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# Chapter 5: Polynomial and Rational Functions MAT 109

Marketplace > Barry University > Mathmatics > MAT 109 > Chapter 5 Polynomial and Rational Functions
Sterling
Barry University
GPA 3.7

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The following bundle contains the recommended sections for this chapter: 5.1 Polynomial Functions and Models 5.2 Properties of Rational Functions 5.3 The Graph of a Rational Function 5.4 Polyno...
COURSE
Precalculus Mathematics 1
PROF.
Dr. Singh
TYPE
Bundle
PAGES
15
WORDS
CONCEPTS
Precalculus
KARMA
75 ?

## Popular in Mathmatics

This 15 page Bundle was uploaded by Sterling on Wednesday September 14, 2016. The Bundle belongs to MAT 109 at Barry University taught by Dr. Singh in Fall 2016. Since its upload, it has received 9 views. For similar materials see Precalculus Mathematics 1 in Mathmatics at Barry University.

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Date Created: 09/14/16
MAT 109 PreCalculus Mathematics 1 5.1 Polynomial Functions and Models Notes L. Sterling September 1st, 2016 Abstract Provide a generalization to each of the key terms listed in this section. 1 Identifying Polynomial Functions f (x) = a x + a x x▯1+ ::: + a x + a n n▯1 1 0 Exponents : Positive and Integers Coefficients : Real Nonnegative Integer : n Domain : All Real Numbers 2 Properties of Polynomial Functions 2.1 Zero Function Example : y = 0 Degree : None Graph : X ▯ Axis 2.2 Constant Function Example : y = 2 Degree : 0 Graph : Horizontal line with "c" as y ▯ intercept 2.3 Linear Function Example : y = x + 5 Degree : 1 Graph : A nonhorizontal and nonvertical line with a slope and y ▯ intercept: 1 2.4 Quadratic Function Example : y = x + 5x + 6 Degree : 2 Aparabolawiththegraphopeningupwhena > 02ndopeningdownwhena < 0: 2 3 Power Function n f (x) = ax a 6= 0 n > 0 3.1 Properties if the n in f(x) = x is Even ▯ f will be an even function, which would make the graph being symmetric, but with respect to the y-axis. ▯ Domain: All Real Numbers ▯ Range: Nonnegative Real Numbers ▯ The graph will always have 3 of the same points: (▯1;1), (0;0), and (1;1). ▯ When the exponent is increasing the graph [whether when x < ▯1 or x > 1] will be more vertical. ▯ When x is getting nearer to the origin, the graph will be tending to become more attened out while also lying closer towards the x-axis. n 3.2 Properties if the n in f(x) = x is Odd ▯ f will be an odd function, which would make the graph being symmetric, but with respect to the origin. ▯ Domain: All Real Numbers ▯ Range: All Real Numbers ▯ The graph will always have 3 of the same points: (▯1;▯1), (0;0), and (1;1). ▯ When the exponent is increasing the graph [whether when x < ▯1 or x > 1] will be more vertical. ▯ When x is getting nearer to the origin, the graph will be tending to become more attened out while also lying closer towards the x-axis. 4 Real Zero The r in f(r) = 0 since r’s an x-intercept of f’s graph and a real zero of a polynomial function while also being a solution to f(x) = 0. x ▯ r’s a factor if f. 2 5 Zero of Multiplicity r when the following: m ▯ (x ▯ r) is a factor of f m+1 ▯ (x ▯ r) is not a factor of f 6 Touching and Crossing 6.1 Zero of Even Multiplicity f(x)’s sign doesn’t change from one side of r to the other side, which will make the graph touch the x-axis at r. 6.2 Zero of Odd Multiplicity f(x)’s sign does change from one side of r to the other side, which will make the graph cross the x-axis at r. 7 Theorem of Turning Points If you have f being a polynomial function of degree n, then f will, at most, have n ▯ 1 turning points. On the other hand, if the graph has n ▯ 1 turning points, then f’s degree will be at least n. 8 End Behavior For all of x’s [either negative or positive] larger values, the polynomial’s graph will be f (x) =na x +n▯1 xx▯1+ ::: +1a x +0a while resembling the power function’s graph, which would be y = a x . n 9 Steps to Analyze a Polynomial Function’s Graph ▯ Find the end behavior. ▯ Find the intercepts. ▯ Find the zeros and their multiplicity. ▯ Find out whether the graph will touch or cross the x-intercept(s). ▯ Graph the function with any graphing utility. ▯ Find the graph’s turning point(s). ▯ Draw the graph by hand. ▯ Find the domain and range. ▯ Find where it’s increasing or decreasing. 