P.5 Rational Expressions

Transcription

1 P.5 Rational Expressions I Domain Domain: Rational expressions : Finding domain a. polynomials: b. Radicals: keep it real! i. sqrt(x-2) x>=2 [2, inf) ii. cubert(x-2) all reals since cube rootscan be positive and negative c. rational expressions: can t divide by zero (undefined) so look at denominator and set it equal to zero. i. x + 1 ( x 2)( x 4) ii. x 2 + 4x + 3 x + 3 II Simplifying rational expressions *simplifying rational expression can have an impact on domain helps determine graphs in this course and calculus. #36 2 x + 8x 20 2 x + 11x + 10 III Operatoins with Rational Expressions remember fraction operations from P.1 notes A) Multiply #54

2 B) Dividing #96 C) Combining rational expressions Ex 7 IV Complex Fractions Complex fraction: separate fractions in numerator and/or denominator EX x x 2 methods: straight division, multiply numerator and denominator by LCD Factoring out negative exponents Ex 10 sign 2 ways: factor out smallest exponent, multiply top/bottom by the smallest exponent with the opposite V Difference Quotients: goal is to eliminate the original denominator Ex 11 x + h x h rationalize numerator

3 I Intro 4.1 Rational Functions and Asymptotes Rational function: = given that and are polynomials. Ex1 Domain of a rational function = II Vertical and Horizontal Asymptotes Definitions p.333 blue box 1) The line = is a vertical Asymptote of the graph of if or. 2) The line = is a horizontal asymptote of the graph of if as or. Examples for domain (from p. 333) = f(x)= f(x)= = P. 334 Vertical and Horizontal Asymptotes of a Rational Function f(x)= = with no common factors! " "! " "!! 1) The graph of has vertical asymptotes at the zeros of 2) The graph of has one or no Horizontal Asymptote determined by comparing the degrees of and a. If, the graph of has the line (x-axis) as a horizontal asymptote. b. If, the graph of has the line (ratio of leading coefficients) as a horizontal asymptote c. If, the graph of has horizontal asymptote. Ex 2 finding VA and HA a) = b) =

4 Ex 3 f(x)= # Additional Example $= % Ex 4 & 5 are Application problems only going to do ex 5 For a person with sensitive skin, the amount of time & hours the person can be exposed to the sun with minimal burning can be modeled by &=.().* 0<- 120 where - is the Sun sore Scale Reading (based on intensity of Ultraviolet ) rays). d. Find the amounts of time a person with sensitive skin can be exposed to the sun with minimal burning when - =10,- =25,& -=100. e. If the model were valid for all - >0, what would be the horizontal asymptote of this function, and what would it represent? Class Discussion: can a horizontal asymptote be crossed? How do we find that? 1) = 2) $= 3) h= 7

5 4.2 Graphs of Rational functions I Analyzing graphs of rational functions Guidelines Let = /, where and are polynomials Steps to graphing rational functions: Ex1 $= #26 $= Ex2 = #20 9= Ex3 = #37 = : 7 $= % #40 = :

6 II Slant Asymptotes Only occurs when the degree of is bigger than the degree of. Use to find the slant asymptote Ex 5 = Similar: #58 $= ;7 < * III Application finding minimum area Ex6 A rectangular page is designed to contain 48 square inches of print. The margins at the top and bottom of the page are 1 inch deep. The margins on each side are 1 inches wide. What should the dimensions of the page be so that the least amount of paper is used?

7 9.4 Partial Fractions Decomposition into partial Fractions (p. 690): 1) If degree of numerator is great than denominator, the divide first, then use remainder to apply steps 2 and 3 2) Factor denominator (over the integers) to get factors of = > or ++@ 3) Linear factors: for each factor of the form = >, the partical decomposition must include the following sum of factors: A = > + A A B = > + + = > B 4) Quadratic factors: for each factor of the form = >, the partical decomposition must include the following sum of factors: C +D ++@ + C +D C E +D E ++@ @ E Ex 1: ( # Ex 3: 7 Ex 4: *7

