Unit 4: Polynomial and Rational Functions

Transcription

1 50 Unit 4: Polynomial and Rational Functions Polynomial Functions A polynomial function y p() is a function of the form p( ) a a a... a a a n n n n n n 0 where an, an,..., a, a, a0 are real constants and are called the coefficients of p ( ). n is the degree of p ( ) and is a positive integer. an is called the leading coefficient and a0 is the constant term of the polynomial. The domain of any polynomial is all real numbers. E. Determine the degree, the leading coefficient and the constant term of the polynomial. a) 4 f ( ) b) g( ) 5 4 End Behavior of a Polynomial There are four scenarios: ) Sketch p( ), p( ) 4 ( n is even, a n 0 ) ) Sketch p( ), p( ) 4 ( n is even, a n 0 ) As, p( ) As, p( ) As, p( ) As, p( )

2 5 ) Sketch p( ), p( ) 5 ( n is odd, a n 0 ) 4) Sketch p( ), p( ) 5 ( n is odd, a n 0 ) As, p( ), p( ) As, p( ), p( ) As and, the graph of the polynomial p( ) a a a... a a a resembles the graph of n n n n n n 0 n y an. E. Use the zeros and the end behavior of the polynomial to sketch an approimation of the graph of the function. a) f ( ) 9 b) 4 g( ) 5 4

3 5 c) 5 f ( ) Repeated Zeros If a polynomial f() has a factor of the form ( c ) k, where k, then cis a repeated zero of multiplicity k. If k is even, the graph of f() flattens and just touches the -ais at c. If k is odd, the graph of f() flattens and crosses the -ais at c. E. 4: Sketch the given graphs 4 f ( ) g( ) ( ) ( )( )

4 5 E. 5: The cubic polynomial p ( ) has a zero of multiplicity two at, a zero of multiplicity one at, and p ( ). Determine p ( ) and sketch the graph. E. 6: An open bo is to be made from a rectangular piece of cardboard that is by 6 feet by cutting out squares of side length feet from each corner and folding up the sides. a) Epress the volume of the bo vas ( ) a function of the size cut out at each corner. b) Use your calculator to approimate the value of which will maimize the volume of the bo. E. 7: The product of two non-negative numbers is 60. What is the minimum sum of the two numbers?

5 54 The Intermediate Value Theorem Suppose that f is continuous on the closed interval [ ab, ] and let N be any number between f( a) and f() b, where f ( a) f ( b ). Then there eists a number c in ( ab, ) such that f () c N. E. : Verify that the Intermediate Value Theorem applies to the indicated interval and find the value of c guaranteed by the theorem. f ( ), [0,5], f( c ) E. : Show that there is a root of the equation 0 in the interval (0,).

6 55 The Division Algorithm Let f() and d ( ) 0 be polynomials where the degree of f() is greater than or equal to the degree of d ( ). Then there eists unique polynomials q () and r ( ) such that f ( ) r( ) q ( ) d( ) d( ) or f ( ) d( ) q( ) r( ). where r ( ) has a degree less than the degree of d ( ). E. : Divide the given polynomials. a) b) c) 6

7 56 Remainder Theorem If a polynomial f() is divided by a linear polynomial c, then the remainder r is the value of f() at c. In other words, f () c r E. : Use the Remainder Theorem to find r when f ( ) 4 4 is divided by. E. : Use the Remainder Theorem to find f() c for 4 f ( ) 5 7 when c Synthetic Division Synthetic division is a shorthand method of dividing a polynomial p ( ) by a linear polynomial uses only the coefficients of p ( ) and must include all 0 coefficients of p ( ) as well. c. It E. 4: Use synthetic division to find the quotient and remainder when a) f ( ) is divided by b) 4 f ( ) is divided by 4 c) is divided by 4

8 57 E. 5: Use synthetic division and the Remainder Theorem to find f() c for f ( ) when c. E. 6: Use synthetic division and the Remainder Theorem to find f() c for f ( ) 7 5 when c 5. The Factor Theorem A number c is a zero of a polynomial p ( ) ( pc ( ) 0) if and only if ( c) is a factor of p ( ). Eamples: Determine whether a) is a factor of 4 f ( ) 5 6 b) is a factor of 4