3 10 Steps to Analyze a Polynomial Function’s Graph by using a Graphing Utility ▯ Find the end behavior. ▯ Graph the function with any graphing utility. ▯ Find the intercepts. ▯ Create a [TABLE] to ▯nd the points on the graph around each and every x-intercept. ▯ Find the graph’s turning point(s). ▯ Draw the graph by hand. ▯ Find the domain and range. ▯ Find where it’s increasing or decreasing. 4 MAT 109 PreCalculus Mathematics 1 5.2 Properties of Rational Functions Notes L. Sterling September 2nd, 2016 Abstract Provide a generalization to each of the key terms listed in this section. 1 Rational Functions Functions that are in the form of R(x) =here you both p(x) and q(x) are q(x) polynomial functions while only q(x) can’t be a zero polynomial, which means q (x) 6= 0. When looking for the domain, besides then q (x) 6= 0, would be all real numbers. If you have a rational function like R(x)is its lowest form, then you q(x) will have R be having a vertical asymptote for each and every value of x for which q(x) = 0. 2 Horizontal Asymptote ▯ A horizontal line that occurs when x is approaching either ▯1 or 1, which would make f(x) approach L. ▯ Can be written in one of two ways. { f (x) ! L as x ! ▯1 { limx!▯1 f (x) = L ▯ Helps describes a function’s end behavior. ▯ A function’s graph can intersect a horizontal asymptote. 3 Vertical Asymptote ▯ A vertical line [x = c] that occurs when x is approaching c, which would make f(x) approach either ▯1 or 1. ▯ A function’s graph can intersect a horizontal asymptote, but it wouldn’t intersect the vertical asymptote. 1 4 Oblique Asymptote An asymptote that is not considered horizontal or vertical, but it is a line like in the form of f(x) = y = mx + b, which notes that f(x) is approaching when x is approaching either ▯1 or 1. p(x) n 5 What if R(x) = = anxm+:::+0 was a rational q(x) bmx +:::+b0 function? 5.1 q(x)’s degree is greater than p(x)’s degree m > n Horizontal Asymptote : y = 0 5.2 q(x)’s degree is one less than p(x)’s degree m = n ▯ 1 Use Long Division Rewritten : R(x) = mx + b + r (x) Oblique Asymptote : y = mx + b 5.3 q(x)’s degree is equaled to p(x)’s degree m = n Horizontal Asymptote : y = bm 5.4 All Other Cases No Horizontal or Oblique Asymptotes 2 MAT 109 Precalculus Mathematics 1 5.3 The Graph of a Rational Function Notes L. Sterling September 6th, 2016 Abstract Provide a generalization to each of the key terms listed in this section. How do you analyze a rational function’s graph? 0.1 Find the function’s domain. 0.2 Reduce to lowest terms. 0.3 Find the graph’s holes [if any]. 0.4 Find the graph’s intercepts [if any]. 0.5 Determine any odd and/or even multiplicities. ▯ Odd: Crossing the x-axis at the given point. ▯ Even: Tangent to the x-axis at the given point. 1 0.6 Find the graph’s vertical, horizontal, and oblique asymp- totes [if any]. 0.7 Find any points that intersect any found asymptotes. 0.8 Divide the real zeros and the vertical asymptotes to divide the x-axis into intervals. 0.9 Pick values of x to ▯nd the interval’s signs while eval- uating the function at the given values. 0.10 Plot all of the points. 0.11 Draw the asymptotes as well as the behavior of the function. 0.12 Sketch the graph and making sure it’s passing the vertical line test. 2 MAT 109 Precalculus Mathematics 1 5.4 Polynomial and Rational Inequalities Notes L. Sterling September 7th, 2016 Abstract Provide a generalization to each of the key terms listed in this section. 1 How do you [algebraically] solve a polynomial inequality? 1.1 Write the given inequality so that the polynomial ex- pression, which is f, is being displayed on the left side with zero being on the right side. 1.2 Find the graph’s real zeroes [if any]. 1.3 Use the zeroes to divide the real number line into intervals. 1 1.4 Choose values between each interval and evaluate at the given values. ▯ If f is positive, then all of the values of f in the interval will be positive. ▯ If f is negative, then all of the values of f in the interval will be negative. 