. Polnomial Functions of Higher Degree 5 Section. Polnomial Functions of Higher Degree Objective: In this lesson ou learned how to sketch and analze graphs of polnomial functions. Course Number Instructor Date Important Vocabular Define each term or concept. Continuous The graph of a polnomial function has no breaks, holes, or gaps. Etrema The minimums and maimums of a function. Relative minimum The least value of a function on an interval. Relative maimum The greatest value of a function on an interval. Repeated zero If ( a) k, k > 1, is a factor of a polnomial, then = a is a repeated zero. Multiplicit The number of times a zero is repeated. Intermediate Value Theorem Let a and b be real numbers such that a < b. If f is a polnomial function such that f(a) f(b), then, in the interval [a, b], f takes on ever value between f(a) and f(b). I. Graphs of Polnomial Functions (Pages ) Name two basic features of the graphs of polnomial functions. 1) continuous ) smooth rounded turns How to use transformations to sketch graphs of polnomial functions Will the graph of 7 g ( ) = look more like the graph of f ( ) = or the graph of f ( ) =? Eplain. The graph will look more like that of f() = 3 because the degree of both is odd. II. The Leading Coefficient Test (Pages ) State the Leading Coefficient Test. As moves without bound to the left or to the right, the graph of the polnomial function f() = a n n a 1 + a 0 eventuall rises or falls in the following manner: 1. When n is odd: 3 How to use the Leading Coefficient Test to determine the end behavior of graphs of polnomial functions a. If the leading coefficient is positive, the graph falls to the left and rises to the right. b. If the leading coefficient is negative, the graph rises to the left and falls to the right.. When n is even: a. If the leading coefficient is positive, the graph rises to the left and right. b. If the leading coefficient is negative, the graph falls to the left and right. Copright Houghton Mifflin Compan. All rights reserved.
4 6 Chapter Polnomial and Rational Functions Eample 1: Describe the left and right behavior of the graph of 6 f ( ) = Because the degree is even and the leading coefficient is negative, the graph falls to the left and right. III. Zeros of Polnomial Functions (Pages ) Let f be a polnomial function of degree n. The function f has at most n real zeros. The graph of f has at most n 1 relative etrema. How to find and use zeros of polnomial functions as sketching aids Let f be a polnomial function and let a be a real number. List four equivalent statements about the real zeros of f. 1) = a is a zero of the function f ) = a is a solution of the polnomial equation f() = 0 3) ( a) is a factor of the polnomial f() 4) (a, 0) is an intercept of the graph of f If a polnomial function f has a repeated zero = 3 with multiplicit 4, the graph of f touches the ais at = 3. If f has a repeated zero = 4 with multiplicit 3, the graph of f crosses the ais at = Eample : Sketch the graph of f ( ) = 3. IV. The Intermediate Value Theorem (Pages 111) Interpret the meaning of the Intermediate Value Theorem. If (a, f(a)) and (b, f(b)) are two points on the graph of a polnomial function f such that f(a) f(b), then for an number d between f(a) and f(b), there must be a number c between a and b such that f(c) = d. How to use the Intermediate Value Theorem to help locate zeros of polnomial functions Describe how the Intermediate Value Theorem can help in locating the real zeros of a polnomial function f. If ou can find a value = a at which f is positive and another value = b at which f is negative, ou can conclude that f has at least one real zero between a and b. Homework Assignment Page(s) Eercises Copright Houghton Mifflin Compan. All rights reserved.
5 Section.3 Real Zeros of Polnomial Functions 7 Section.3 Real Zeros of Polnomial Functions Objective: In this lesson ou learned how to use long division and snthetic division to divide polnomials b other polnomials and how to find the rational and real zeros of polnomial functions. Course Number Instructor Date Important Vocabular Define each term or concept. Long division of polnomials A procedure for dividing two polnomials, which is similar to long division in arithmetic. Division Algorithm If f() and d() are polnomials such that d() 0, and the degree of d() is less than or equal to the degree of f(), there eist unique polnomials q() and r() such that f() = d()q() + r() where r() = 0 or the degree of r() is less than the degree of d(). Snthetic division A shortcut for long division of polnomials when dividing b divisors of the form k. Remainder Theorem If a polnomial f() is divided b k, then the remainder is r = f(k). Factor Theorem A polnomial f() has a factor ( k) if and onl if f(k) = 0. Upper bound A real number b is an upper bound for the real zeros of f if no real zeros of f are greater than b. Lower bound A real number b is a lower bound for the real zeros of f if no real zeros of f are less than b. I. Long Division of Polnomials (Pages ) When dividing a polnomial f() b another polnomial d(), if the remainder r() = 0, d() divides evenl into f(). How to use long division to divide polnomials b other polnomials The rational epression f()/d() is improper if... the degree of f() is greater than or equal to the degree of d(). The rational epression r()/d() is proper if... of r() is less than the degree of d(). the degree Before appling the Division Algorithm, ou should... write the dividend and divisor in descending powers of the variable and insert placeholders with zero coefficients for missing powers of the variable. Eample 1: Divide b (13 + 4)/( + + 1) Copright Houghton Mifflin Compan. All rights reserved.
6 8 Chapter Polnomial and Rational Functions II. Snthetic Division (Page 119) Can snthetic division be used to divide a polnomial b 5? Eplain. No, the divisor must be in the form k. How to use snthetic division to divide polnomials b binomials of the form ( k) Can snthetic division be used to divide a polnomial b + 4? Eplain. Yes, rewrite + 4 as ( 4). Eample : Fill in the following snthetic division arra to 4 divide b 5. Then carr out the snthetic division and indicate which entr represents the remainder remainder III. The Remainder and Factor Theorems (Pages 10 11) To use the Remainder Theorem to evaluate a polnomial function f() at = k,... use snthetic division to divide f() b k. The remainder will be f(k). How to use the Remainder and Factor Theorems Eample 3: Use the Remainder Theorem to evaluate the 4 function f ( ) = at = To use the Factor Theorem to show that ( k) is a factor of a polnomial function f(),... use snthetic division on f() with the factor ( k). If the remainder is 0, then ( k) is a factor. Or, alternativel, evaluate f() at = k. If the result is 0, then ( k) is a factor. Copright Houghton Mifflin Compan. All rights reserved.