8 4.3A Parabolas (p ) Image credit: From previous sections: = ++@ can become = h +F If a>0 then opens If a<0 then opens Conic section format: h =4= F vertical F =4= h horizontal Focus: equidistant from this point, a point Directrix: equidistant from this line, a line P: the distance from the vertex to the focus, a number Vertex: is half way between the focus and directrix, a point D of O: direction of opening (up/down/left/right if =>0 then it opens right/up A of S: axis of symmetry, a line FD: focal diameter, found by finding the absolute value of 4= Basic Vertical Parabola 4=: D of O: Vertex: A of S: Focus: Directrix: FD: Focal diameter is 8; so at the focus we know that each side there is a point 8/2=4 away. Then points are (-4,2) and (4,2) to help us get the shape of the parabola. Basic Horizontal Parabola 4=: D of O: Vertex: A of S: Focus: Directrix: FD: Example 2 D of O: Vertex: A of S: Focus: Directrix: FD: Example: Find the equation of a parabola that has a vertex at 0,0 and its focus at 5,0 D of O: Vertex: A of S: Focus: Directrix: FD: Example: A parabola s vertex is (0,0) and its FD=10 and it opens vertically. What is its equation? D of O: Vertex: A of S: Focus: Directrix: FD: Example 6 0 D of O: Vertex: A of S: Focus: Directrix: FD: Example: Find the equation of a parabola that has its vertex at (0,0) and the directrix is 6 D of O: Vertex: A of S: Focus: Directrix: FD:

9 4.3B Ellipses and circles Circles (p. 349) h + F =I means the circle has its center at (h,k) and its radius is r. Ex: Ellipses (p ) Assume J5K L Horizontal? MN! 1 Center: (0,0) Foci O@,0 Vertices O,0 Major axis 2 Minor axis 2 Images credit: Vertical L!? MN 1 Center: (0,0) Foci 0,O@ Vertices 0,O Major axis 2 Minor axis 2 Eccentricity of Ellipse is how stretched it is. We measure it by P Q. is 0, then the relation is a circle. is 1, then it is very elongated/stretched. Example: 4? Standard form: Center a= b= c= Vertices Foci Major Axis Minor Axis Eccentricity Sketch Example: 9?4 1 Standard form: Center a= b= c= Vertices Foci Major Axis Minor Axis Eccentricity Sketch Example: 9?9 81 Standard form: Center a= b= c= Vertices Foci Major Axis Minor Axis Eccentricity Sketch Example: The major axis length is 6; the minor axis length is 4. The foci are on the x-axis. What is the equation of the ellipse? Example: The foci are O8,0 and the eccentricity is 0.8; what is the equation of the ellipse?

10 4.3C Hyperbolas Image credit: The first term indicates the type of opening. Find = + Horizontal L MN! =1 Vertical MN L! =1 Center (0,0) Foci ±@,0 Vertices ±,0 Branches (this is the hyperbola itself) Transverse axis (vertex to vertex distance) = 2a Conjugate axis =2b Asymptotes =±! Center (0,0) Foci 0,±@ Vertices 0,± Branches (this is the hyperbola itself) Transverse axis (vertex to vertex distance) = 2a Conjugate axis =2b Asymptotes =±! Examples: 9 16 =144 Standard form: Center Find c: Foci Vertices Transverse axis Conjugate axis Asymptotes Sketch Example: 9 +9=0 Standard form: Center Find c: Foci Vertices Transverse axis Conjugate axis Asymptotes Sketch Example: What is the equation of the hyperbola with (0,±6) and asymptotes =±?

11 Circle h + F =I with center (h,k) Ellipse L + MN =1! With center (h,k) Vertices at h±,f Foci h±@,f h + F =1 With center (h,k) Vertices h,f± Foci h,f±@ 4.4 Moving Conic Sections Hyperbola h F =1 Center (h,k) Foci h±@,f Vertices h±,f Asymptotes F =±! h MN L! =1 Center (h,k) Foci h,f±@ Vertices h,f± Asymptotes F =±! h Parabola h =4=( F) vertical Center (h,k) Focus (h, k+p) ( F) =4=( h) horizontal Center (h,k) Focus (h+p, k Conic general equation A ++C+D ++T+U=0 (Usually B is zero) 1. If A or D is zero, then it s usually a 2. If A and C have the same sign then a. A=C means it s a b. A D means it s an 3. If A and C have different signs then it s a Name that conic section! Conic: Center: ( 3) +(+2) =16 Notable characteristics/sketch: Standard form: =0 Center: Notable characteristics/sketch:

12 =0 Conic: Standard form: Center: Notable characteristics/sketch: Conic: Standard form: =0 Center: Notable characteristics/sketch: Degenerate conics: give a reason why each conic is degenerate 9(+1) ( 3) =0 ( 1) 1 + (+1) =1 1 4 ( 1) (+1) 1 =1