9 58 Fundamental Theorem of Algebra A polynomial function p ( ) of degree n 0 has at least one zero. In fact, every polynomial function p ( ) of degree n 0 has at eactly n zeros. Complete Factorization Theorem Let c, c,... c n be the n (not necessary distinct) zeros of the polynomial function of degree n 0 : p( ) a a a... a a a. n n n n n n 0 Then p ( ) can be written as the product of n linear factors p( ) a ( c )( c ) ( c ). n n E. : Give the complete factorization of the given polynomial p ( ) with given information: a) p( ) 9 6 ; is a zero. b) 4 p( ) ;, 5 are both zeros.

10 59 c) p( ) ; ( ) is a factor. d) 4 p( ) 7 5 ; ( ) is a factor. E. : Find a polynomial function f() of degree three, with zeros,-4, 5 such that the graph possesses the y - intercept (0,5).

11 60 The Rational Zero Test Suppose p q is a rational zero of f ( ) a n a n a n... a a a, n n n 0 where a0, a..., an are integers and a n 0. Then p divides a0 and q divides a n. The Rational Zero Test provides a list of possible rational zeros. Eamples: Find all the rational zeros of f() then factor the polynomial completely. a) 4 f ( ) 0 8. b) 4 f ( )

12 6 Comple Roots of Polynomials Consider factoring the function: f ( ) The Square Root of - We define i so that i. Comple Numbers A comple number is a number of the form a bi where a and b are real numbers. The number a is called the real part and the number b is called the imaginary part. Comple Arithmetic E. : a) ( i) (6 i ) b) ( i)(4 i ) c) ( 6 i)( 6 i ) d) (4 5 i)(4 5 i ) Comple Conjugates The comple conjugate for a comple number z a bi is z a bi. In general, ( a bi)( a bi )

13 6 E. : Simplify. a) ( i) ( 6 i) ( i) ( 7 i) E. : Simplify. a) 4 b) 8 E. 4: Determine all solutions to the equation 4 0 E. 5: Completely factor f ( ).

14 E. 6: Find the complete factorization of multiplicity two. 4 f ( ) given that is a zero of 6 Conjugate Pairs of Zeros of Real Polynomials If the comple number z a bi is a zero of some polynomial p ( ) with real coefficients, then its conjugate z a bi is also a zero of p ( ). E. 7: Find a rd degree polynomial g ( ) with real coefficients and a leading coefficient of with zeros and i. E. 8: i is a zero of complete factorization of f(). 4 f ( ) Find all other zeros and then give the

15 64 Rational Functions A rational function y f () is a function of the form functions. p ( ) f( ), where p and q are polynomial q ( ) E. : Recall the parent function f( ). Use transformations to sketch g ( ) Asymptotes of Rational Functions The line ais a vertical asymptote of the graph of f() if f() or f() as a (from the right) or Vertical Asymptotes The graph of f( ) p ( ) q ( ) a (from the left). has vertical asymptotes at the zeros of q () after all of the common factors of p ( ) and q () have been canceled out; the values of where q ( ) 0 and p ( ) 0. Holes The graph of f( ) p ( ) has a hole at the values of where q ( ) 0 and ( ) 0 q ( ) p.

16 65 Horizontal Asymptotes The line y bis a horizontal asymptote of the graph of f() if f () b when or. In particular, with a rational function There are three cases: f( ) n p ( ) an a... a a m q( ) b b... b b m n n 0 m m 0. If n m, then y 0is the horizontal asymptote. E: f( ) 7 an. If n m, then y b m is the horizontal asymptote. E: f( ) 6 4. If n m, then there is no horizontal asymptote. E: f( ) Slant Asymptote If the degree of numerator is eactly one more than the degree of the denominator, the graph of f() has a slant asymptote of the form y m b. The slant asymptote is the linear quotient found by dividing p ( ) by q () and essentially disregarding the remainder. Eample: f( ) 8