7 Section.3 Real Zeros of Polnomial Functions 9 List three facts about the remainder r, obtained in the snthetic division of f() b k: 1) The remainder r gives the value of f at = k. That is, r = f(k). ) If r = 0, ( k) is a factor of f(). 3) If r = 0, (k, 0) is an intercept of the graph of f. IV. The Rational Zero Test (Pages 1 14) Describe the purpose of the Rational Zero Test. The Rational Zero Test relates the possible rational zeros of a polnomial with integer coefficients to the leading coefficient and to the constant term of the polnomial. How to use the Rational Zero Test to determine possible rational zeros of polnomial functions State the Rational Zero Test. If the polnomial f() = a n n + a n 1 n a + a 1 + a 0 has integer coefficients, ever rational zero of f has the form: rational zero = p/q, where p and q have no common factors other than 1, p is a factor of the constant term a 0, and q is a factor of the leading coefficient a n. To use the Rational Zero Test,... first list all rational numbers whose numerators are factors of the constant term and whose denominators are factors of the leading coefficient. Then use trial and error to determine which of these possible rational zeros, if an, are actual zeros of the polnomial. Eample 4: List the possible rational zeros of the polnomial 5 function f ( ) = ± 1, ± 5, ± 1/3, ± 5/3 Some strategies that can be used to shorten the search for actual zeros among a list of possible rational zeros include... using a programmable calculator to speed up the calculations, using a graphing utilit to estimate the locations of zeros, using the Intermediate Value Theorem (along with a table generated b a graphing utilit) to give approimations of zeros, or using the Factor Theorem and snthetic division to test possible rational zeros, etc. 4 3 Copright Houghton Mifflin Compan. All rights reserved.
8 30 Chapter Polnomial and Rational Functions V. Other Tests for Zeros of Polnomials (Pages 14 16) State the Upper and Lower Bound Rules. Let f() be a polnomial with real coefficients and a positive leading coefficient. Suppose f() is divided b c, using snthetic division. 1. If c > 0 and each number in the last row is either positive or zero, c is an upper bound for the real zeros of f.. If c < 0 and the numbers in the last row are alternatel positive and negative (zero entries count as positive or negative), c is a lower bound for the real zeros of f. How to use Descartes s Rule of Signs and the Upper and Lower Bound Rules to find zeros of polnomials Eplain how the Upper and Lower Bound Rules can be useful in the search for the real zeros of a polnomial function. Eplanations will var. For instance, suppose ou are checking a list of possible rational zeros. When checking the possible rational zero with snthetic division, each number in the last row is positive or zero. Then ou need not check an of the other possible rational zeros that are greater than and can concentrate on checking onl values less than. Additional notes Homework Assignment Page(s) Eercises Copright Houghton Mifflin Compan. All rights reserved.
9 Section.4 Comple Numbers 31 Section.4 Comple Numbers Objective: In this lesson ou learned how to perform operations with comple numbers and plot comple numbers in the comple plane. Course Number Instructor Date Important Vocabular Define each term or concept. Comple number If a and b are real numbers, the number a + bi, where the number a is called the real part and the number bi is called the imaginar part, is a comple number written in standard form. Comple conjugates A pair of comple numbers of the form a + bi and a bi. Imaginar ais The vertical ais in the comple plane. Real ais The horizontal ais in the comple plane. Bounded A description for a sequence in which the absolute value of each number in the sequence is less than some fied number N. Unbounded A description for a sequence in which the absolute values of the terms of the sequence become infinitel large. I. The Imaginar Unit i (Page 131) Mathematicians created an epanded sstem of numbers using the imaginar unit i, defined as i = 1, because... there is no real number that can be squared to produce 1. How to use the imaginar unit i to write comple numbers B definition, i = 1. For the comple number a + bi, if b = 0, the number a + bi = a is a(n) real number. If b 0, the number a + bi is a(n) imaginar number. If a = 0, the number a + bi = b, where b 0, is called a(n) pure imaginar number. The set of comple numbers consists of the set of real numbers and the set of imaginar numbers. Two comple numbers a + bi and c + di, written in standard form, are equal to each other if... and onl if a = c and b = d. Copright Houghton Mifflin Compan. All rights reserved.
10 3 Chapter Polnomial and Rational Functions II. Operations with Comple Numbers (Pages ) To add two comple numbers,... add the real parts and the imaginar parts of the numbers separatel. How to add, subtract, and multipl comple numbers To subtract two comple numbers,... subtract the real parts and the imaginar parts of the numbers separatel. The additive identit in the comple number sstem is 0. The additive inverse of the comple number a + bi is (a + bi) = a bi. Eample 1: Perform the operations: (5 6i) (3 i) + 4i To multipl two comple numbers a + bi and c + di,... the multiplication rule (ac bd) + (ad + bc)i or use the use Distributive Propert to multipl the two comple numbers, similar to using the FOIL method for multipling two binomials. Eample : Multipl: (5 6i)(3 i) 3 8i III. Comple Conjugates (Page 134) The product of a pair of comple conjugates is a(n) real number. To find the quotient of the comple numbers a + bi and c + di, where c and d are not both zero,... multipl the numerator and denominator b the comple conjugate of the denominator. How to use comple conjugates to write the quotient of two comple numbers in standard form Eample 3: Divide (1 + i) b ( i). Write the result in standard form. 1/5 + 3/5i Copright Houghton Mifflin Compan. All rights reserved.