2 How do you [algebraically] solve a rational in- equality? 2.1 Write the given inequality so that the polynomial ex- pression, which is f, is being displayed on the left side with zero being on the right side. 2.2 Find the graph’s real zeroes [if any]. 2.3 Find the graph’s real numbers [if any]. 2.4 Use the zeroes and unde▯ned values to divide the real number line into intervals. 2.5 Choose values between each interval and evaluate at the given values. ▯ If f is positive, then all of the values of f in the interval will be positive. ▯ If f is negative, then all of the values of f in the interval will be negative. 2 MAT 109 Precalculus Mathematics 1 5.5 The Real Zeros of a Polynomial Function Notes L. Sterling September 8th, 2016 Abstract Provide a generalization to each of the key terms listed in this section. 1 Remainder and Factor Theorems (Quotient)(Divisor) + Remainder = Dividend 2 Division Algorithm for Polynomials f (x) r (x) = q (x) + g (x) g (x) f (x) = q (x)g (x) + r (x) f(x) = Dividend 1 q(x) = Quotient g(x) = Divisor r(x) = Remainder 3 Remainder Theorem First o▯, let f be a polynomial function. Since f(x) is the dividend, if f(x) is f(x) being divided by x ▯ c, which would lookx▯c, then the remainder would technically be f(c). 4 Factor Theorem First o▯, let f be a polynomial function. Since x ▯ c is a factor of f(x) i▯ [if and only if] f(c) be equaled to 0. ▯ If f(c) = 0, then x ▯ c would be a f(x)s factor. ▯ If x ▯ c would be a f(x)s factor, then f(c) = 0. 5 Number of Real Zeros A polynomial function and cant have more actual real zeroes than its own degree or even of degree n, like n ▯ 1, has at most n real zeros. 2 6 Rational Zeros Theorem Since you are letting f is a polynomial function of degree 1 or any higher in the following form that note that each coe▯cient is an integer: n n▯1 f (x) = n x + an▯1 x + ::: + 1 x + 0 an 6= 0 a 06= 0 If you havep [a rational zero of f] in its lowest terms, then you would have q p being an a factor0a with q being a factor nf a . 7 How do you ▯nd a polynomial functions real zeroes? 7.1 Find the maximum number of real zeroes thanks to using the polynomials degree. ▯ Use the Rational Zeros Theorem to identify the rational numbers that potentially could be zeros if you have the polynomial having integer coef- ▯cients. ▯ Use techniques like synthetic division, long division, or even substitution to test each of the given rational zeros. Each time that a zero [and therefore a factor] is found, it can be by repeating any certain step on the depressed equation. { Use any of the factoring techniques [grouping, special products, etc.] when you are trying to ▯nd the zeros. 3 8 Polynomial Functions ▯ Every polynomial function with real coe▯cients can be uniquely factored into a product of linear factors and/or irreducible (prime) quadratic fac- tors. ▯ A polynomial function of odd degree that has real coe▯cients has at least one real zero. 9 Bounds of Zero You should let f be a denoted polynomial function by having 1 as their leading coe▯cients in the following form: f (x) = n x + an▯1 xn▯1+ ::: + 1 x + 0 The bound of M on any of fs real zeros of f would be the smaller of any of the two numbers, which would make choosing the largest possible entry in fg equals Max fg, which is in the following form: Maxf1; ja 0 + j1 j + ::: n▯1ajg; +Maxfja 0 + ja1j + ::: +n▯1jg 10 Intermediate Value Theorem This is denoting f to be a polynomial function while if both a < b with f(a) and f(b) being the opposite sign, then theres fs at least one actual [real] zero between for a and b. 4 11 What are the steps to the real zeros of a polynomial function? 11.1 Find the two consecutive integers, which can be both a and a + 1, for f being a zero between them. 11.2 Divide the interval, which can be [a; a + 1] into ex- actly 10 equivalent subintervals. 11.3 Evaluate f at each of the given subintervals end- point until applying the Intermediate Value Theo- rem, which would make the subinterval containing a zero. 11.4 Divide the interval again until you have the appro- priate accuracy. 5

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