17 66 E. : Find all asymptotes and intercepts and sketch the graphs of the given rational functions: a) f( ) Domain: Range: Equation(s) of vertical asymptotes: Equation(s) of horizontal asymptotes: Equation of slant asymptote: - intercepts: y - intercept: b) f( ) 4 Domain: Range: Equation(s) of vertical asymptotes: Equation(s) of horizontal asymptotes: Equation of slant asymptote: - intercepts: y - intercept:

18 67 c) f( ) ( )( 5) Domain: Range: Equation(s) of vertical asymptotes: Equation(s) of horizontal asymptotes: Equation of slant asymptote: - intercepts: y - intercept: d) f( ) Domain: Equation(s) of vertical asymptotes: Equation(s) of horizontal asymptotes: Equation of slant asymptote: - intercepts: y - intercept:

19 68 e) f( ) Domain: Range: Equation(s) of vertical asymptotes: Equation(s) of horizontal asymptotes: Equation of slant asymptote: - intercepts: y - intercept: f) f( ) ( )( ) ( )( ) Domain: Range: Equation(s) of vertical asymptotes: Equation(s) of horizontal asymptotes: Equation of slant asymptote: - intercepts: y - intercept: E. : Sketch the graph of a rational function that satisfies all of the following conditions: f() as and f() as f() as and f() as f() has a horizontal asymptote y 0 f() has no -intercepts Has a local maimum at (, )

20 Honors Precalculus Academic Magnet High School 69 Name Mandelbrot Set Activity using Fractint fractal generator STEP - CREATE, SAVE, and PRINT an inspirational, visually pleasing area of the Mandelbrot set. Important Menu Items: VIEW- Image Settings, Zoom In/ Out bo, Coordinate Bo FRACTALS- Fractal Formula, Basic Options, Fractal Parameters COLORS- Load Color- Map FILE- Save As ) Start Fractint by clicking on the desktop icon. Fractint always starts with the Mandelbrot set, but in case things get weird, ALWAYS make sure mandel is selected in the Fractal-Fractal Formula menu item. Use the Image Settings bo to set the size of the picture ( should work fine). ) Use the Zoom In/Out feature along with the Colors-Load Color Map to create a variation of the Mandelbrot set. If the color palettes do not load, double click on the bo that is labeled Pallette Files (*.Map) If you zoom in a few times you lose detail, you can increase the iterations in the Fractals-Basic Options Bo- Remember that the more iterations the computer has to perform, the longer it will take ) Use the Fractals-Fractal Params window to record the and y mins and mas of the viewing rectangle on the imaginary plane. 4) Using the Coordinates bo, point your arrow to a point you think is in the Mandelbrot set and record the and y values. 5) Repeat #4 for a point you think is NOT in the set. 6) SAVE the fractal. Write down the coordinates ( and y mins and mas) and number of iterations of your current position in the Mandelbrot set. 7) Print your fractal. STEP - Create a typed tet document ( page or so) including, but not limited to: The NAME of your group s fractal and the name of everyone in your group A short story about your creation (what it makes you think of, color, choice, etc.)

21 70 STEP Typed: ) List the and y mins and mas for your viewing rectangle from Step ) Recall the coordinates of the point you thought was in the Mandelbrot set from Step. Let = a and y = b for the comple number a + bi Let this number a + bi = c iterate this value 00 or more times using the Mandelbrot sequence: 0 = c = 0 + c = + c Etc You will be using decimals and your calculator. Unlike the fractals, these calculations will not be pretty. Let your TI-84 do the work for you (i is above the decimal point). ) Record the last 0 iterations for analysis. Remember that you may need to scroll the TI- 84 to the right to get the entire number 4) Were your predictions right about this point? Do you need more information to determine if it is in the set? 5) Repeat for the point you thought was not in the set. 6) Summarize your findings. TURN IN ALL STEPS PAPER-CLIPPED together in order. Etra Credit: Create your own color map. - for info on Fractint