11 Section.4 Comple Numbers 33 IV. Fractals and the Mandelbrot Set (Pages ) The comple plane is... a coordinate sstem in which ever point corresponds to a comple number a + bi. How to plot comple numbers in the comple plane On the comple plane shown below, (a) label the real ais, (b) label the imaginar ais, and (c) plot and label the comple numbers 3i and 4 + i. Imaginar ais i Real ais 3i 35 Let c represent a comple number. Describe how to tell whether or not c belongs to the Mandelbrot Set. Use the value of c to find the sequence c, c + c, (c + c) + c, [(c + c) + c] + c,. If the sequence is bounded, the comple number c is in the Mandelbrot Set. If the sequence is unbounded, the comple number c is not in the Mandelbrot Set. Describe how the Mandelbrot Set could be graphed. Descriptions ma var. For instance, one wa to graph the Mandelbrot Set would be to plot points in the comple plane with one of two colors: one color for points that are in the set and another color for points that are not in the set. Copright Houghton Mifflin Compan. All rights reserved.
12 34 Chapter Polnomial and Rational Functions Additional notes Homework Assignment Page(s) Eercises Copright Houghton Mifflin Compan. All rights reserved.
13 Section.5 The Fundamental Theorem of Algebra 35 Section.5 The Fundamental Theorem of Algebra Objective: In this lesson ou learned how to determine the numbers of zeros of polnomial functions and find them. Course Number Instructor Date Important Vocabular Define each term or concept. Fundamental Theorem of Algebra If f() is a polnomial of degree n, where n > 0, then f has at least one zero in the comple number sstem. Linear Factorization Theorem If f() is a polnomial of degree n, where n > 0, f has precisel n linear factors f() = a n ( c 1 )( c )... ( c n ) where c 1, c,..., c n are comple numbers. Conjugates A pair of comple numbers of the form a + bi and a bi. I. The Fundamental Theorem of Algebra (Pages ) In the comple number sstem, ever nthdegree polnomial function has precisel n zeros. Eample 1: How man zeros does the polnomial function f ( ) = have? How to use the Fundamental Theorem of Algebra to determine the number of zeros of a polnomial function and find all zeros of polnomial functions, including comple zeros An nthdegree polnomial can be factored into precisel n linear factors. Eample : List all of the zeros of the polnomial function 3 f ( ) = , 6i, 6i II. Conjugate Pairs (Page 141) Let f() be a polnomial function that has real coefficients. If a + bi, where b 0, is a zero of the function, then we know that a bi is also a zero of the function. How to find conjugate pairs of comple zeros Copright Houghton Mifflin Compan. All rights reserved.
14 36 Chapter Polnomial and Rational Functions III. Factoring a Polnomial (Pages ) To write a polnomial of degree n > 0 with real coefficients as a product without comple factors, write the polnomial as... the product of linear and/or quadratic factors with real coefficients, where the quadratic factors have no real zeros. How to find zeros of polnomials b factoring A quadratic factor with no real zeros is said to be prime or irreducible over the reals. Eample 3: Write the polnomial f ( ) = (a) as the product of linear factors and quadratic factors that are irreducible over the reals, and (b) in completel factored form. (a) f() = ( + )( )( + 9) (b) f() = ( + )( )( + 3i)( 3i) 4 Eplain wh a graph cannot be used to locate comple zeros. Real zeros are the onl zeros that appear as intercepts on a graph. A polnomial function s comple zeros must be found algebraicall. Additional notes Homework Assignment Page(s) Eercises Copright Houghton Mifflin Compan. All rights reserved.
15 Section.6 Rational Functions and Asmptotes 37 Section.6 Rational Functions and Asmptotes Objective: In this lesson ou learned how to determine the domains and find asmptotes of rational functions. Course Number Instructor Date Important Vocabular Define each term or concept. Rational function A function that can be written in the form: f() = N()/D(), where N() and D() are polnomials and D() is not the zero polnomial. Vertical asmptote The line = a is a vertical asmptote of the graph of f if f() or f() as a, either from the right or from the left. Horizontal asmptote The line = b is a horizontal asmptote of the graph of f if f() b as or. I. Introduction to Rational Functions (Pages ) The domain of a rational function of includes all real numbers ecept... values that make the denominator zero. How to find the domains of rational functions To find the domain of a rational function of,... set the denominator of the rational function equal to zero and solve for. These values of must be ecluded from the domain of the function. 1 Eample 1: Find the domain of the function f ( ) =. 9 The domain of f is all real numbers ecept = 3 and = 3. II. Horizontal and Vertical Asmptotes (Pages ) The notation f() 5 as means... that f() approaches 5 as increases without bound. How to find horizontal and vertical asmptotes of graphs of rational functions Let f be the rational function given b N( ) an f ( ) = = D( ) b m n m + a + b n 1 m 1 n 1 m a + a b + b where N() and D() have no common factors. 1) The graph of f has vertical asmptotes at the zeros of D(). 0 Copright Houghton Mifflin Compan. All rights reserved.
16 38 Chapter Polnomial and Rational Functions ) The graph of f has at most one horizontal asmptote determined b comparing the degrees of N() and D(). a) If n < m, the line = 0 (the ais) is a horizontal asmptote. b) If n = m, the line = a n /b m is a horizontal asmptote. c) If n > m, the graph of f has no horizontal asmptote. Eample : Find the asmptotes of the function 1 f ( ) =. 6 Vertical: =, = 3; Horizontal: = 0 III. Applications of Rational Functions (Pages ) Give an eample of asmptotic behavior that occurs in real life. Answers will var. How to use rational functions to model and solve reallife problems Homework Assignment Page(s) Eercises Copright Houghton Mifflin Compan. All rights reserved.
