P.5 Rational Expressions

I Domain

Domain:

Rational expressions :

Finding domain

a. :

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 .

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 : given that and are polynomials. = Ex1 Domain of a =

II Vertical and Horizontal Asymptotes Definitions p.333 blue box 1) The = 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 < ≤ 120 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 =

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 = = = =

Ex3 #37 #40 = = = =

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 = =

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:

+ + ⋯ + − − −

4) Quadratic factors: for each factor of the form − , the partical decomposition must include the following sum of factors:

+ + + + + ⋯ + + + + + + +

Ex 1: Ex 3: Ex 4:

Image credit: http://www.varsitytutors.com/hotmath/hotmath_help/topics/latus-rectum.html 4.3A (p. 350-351)

From previous sections: = + + can become = − ℎ + If a>0 then opens ____ If a<0 then opens____

Conic section format: − ℎ = 4 − vertical − = 4 − ℎ 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 . 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 : 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 2 Example: Find the equation of a Example: A parabola’s vertex is D of O: parabola that has a vertex at 0,0 (0,0) and its FD=10 and it opens Vertex: and its focus at 5,0 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) − ℎ + − = means the circle has its center at (h,k) and its radius is r. Ex: − 4 + − 5 1

Ellipses (p. 352-353) Assume credit:Images Horizontal 1 Vertical 1 http://www.varsitytutors.com/hotmath/hotmath_help/t

opics/ellipse.html

Center: (0,0) Foci , 0 Vertices , 0 Center: (0,0) Foci 0, Vertices 0,

Major axis 2 Minor axis 2 To get : Major axis 2 Minor axis 2 To get :

Eccentricity of Ellipse is how stretched it is. We measure it by . If is 0, then the relation is a circle. If is 1, then it is very elongated/stretched.

Example: 4 25 100 Example: 9 4 1 Example: 9 9 81 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 8,0 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

Image credit: https://sites.google.com/site/jennybethshyperbolas/gamebattles The first term indicates the type of opening. Find c: = +

Horizontal Vertical − = 1 − = 1

Center (0,0) Center (0,0) Foci ±, 0 Foci 0, ± Vertices ±, 0 Vertices 0, ± Branches (this is the 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 = ± = ±

Examples: 9 − 16 = 144 Example: − 9 + 9 = 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 @ (0, ±6) and asymptotes = ± ?

4.4 Moving Conic Sections

Circle Hyperbola Parabola − ℎ + − = with − ℎ = 4( − ) center (h,k) − ℎ − vertical − = 1 Center (h,k) Focus (h, k+p) Ellipse Center (h,k) Foci ( − ) = 4( − ℎ) + = 1 ℎ ± , Vertices horizontal With center (h,k) ℎ ± , Center (h,k) Focus (h+p, k Asymptotes − = ± − ℎ Vertices at ℎ ± ,

Foci ℎ ± , − = 1 − ℎ − + = 1 With center (h,k) Center (h,k) Foci ℎ, ± Vertices Vertices ℎ, ± ℎ, ± Foci Asymptotes ℎ, ± − = ± − ℎ

Conic general equation + + + + + + = 0 (Usually B is zero) 1. If or 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. ≠ means it’s an ______

3. If A and C have different signs then it’s a ______

Name that ! ( − 3) + ( + 2) = 16 Conic: Center:

Notable characteristics/sketch:

3 + 2 + 8 − 4 = 0 Standard form:

Center:

Notable characteristics/sketch:

+ 16 −6+96+137=0

Conic: Standard form:

Center:

Notable characteristics/sketch:

4 − 9 −16+54−29=0 Conic: Standard form:

Center:

Notable characteristics/sketch:

Degenerate conics: give a reason why each conic is degenerate 9( + 1) − ( − 3) = 0 ( − 1) ( + 1) ( − 1) ( + 1) + = 1 + = 1 1 1 − −1 −1 4 4

P.2 Exponent properties Exponent Rules a) Properties (p. 15) amn a= a mn+ m a − = am n an

− 1 a n = an 1 an = a−n a0 =1 (ab ) m= ab m m (amn ) = a mn* a  m a m   = b  b m

EX 3 a & c 3− 4 − 12 x y (− 3ab4 )(4 ab 3 ) 4x−2 y

b) Scientific notation – an extension of the exponent rules (can take any number and create it as a power of 10) a. .0004 = 4 x 10^-4 b. # always between 1 & 10

Divide 5.6 × 10 by 1.4 × 10

5.1 Exponential Functions and their graphs

I Exponential Functions The 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 ______!

EX4a EX4b 9 = 3 = 8

Transformations of exponentials : = 2 + 5 = −5 + 1

III The natural base

≈______know the first 5 digits minimum!

EX 6: use calculator to evaluate = at = −2,& = .25

EX 7a Graphing . . p. 384 for exact picture = 2 =

IV Application: Continuously compounded interest

Yearly = 1 + P = principal, annual interest rate r, compounded once a year

p. 386 N compoundings per year = 1 +

Continuous = (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)

5.4 use the one to one property to solve exponential equations = = 81 2 = 8 = 16

5.5 Modeling with Exponential equations 1) Exponential growth = b>0 2) Exponential decay model =

Decay – carbon dating p. 422 / Estimate the age of a newly discovered fossil win which the ratio of = carbon 14 to carbon 12 is . =

Graph of Gaussian model is the ______– used a lot in ______.

Standard normal is = / √

( )/, Ex4 SAT Scores = 0.0034 200 ≤ ≤ 800 where is the math score. Estimate the average!

Review of Topics for Exam 3

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