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. x +1 x2 +4 x + 3 i. ii. ()()x−2 x − 4 x + 3 II Simplifying rational expressions *simplifying rational expression can have an impact on domain – helps determine graphs in this course and calculus. x2 +8 x − 20 #36 x2 +11 x + 10 III Operatoins with Rational Expressions – remember fraction operations from P.1 notes A) Multiply #54 B) Dividing #96 C) Combining rational expressions Ex 7 IV Complex Fractions Complex fraction: separate fractions in numerator and/or denominator x + 3 EX 4 2 3 2 − x 2 methods: straight division, multiply numerator and denominator by LCD Factoring out negative exponents Ex 10 2 ways: factor out smallest exponent, multiply top/bottom by the smallest exponent with the opposite sign V Difference Quotients: goal is to eliminate the original denominator x+ h − x Ex 11 rationalize numerator h 4.1 Rational Functions and Asymptotes I Intro 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) = = 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 where is the Sun sore Scale Reading (based on intensity of Ultraviolet & = ) 0 , - . 100 - rays). d. Find the amounts of time a person with sensitive skin can be exposed to the sun with minimal burning when - = 101- = 02134- = 100 . e. If the model were valid for all - 5 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) 6 = 7 4.2 Graphs of Rational functions I Analyzing graphs of rational functions Guidelines Let = 8, where and are polynomials Steps to graphing rational functions: Ex1 #26 Ex2 #20 $ = $ = = 9 = : % : Ex3 #37 #40 = = 7 $ = = II Slant Asymptotes Only occurs when the degree of is bigger than the degree of . Use _____________________ to find the slant asymptote ;7< Ex 5 Similar: #58 = $ = * 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 inches wide. What should the dimensions of the page be so that the least 1 amount of paper is used? 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 AB ? ? ? = − > = − > = − >B 4) Quadratic factors: for each factor of the form = − >, the partical decomposition must include the following sum of factors: C ? D C ? D CE ?DE ? ? ? ? ? @ ? ? @ ? ? @E 7 Ex 1: ( Ex 3: Ex 4: * # 7 Image credit: http://www.varsitytutors.com/hotmath/hotmath_help/topics/latus-rectum.html 4.3A Parabolas (p. 350-351) From previous sections: = ? ? @ can become = − 6 ?F If a>0 then opens ____ If a<0 then opens____ Conic section format: − 6 = 4= − F vertical − F = 4= − 6 horizontal Focus : equidistant from this point, a point Focal diameter is 8; so at the focus we Directrix : equidistant from this line, a line know that each side there is a point P: the distance from the vertex to the focus, a number 8/2=4 away. Then points are (-4,2) Vertex : is half way between the focus and directrix, a point and (4,2) to help us get the shape of D of O : direction of opening (up/down/left/right the parabola. if = 5 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= : Basic Horizontal Parabola 4= : D of O: D of O: Vertex: Vertex: A of S: A of S: Focus: Focus: Directrix: Directrix: FD: FD: Example 0 Example: Find the equation of a Example: A parabola’s vertex is D of O: parabola that has a vertex at 010 (0,0) and its FD=10 and it opens Vertex: and its focus at 210 vertically. What is its equation? A of S: D of O: D of O: Focus: Vertex: Vertex: Directrix: A of S: A of S: FD: Focus: Focus: Directrix: Directrix: FD: FD: Example 6 0 Example: Find the equation of a D of O: parabola that has its vertex at Vertex: (0,0) and the directrix is 6 A of S: D of O: Focus: Vertex: Directrix: A of S: FD: Focus: Directrix: FD: 4.3B Ellipses and circles Circles (p. 349) − 6 ? − F = I means the circle has its center at (h,k) and its radius is r. Ex: − 4 ? − 2 1 Ellipses (p. 352-353) Assume J 5 K Images credit: L MN L MN Horizontal ? 1 Vertical ? 1 ! http://www.varsitytutors.com/hotmath/hotmath_help/t ! opics/ellipse.html Center: (0,0) Foci O@1 0 Vertices O1 0 Center: (0,0) Foci 01 O@ Vertices 01 O Major axis 0 Minor axis 0 To get @: @ Major axis 0 Minor axis 0 To get @: @ Q Eccentricity of Ellipse is how stretched it is. We measure it by P . If @ is 0, then the relation is a circle. If @ is 1, then it is very elongated/stretched. Example: 4 ? 02 100 Example: R ? 4 1 Example: R ? R S1 Standard form: Standard form: Standard form: Center Center Center a= b= c= a= b= c= a= b= c= Vertices Vertices Vertices Foci Foci Foci Major Axis Major Axis Major Axis Minor Axis Minor Axis Minor Axis Eccentricity Eccentricity Eccentricity Sketch Sketch Sketch Example: The major axis length is 6; the minor axis Example: The foci are OS10 and the eccentricity is 0.8; length is 4. The foci are on the x-axis. What is the what is the equation of the ellipse? equation of the ellipse? 4.3C Hyperbolas Image credit: https://sites.google.com/site/jennybethshyperbolas/gamebattles The first term indicates the type of opening. Find c: @ = ? L MN MN L Horizontal Vertical − ! = 1 − ! = 1 Center (0,0) Center (0,0) Foci O@1 0 Foci 01 O@ Vertices O1 0 Vertices 01 O Branches (this is the hyperbola itself) Branches (this is the hyperbola itself) Transverse axis (vertex to vertex distance) = 2a Transverse axis (vertex to vertex distance) = 2a Conjugate axis =2b Conjugate axis =2b Asymptotes ! Asymptotes = O = O ! Examples: R − 16 = 144 Example: − R ? R = 0 Standard form: Standard form: Center Center Find c: Find c: Foci Foci Vertices Vertices Transverse axis Transverse axis Conjugate axis Conjugate axis Asymptotes Asymptotes Sketch Sketch Example: What is the equation of the hyperbola with vertices @ (01 O6) and asymptotes = O ? 4.4 Moving Conic Sections Circle Hyperbola Parabola − 6 ? − F = I with − 6 = 4=( − F) center (h,k) − 6 − F vertical − = 1 Center (h,k) Focus (h, k+p) Ellipse L MN Center (h,k) Foci ( − F) = 4=( − 6) ? = 1 6 O @1 F ! Vertices horizontal With center (h,k) 6 O 1 F ! Center (h,k) Focus (h+p, k Asymptotes − F = O − 6 Vertices at 6 O 1 F Foci 6 O @1 F MN L − = 1 − 6 − F ! 4 ? = 1 With center (h,k) Center (h,k) Foci 61 F O @ Vertices Vertices 61 F O 61 F O Foci Asymptotes 61 F O @ − F = O ! − 6 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. AVD means it’s an _____________________ 3. If A and C have different signs then it’s a _____________________ Name that conic section! ( − 3) ? ( ? 0) = 16 Conic: Center: Notable characteristics/sketch: 3 ? 0 ? S − 4 = 0 Standard form: Center: Notable characteristics/sketch: ? 16 −6?R6?137=0 Conic: Standard form: Center: Notable characteristics/sketch: 4 − R −16?24−0R=0 Conic: Standard form: Center: Notable characteristics/sketch: Degenerate conics: give a reason why each conic is degenerate R( ? 1) − ( − 3) = 0 ( − 1) ( ? 1) ( − 1) ( ? 1) ? = 1 ? = 1 1 1 − −1 −1 4 4 P.2 Exponent properties Exponent Rules a) Properties (p.