17 Section.7 Graphs of Rational Functions 39 Section.7 Graphs of Rational Functions Objective: In this lesson ou learned how to sketch graphs of rational functions. Course Number Instructor Date Important Vocabular Define each term or concept. Slant (or oblique) asmptote If the degree of the numerator of a rational function is eactl one more than the degree of the denominator, then the line determined b the quotient of the denominator into the numerator is a slant asmptote of the graph of the rational function. I. The Graph of a Rational Function (Pages ) To sketch the graph of the rational function f() = N()/D(), where N() and D() are polnomials with no common factors,... 1) Simplif f, if possible. An restrictions on the domain of f not in the simplified function should be listed. ) Find and plot the intercept (if an) b evaluating f(0). 3) Find the zeros of the numerator (if an) b setting the numerator equal to zero. Then plot the corresponding  intercepts. 4) Find the zeros of the denominator (if an) b setting the denominator equal to zero. Then sketch the corresponding vertical asmptotes using dashed vertical lines and plot the corresponding holes using open circles. 5) Find and sketch an other asmptotes of the graph using dashed lines. 6) Plot at least one point between and one point beond each intercept and vertical asmptote. 7) Use smooth curves to complete the graph between and beond the vertical asmptotes, ecluding an points where f is not defined. How to analze and sketch graphs of rational functions Eample 1: Sketch the graph of 3 f ( ) =. + 4 Copright Houghton Mifflin Compan. All rights reserved.
18 40 Chapter Polnomial and Rational Functions II. Slant Asmptotes (Page 159) To find the equation of a slant asmptote,... use long division to divide the denominator of the rational function into the numerator. The equation of the slant asmptote is the quotient, ecluding the remainder. How to sketch graphs of rational functions that have slant asmptotes Eample : Decide whether each of the following rational functions has a slant asmptote. If so, find the equation of the slant asmptote (a) f ( ) = (b) f ( ) = (a) Yes, = 3 (b) No III. Applications of Graphs of Rational Functions (Page 160) Describe a reallife situation in which a graph of a rational function would be helpful when solving a problem. How to use rational functions to model and solve reallife problems Answers will var. Homework Assignment Page(s) Eercises Copright Houghton Mifflin Compan. All rights reserved.
19 Section.8 Quadratic Models 41 Section.8 Quadratic Models Objective: In this lesson ou learned how to classif scatter plots and use a graphing utilit to find quadratic models for data. Course Number Instructor Date I. Classifing Scatter Plots (Page 165) Describe how to decide whether a set of data can be modeled b a linear model. How to classif scatter plots Make a scatter plot of the ordered pairs, either b hand or b entering the data into a graphing utilit and displaing a scatter plot. Eamine the shape of the scatter plot. If it appears that the data follows a linear pattern, it can be modeled b a linear function. Describe how to decide whether a set of data can be modeled b a quadratic model. Make a scatter plot of the ordered pairs, either b hand or b entering the data into a graphing utilit and displaing a scatter plot. Eamine the shape of the scatter plot. If it appears that the data follows a parabolic pattern, it can be modeled b a quadratic function. II. Fitting a Quadratic Model to Data (Pages ) Once it has been determined that a quadratic model is appropriate for a set of data, a quadratic model can be fit to data b... entering the data into a graphing utilit and using the regression feature. How to use scatter plots and a graphing utilit to find quadratic models for data Eample 1: Find a model that best fits the data given in the table = Copright Houghton Mifflin Compan. All rights reserved.
20 4 Chapter Polnomial and Rational Functions III. Choosing a Model (Pages ) If it isn t eas to tell from a scatter plot which tpe of model a set of data would best be modeled b, ou should... first find several models for the data and then choose the model that best fits the data b comparing the values of each model with the actual values. How to choose a model that best fits a set of data Homework Assignment Page(s) Eercises Copright Houghton Mifflin Compan. All rights reserved.
Zeros of Polynomial Functions. The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra. zero in the complex number system.