13 Exponent Rules a) Properties (p. 15) a a a a a a a m n m+ n m 0 n n n ( ab) ( a ) m n 1 = n a 1 = n a = 1 m m m m n m* n a b m = a = a = a b = a a = b m m P.2 Exponent properties EX 3 a & c 4 3 ( 3 ab )(4 ab ) 12x y 2 4x y 3 4 b) Scientific notation an extension of the exponent rules (can take any number and create it as a power of 10) a = 4 x 10^-4 b. # always between 1 & 10 Divide ( by

14 5.1 Exponential Functions and their graphs I Exponential Functions The exponential function with base is denoted by where >0, 1, and is any real number. =2 for x=2 =2 for x=2 =0.6 for x=2/3 II Graphs of exponential functions EX2a = =2 EX3b = =4 Make a table and plot points (-3 to 3) General conclusion on pare 382 see if we can get these from students! = = Domain range y-int Increasing/decreasing Horizontal Asymptote Continuous These are also one-to-one One-to-one functions help us realize that they have and tells us that we can!

15 EX4a 9=3 EX4b Z [ =8 Transformations of exponentials : =2 +5 = 5 +1 III The natural base \ P know the first 5 digits minimum! EX 6: use calculator to evaluate =P at = 2,& =.25 EX 7a Graphing P =2P. $= P. p. 384 for exact picture IV Application: Continuously compounded interest Yearly A=91+I^ P = principal, annual interest rate r, compounded once a year p. 386 N compoundings per year A=9Z1+ _ E [E^ Continuous A=9P _^ (Proof on page 385) Ex 8 a,c A total of $12000 is invested at an annual interest rate of 9%. Find the balance after 5 years if it is compounded (a) quarterly (c) continuously Radio active decay (Ex9)

16 5.4 use the one to one property to solve exponential equations P =P Z [( =81 2 =8 Z [ = Modeling with Exponential equations 1) Exponential growth =P! b>0 2) Exponential decay model =P! Decay carbon dating p. 422 ` = P^/* Estimate the age of a newly discovered fossil win which the ratio of carbon 14 to carbon 12 is ` = 7. Graph of Gaussian model is the used a lot in. Standard normal curve is = b P / Ex4 SAT Scores =0.0034P (::) /#,% where is the math score. Estimate the average!

17 Review of Topics for Exam 3 Section P.5 Find domains of algebraic expressions Simplify rational expressions Add, subtract, multiply, & divide rational expressions Simplify complex fractions Rewrite difference quotients Section 4.1 Find the domain of a rational function. Find the vertical asymptotes of a rational function. Find the horizontal asymptotes of a rational function. Section 4.2 Sketch the graph of a rational function. Find a slant asymptote of a rational function. Sketch the graph of a rational function with a slant asymptote. Section 9.4 Divide polynomials using polynomial long division. Factor a polynomial. Find the partial fraction decomposition of a rational expression that contains distinct or repeated linear factors and/or distinct or repeated quadratic factors. Section 4.3 Write the equation for a parabola, ellipse, or hyperbola with center or vertex (0,0). Graph a circle, parabola, ellipse, or hyperbola with center or vertex (0,0) Find all the pertinent information (vertices, foci, endpoints, vertex, p, asymptotes, directrix) about a parabola, ellipse, or hyperbola from the graph or the equation with center/vertex (0,0). Section 4.4 Write the equation for a parabola, ellipse, or hyperbola in standard form by completing the square. Determine the type of conic section once the conic is in standard form. Write the equation for a parabola, ellipse, or hyperbola with center or vertex (h,k). Graph a parabola, ellipse, or hyperbola with center or vertex (h, k). Find all the pertinent information (vertices, foci, endpoints, vertex, p, asymptotes, directrix) about a parabola, ellipse, or hyperbola from the graph or the equation with center/vertex (h,k) Section P.2 Use properties of exponents Use scientific notation to represent real numbers Use properties of radicals Simplify and combine radicals Rationalize numerators and denominators Use properties of rational exponents Section 5.1 Recognize an exponential function Graph an exponential function with base a or base e Evaluate an exponential function with base a or base e Solve a real world problem using an exponential function Section 5.5 Recognize common exponential models Use the common exponential models to solve real world problems