_.qd /7/ 9:6 AM Page 69 Section. Zeros of Polnomial Functions 69. Zeros of Polnomial Functions What ou should learn Use the Fundamental Theorem of Algebra to determine the number of zeros of polnomial
More informationAnalyzing the Graph of a Function
SECTION A Summar of Curve Sketching 09 0 00 Section 0 0 00 0 Different viewing windows for the graph of f 5 7 0 Figure 5 A Summar of Curve Sketching Analze and sketch the graph of a function Analzing the
More informationA Summary of Curve Sketching. Analyzing the Graph of a Function
0_00.qd //0 :5 PM Page 09 SECTION. A Summar of Curve Sketching 09 0 00 Section. 0 0 00 0 Different viewing windows for the graph of f 5 7 0 Figure. 5 A Summar of Curve Sketching Analze and sketch the graph
More informationPolynomial 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 informationIdentify a pattern and find the next three numbers in the pattern. 5. 5(2s 2 1) 2 3(s 1 2); s 5 4
Chapter 1 Test Do ou know HOW? Identif a pattern and find the net three numbers in the pattern. 1. 5, 1, 3, 7, c. 6, 3, 16, 8, c Each term is more than the previous Each term is half of the previous term;
More informationSAMPLE. Polynomial functions
Objectives C H A P T E R 4 Polnomial functions To be able to use the technique of equating coefficients. To introduce the functions of the form f () = a( + h) n + k and to sketch graphs of this form through
More information5.3 Graphing Cubic Functions
Name Class Date 5.3 Graphing Cubic Functions Essential Question: How are the graphs of f () = a (  h) 3 + k and f () = ( 1_ related to the graph of f () = 3? b (  h) 3 ) + k Resource Locker Eplore 1
More informationPolynomial Degree and Finite Differences
CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial
More informationMath 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 informationHigher. Polynomials and Quadratics 64
hsn.uk.net Higher Mathematics UNIT OUTCOME 1 Polnomials and Quadratics Contents Polnomials and Quadratics 64 1 Quadratics 64 The Discriminant 66 3 Completing the Square 67 4 Sketching Parabolas 70 5 Determining
More informationThe Graph of a Linear Equation
4.1 The Graph of a Linear Equation 4.1 OBJECTIVES 1. Find three ordered pairs for an equation in two variables 2. Graph a line from three points 3. Graph a line b the intercept method 4. Graph a line that
More informationPOLYNOMIAL 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 informationSolving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form
SECTION 11.3 Solving Quadratic Equations b Graphing 11.3 OBJECTIVES 1. Find an ais of smmetr 2. Find a verte 3. Graph a parabola 4. Solve quadratic equations b graphing 5. Solve an application involving
More informationChapter 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 information1. a. standard form of a parabola with. 2 b 1 2 horizontal axis of symmetry 2. x 2 y 2 r 2 o. standard form of an ellipse centered
Conic Sections. Distance Formula and Circles. More on the Parabola. The Ellipse and Hperbola. Nonlinear Sstems of Equations in Two Variables. Nonlinear Inequalities and Sstems of Inequalities In Chapter,
More informationMethods to Solve Quadratic Equations
Methods to Solve Quadratic Equations We have been learning how to factor epressions. Now we will apply factoring to another skill you must learn solving quadratic equations. a b c 0 is a seconddegree
More information5.2 Inverse Functions
78 Further Topics in Functions. Inverse Functions Thinking of a function as a process like we did in Section., in this section we seek another function which might reverse that process. As in real life,
More informationGRAPHS OF RATIONAL FUNCTIONS
0 (0) Chapter 0 Polnomial and Rational Functions. f() ( 0) ( 0). f() ( 0) ( 0). f() ( 0) ( 0). f() ( 0) ( 0) 0. GRAPHS OF RATIONAL FUNCTIONS In this section Domain Horizontal and Vertical Asmptotes Oblique
More informationWhen I was 3.1 POLYNOMIAL FUNCTIONS
146 Chapter 3 Polnomial and Rational Functions Section 3.1 begins with basic definitions and graphical concepts and gives an overview of ke properties of polnomial functions. In Sections 3.2 and 3.3 we
More informationPolynomial and Synthetic Division. Long Division of Polynomials. Example 1. 6x 2 7x 2 x 2) 19x 2 16x 4 6x3 12x 2 7x 2 16x 7x 2 14x. 2x 4.
_.qd /7/5 9: AM Page 5 Section.. Polynomial and Synthetic Division 5 Polynomial and Synthetic Division What you should learn Use long division to divide polynomials by other polynomials. Use synthetic
More informationInequalities and Absolute Values. Assignment Guide: EOO = every other odd, 1, 5, 9, 13, EOP = every other pair, 1, 2, 5, 6, 9, 10,
Chapter 4 Inequalities and Absolute Values Assignment Guide: E = ever other odd,, 5, 9, 3, EP = ever other pair,, 2, 5, 6, 9, 0, Lesson 4. Page 7577 Es. 420. 2328, 2939 odd, 4043, 4952, 5973 odd
More informationINVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1
Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.
More informationGraphing Quadratic Equations
.4 Graphing Quadratic Equations.4 OBJECTIVE. Graph a quadratic equation b plotting points In Section 6.3 ou learned to graph firstdegree equations. Similar methods will allow ou to graph quadratic equations
More information135 Final Review. Determine whether the graph is symmetric with respect to the xaxis, the yaxis, and/or the origin.
13 Final Review Find the distance d(p1, P2) between the points P1 and P2. 1) P1 = (, 6); P2 = (7, 2) 2 12 2 12 3 Determine whether the graph is smmetric with respect to the ais, the ais, and/or the
More informationSection 5.0A Factoring Part 1
Section 5.0A Factoring Part 1 I. Work Together A. Multiply the following binomials into trinomials. (Write the final result in descending order, i.e., a + b + c ). ( 7)( + 5) ( + 7)( + ) ( + 7)( + 5) (
More informationDownloaded from www.heinemann.co.uk/ib. equations. 2.4 The reciprocal function x 1 x
Functions and equations Assessment statements. Concept of function f : f (); domain, range, image (value). Composite functions (f g); identit function. Inverse function f.. The graph of a function; its
More information1.6. Piecewise Functions. LEARN ABOUT the Math. Representing the problem using a graphical model
. Piecewise Functions YOU WILL NEED graph paper graphing calculator GOAL Understand, interpret, and graph situations that are described b piecewise functions. LEARN ABOUT the Math A cit parking lot uses
More informationQ (x 1, y 1 ) m = y 1 y 0
. Linear Functions We now begin the stud of families of functions. Our first famil, linear functions, are old friends as we shall soon see. Recall from Geometr that two distinct points in the plane determine
More informationP1. Plot the following points on the real. P2. Determine which of the following are solutions
Section 1.5 Rectangular Coordinates and Graphs of Equations 9 PART II: LINEAR EQUATIONS AND INEQUALITIES IN TWO VARIABLES 1.5 Rectangular Coordinates and Graphs of Equations OBJECTIVES 1 Plot Points in
More informationMore Equations and Inequalities
Section. Sets of Numbers and Interval Notation 9 More Equations and Inequalities 9 9. Compound Inequalities 9. Polnomial and Rational Inequalities 9. Absolute Value Equations 9. Absolute Value Inequalities
More informationComplex Numbers. (x 1) (4x 8) n 2 4 x 1 2 23 No realnumber solutions. From the definition, it follows that i 2 1.
7_Ch09_online 7// 0:7 AM Page 99. Comple Numbers 9 SECTION 9. OBJECTIVES Epress square roots of negative numbers in terms of i. Write comple numbers in a bi form. Add and subtract comple numbers. Multipl
More informationTranslating Points. Subtract 2 from the ycoordinates
CONDENSED L E S S O N 9. Translating Points In this lesson ou will translate figures on the coordinate plane define a translation b describing how it affects a general point (, ) A mathematical rule that
More informationLesson 2.3 Exercises, pages 114 121
Lesson.3 Eercises, pages 11 11 A. For the graph of each rational function below: i) Write the equations of an asmptotes. ii) State the domain. a) b) 0 6 8 8 0 8 16 i) There is no vertical asmptote. The
More information1.2 GRAPHS OF EQUATIONS
000_00.qd /5/05 : AM Page SECTION. Graphs of Equations. GRAPHS OF EQUATIONS Sketch graphs of equations b hand. Find the  and intercepts of graphs of equations. Write the standard forms of equations of
More informationQUADRATIC FUNCTIONS AND COMPLEX NUMBERS
CHAPTER 86 5 CHAPTER TABLE F CNTENTS 5 Real Roots of a Quadratic Equation 52 The Quadratic Formula 53 The Discriminant 54 The Comple Numbers 55 perations with Comple Numbers 56 Comple Roots of a
More informationExponential and Logarithmic Functions
Chapter 6 Eponential and Logarithmic Functions Section summaries Section 6.1 Composite Functions Some functions are constructed in several steps, where each of the individual steps is a function. For eample,
More informationPower 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 n1 n1 + + a 1 + a 0 Eample: = 3 3 + 5  The domain o a polynomial unction is the set o all real numbers. The intercepts
More informationLesson 9.1 Solving Quadratic Equations
Lesson 9.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with a. One intercept and all nonnegative yvalues. b. The verte in the third quadrant and no intercepts. c. The verte
More informationSECTION 2.2. Distance and Midpoint Formulas; Circles
SECTION. Objectives. Find the distance between two points.. Find the midpoint of a line segment.. Write the standard form of a circle s equation.. Give the center and radius of a circle whose equation
More information6. The given function is only drawn for x > 0. Complete the function for x < 0 with the following conditions:
Precalculus Worksheet 1. Da 1 1. The relation described b the set of points {(, 5 ),( 0, 5 ),(,8 ),(, 9) } is NOT a function. Eplain wh. For questions  4, use the graph at the right.. Eplain wh the graph
More information4.9 Graph and Solve Quadratic
4.9 Graph and Solve Quadratic Inequalities Goal p Graph and solve quadratic inequalities. Your Notes VOCABULARY Quadratic inequalit in two variables Quadratic inequalit in one variable GRAPHING A QUADRATIC
More informationPolynomials. Jackie Nicholas Jacquie Hargreaves Janet Hunter
Mathematics Learning Centre Polnomials Jackie Nicholas Jacquie Hargreaves Janet Hunter c 26 Universit of Sdne Mathematics Learning Centre, Universit of Sdne 1 1 Polnomials Man of the functions we will
More informationC3: Functions. Learning objectives
CHAPTER C3: Functions Learning objectives After studing this chapter ou should: be familiar with the terms oneone and manone mappings understand the terms domain and range for a mapping understand the
More information2.5 Library of Functions; Piecewisedefined Functions
SECTION.5 Librar of Functions; Piecewisedefined Functions 07.5 Librar of Functions; Piecewisedefined Functions PREPARING FOR THIS SECTION Before getting started, review the following: Intercepts (Section.,
More informationCore Maths C2. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...
More informationChapter 3A  Rectangular Coordinate System
 Chapter A Chapter A  Rectangular Coordinate Sstem Introduction: Rectangular Coordinate Sstem Although the use of rectangular coordinates in such geometric applications as surveing and planning has been
More informationFunctions and Graphs CHAPTER INTRODUCTION. The function concept is one of the most important ideas in mathematics. The study
Functions and Graphs CHAPTER 2 INTRODUCTION The function concept is one of the most important ideas in mathematics. The stud 21 Functions 22 Elementar Functions: Graphs and Transformations 23 Quadratic
More informationZeros of Polynomial Functions
Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n1 x n1 + + 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 informationr(x) = p(x) q(x), 4. r(x) = 2x2 1
Chapter 4 Rational Functions 4. Introduction to Rational Functions If we add, subtract or multipl polnomial functions according to the function arithmetic rules defined in Section.5, we will produce another
More informationQuadratic Functions and Parabolas
MATH 11 Quadratic Functions and Parabolas A quadratic function has the form Dr. Neal, Fall 2008 f () = a 2 + b + c where a 0. The graph of the function is a parabola that opens upward if a > 0, and opens
More informationC1: Coordinate geometry of straight lines
B_Chap0_0805.qd 5/6/04 0:4 am Page 8 CHAPTER C: Coordinate geometr of straight lines Learning objectives After studing this chapter, ou should be able to: use the language of coordinate geometr find the
More informationSECTION P.5 Factoring Polynomials
BLITMCPB.QXP.0599_4874 /0/0 0:4 AM Page 48 48 Chapter P Prerequisites: Fundamental Concepts of Algebra Technology Eercises Critical Thinking Eercises 98. The common cold is caused by a rhinovirus. The
More informationReteaching Masters. To jump to a location in this book. 1. Click a bookmark on the left. To print a part of the book. 1. Click the Print button.
Reteaching Masters To jump to a location in this book. Click a bookmark on the left. To print a part of the book. Click the Print button.. When the Print window opens, tpe in a range of pages to print.
More informationUse order of operations to simplify. Show all steps in the space provided below each problem. INTEGER OPERATIONS
ORDER OF OPERATIONS In the following order: 1) Work inside the grouping smbols such as parenthesis and brackets. ) Evaluate the powers. 3) Do the multiplication and/or division in order from left to right.
More information10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED
CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations
More informationQuadratic Equations and Functions
Quadratic Equations and Functions. Square Root Propert and Completing the Square. Quadratic Formula. Equations in Quadratic Form. Graphs of Quadratic Functions. Verte of a Parabola and Applications In
More information25 Rational Functions
5 Rational Functions Find the domain of each function and the equations of the vertical or horizontal asymptotes, if any 1 f () = The function is undefined at the real zeros of the denominator b() = 4
More information5.1 The Remainder and Factor Theorems; Synthetic Division
5.1 The Remainder and Factor Theorems; Synthetic Division In this section you will learn to: understand the definition of a zero of a polynomial function use long and synthetic division to divide polynomials
More informationSection 4.4 Rational Functions and Their Graphs
Section 4.4 Rational Functions and Their Graphs p( ) A rational function can be epressed as where p() and q() are q( ) 3 polynomial functions and q() is not equal to 0. For eample, is a 16 rational function.
More informationZeros 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 informationLIMITS AND CONTINUITY
LIMITS AND CONTINUITY 1 The concept of it Eample 11 Let f() = 2 4 Eamine the behavior of f() as approaches 2 2 Solution Let us compute some values of f() for close to 2, as in the tables below We see from
More informationPolynomial and Rational Functions
Polnomial and Rational Functions 3 A LOOK BACK In Chapter, we began our discussion of functions. We defined domain and range and independent and dependent variables; we found the value of a function and
More informationACT Math Vocabulary. Altitude The height of a triangle that makes a 90degree angle with the base of the triangle. Altitude
ACT Math Vocabular Acute When referring to an angle acute means less than 90 degrees. When referring to a triangle, acute means that all angles are less than 90 degrees. For eample: Altitude The height
More informationImagine a cube with any side length. Imagine increasing the height by 2 cm, the. Imagine a cube. x x
OBJECTIVES Eplore functions defined b rddegree polnomials (cubic functions) Use graphs of polnomial equations to find the roots and write the equations in factored form Relate the graphs of polnomial equations
More informationChapter 6 Quadratic Functions
Chapter 6 Quadratic Functions Determine the characteristics of quadratic functions Sketch Quadratics Solve problems modelled b Quadratics 6.1Quadratic Functions A quadratic function is of the form where
More informationPartial Fractions. and Logistic Growth. Section 6.2. Partial Fractions
SECTION 6. Partial Fractions and Logistic Growth 9 Section 6. Partial Fractions and Logistic Growth Use partial fractions to find indefinite integrals. Use logistic growth functions to model reallife
More informationReview of Intermediate Algebra Content
Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6
More informationThe degree of a polynomial function is equal to the highest exponent found on the independent variables.
DETAILED SOLUTIONS AND CONCEPTS  POLYNOMIAL FUNCTIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE NOTE
More informationChapter 3. Curve Sketching. By the end of this chapter, you will
Chapter 3 Curve Sketching How much metal would be required to make a ml soup can? What is the least amount of cardboard needed to build a bo that holds 3 cm 3 of cereal? The answers to questions like
More informationy intercept Gradient Facts Lines that have the same gradient are PARALLEL
CORE Summar Notes Linear Graphs and Equations = m + c gradient = increase in increase in intercept Gradient Facts Lines that have the same gradient are PARALLEL If lines are PERPENDICULAR then m m = or
More information7.3 Graphing Rational Functions
Section 7.3 Graphing Rational Functions 639 7.3 Graphing Rational Functions We ve seen that the denominator of a rational function is never allowed to equal zero; division b zero is not defined. So, with
More informationIn this lesson you will learn to find zeros of polynomial functions that are not factorable.
2.6. Rational zeros of polynomial functions. In this lesson you will learn to find zeros of polynomial functions that are not factorable. REVIEW OF PREREQUISITE CONCEPTS: A polynomial of n th degree has
More informationD.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review
D0 APPENDIX D Precalculus Review APPENDIX D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane Just as ou can represent real numbers b
More informationThe Distance Formula and the Circle
10.2 The Distance Formula and the Circle 10.2 OBJECTIVES 1. Given a center and radius, find the equation of a circle 2. Given an equation for a circle, find the center and radius 3. Given an equation,
More informationObjectives. By the time the student is finished with this section of the workbook, he/she should be able
QUADRATIC FUNCTIONS Completing the Square..95 The Quadratic Formula....99 The Discriminant... 0 Equations in Quadratic Form.. 04 The Standard Form of a Parabola...06 Working with the Standard Form of a
More informationax 2 by 2 cxy dx ey f 0 The Distance Formula The distance d between two points (x 1, y 1 ) and (x 2, y 2 ) is given by d (x 2 x 1 )
SECTION 1. The Circle 1. OBJECTIVES The second conic section we look at is the circle. The circle can be described b using the standard form for a conic section, 1. Identif the graph of an equation as
More informationFive 5. Rational Expressions and Equations C H A P T E R
Five C H A P T E R Rational Epressions and Equations. Rational Epressions and Functions. Multiplication and Division of Rational Epressions. Addition and Subtraction of Rational Epressions.4 Comple Fractions.
More informationTHE PARABOLA 13.2. section
698 (3 0) Chapter 3 Nonlinear Sstems and the Conic Sections 49. Fencing a rectangle. If 34 ft of fencing are used to enclose a rectangular area of 72 ft 2, then what are the dimensions of the area? 50.
More informationAx 2 Cy 2 Dx Ey F 0. Here we show that the general seconddegree equation. Ax 2 Bxy Cy 2 Dx Ey F 0. y X sin Y cos P(X, Y) X
Rotation of Aes ROTATION OF AES Rotation of Aes For a discussion of conic sections, see Calculus, Fourth Edition, Section 11.6 Calculus, Earl Transcendentals, Fourth Edition, Section 1.6 In precalculus
More informationAlex and Morgan were asked to graph the equation y = 2x + 1
Which is better? Ale and Morgan were asked to graph the equation = 2 + 1 Ale s make a table of values wa Morgan s use the slope and intercept wa First, I made a table. I chose some values, then plugged
More informationI think that starting
. Graphs of Functions 69. GRAPHS OF FUNCTIONS One can envisage that mathematical theor will go on being elaborated and etended indefinitel. How strange that the results of just the first few centuries
More information4.4 Concavity and Curve Sketching
Concavity and Curve Sketching Section Notes Page We can use the second derivative to tell us if a graph is concave up or concave down To see if something is concave down or concave up we need to look at
More informationD.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review
D0 APPENDIX D Precalculus Review SECTION D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane An ordered pair, of real numbers has as its
More informationModuMath Algebra Lessons
ModuMath Algebra Lessons Program Title 1 Getting Acquainted With Algebra 2 Order of Operations 3 Adding & Subtracting Algebraic Expressions 4 Multiplying Polynomials 5 Laws of Algebra 6 Solving Equations
More informationPolynomial and Rational Functions
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
More informationUnit 7 Polynomials. 7 1 Naming Polynomials. 7 2 Adding/Subtracting Polynomials. 7 3 Multiplying Monomials. 7 4 Dividing Monomials
Unit 7 Polnomials 7 1 Naming Polnomials 7 Adding/Subtracting Polnomials 7 Multipling Monomials 7 Dividing Monomials 7 Multipling (Monomial b Pol) 7 6 Multipling Polnomials 7 7 Special Products 0 Section
More informationFunctions and Their Graphs
3 Functions and Their Graphs On a sales rack of clothes at a department store, ou see a shirt ou like. The original price of the shirt was $00, but it has been discounted 30%. As a preferred shopper, ou
More informationDISTANCE, CIRCLES, AND QUADRATIC EQUATIONS
a p p e n d i g DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS DISTANCE BETWEEN TWO POINTS IN THE PLANE Suppose that we are interested in finding the distance d between two points P (, ) and P (, ) in the
More informationLINEAR FUNCTIONS. Form Equation Note Standard Ax + By = C A and B are not 0. A > 0
LINEAR FUNCTIONS As previousl described, a linear equation can be defined as an equation in which the highest eponent of the equation variable is one. A linear function is a function of the form f ( )
More informationSOLVING SYSTEMS OF EQUATIONS
SOLVING SYSTEMS OF EQUATIONS 4.. 4..4 Students have been solving equations even before Algebra. Now the focus on what a solution means, both algebraicall and graphicall. B understanding the nature of solutions,
More informationStudents Currently in Algebra 2 Maine East Math Placement Exam Review Problems
Students Currently in Algebra Maine East Math Placement Eam Review Problems The actual placement eam has 100 questions 3 hours. The placement eam is free response students must solve questions and write
More informationTHE PARABOLA section. Developing the Equation
80 (0) Chapter Nonlinear Sstems and the Conic Sections. THE PARABOLA In this section Developing the Equation Identifing the Verte from Standard Form Smmetr and Intercepts Graphing a Parabola Maimum or
More informationArithmetic Operations. The real numbers have the following properties: In particular, putting a 1 in the Distributive Law, we get
Review of Algebra REVIEW OF ALGEBRA Review of Algebra Here we review the basic rules and procedures of algebra that you need to know in order to be successful in calculus. Arithmetic Operations The real
More informationPacket 1 for Unit 2 Intercept Form of a Quadratic Function. M2 Alg 2
Packet 1 for Unit Intercept Form of a Quadratic Function M Alg 1 Assignment A: Graphs of Quadratic Functions in Intercept Form (Section 4.) In this lesson, you will: Determine whether a function is linear
More information6.3 PARTIAL FRACTIONS AND LOGISTIC GROWTH
6 CHAPTER 6 Techniques of Integration 6. PARTIAL FRACTIONS AND LOGISTIC GROWTH Use partial fractions to find indefinite integrals. Use logistic growth functions to model reallife situations. Partial Fractions
More informationRational Exponents and Radical Functions
Rational Eponents and Radical Functions.1 nth Roots and Rational Eponents. Properties of Rational Eponents and Radicals. Graphing Radical Functions. Solving Radical Equations and Inequalities. Performing
More informationGraphing Linear Equations in SlopeIntercept Form
4.4. Graphing Linear Equations in SlopeIntercept Form equation = m + b? How can ou describe the graph of the ACTIVITY: Analzing Graphs of Lines Work with a partner. Graph each equation. Find the slope
More informationSECTION 51 Exponential Functions
354 5 Eponential and Logarithmic Functions Most of the functions we have considered so far have been polnomial and rational functions, with a few others involving roots or powers of polnomial or rational
More informationFINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
FINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether or not the relationship shown in the table is a function. 1) 
More informationGRAPH OF A RATIONAL FUNCTION
GRAPH OF A RATIONAL FUNCTION Find vertical asmptotes and draw them. Look for common factors first. Vertical asmptotes occur where the denominator becomes zero as long as there are no common factors. Find
More information