Unit 4 Rational Expressions

Mrs.Valen+neMathIII

4.1 Simplifying Rational Expressions•  Simplifying Rational Expressions

–  Expression in the form

–  Simplifying a rational expression is like simplifying any other fraction:

•  Factor •  Divide out common factors •  Simplify

–  Examples:

4.1 Simplifying Rational Expressions•  Simplifying a Rational Expression Containing a Trinomial

–  Factor both the numerator and the denominator –  Divide out the common factor. Simplify. –  State the simplified form with any restrictions on the variable

•  Excluded value – a value of a variable for which a rational expression is undefined

•  Denominator ≠0 •  Example:

x≠–3or2becausetheoriginaldenominatorwouldthenequal0.

wherex≠–3or2

4.1 Simplifying Rational Expressions•  Recognizing Opposite Factors

–  If the numerator and denominator are opposites, factor –1 from one of the terms.

–  Examples:

4.1 Simplifying Rational Expressions•  Using a Rational Expression

–  You are choosing between two wastebaskets that have the shape of the figures in the diagram. They both have the same volume. What is the height h of the rectangular wastebasket? Give your answer in terms of a.

V=πr2hforacylinderandV=l*w*hforarectangularprism

V=πa2(2a–8)basedonthecylinder

V=4a2*hbasedontheprism

πa2(2a–8)=4a2*hsamevolume

4.2 Multiplying and Dividing Rational Expressions•  Multiplying Rational Expressions

–  Multiply and divide the rational expressions using the same properties as numerical fractions.

–  In both cases, the property can be used.

–  Examples:

4.2 Multiplying and Dividing Rational Expressions•  Using Factoring

–  Sometimes, you may have to factor the polynomial before you can multiply and simplify.

–  Examples

4.2 Multiplying and Dividing Rational Expressions•  Multiplying a Rational Expression by a Polynomial

–  Follows the same rules –  Treat the polynomial as a fraction with 1 as the denominator –  Examples:

4.2 Multiplying and Dividing Rational Expressions•  Dividing Rational Expressions

–  When dividing rational expressions, take the reciprocal of the divisor, then multiply.

–  Examples:

4.2 Multiplying and Dividing Rational Expressions•  Dividing a Rational Expression by a Polynomial

–  Like with multiplication, treat the polynomial as a fraction over 1. –  Example:

4.2 Multiplying and Dividing Rational Expressions•  Simplifying a Complex Fraction

–  Complex fraction: a fraction with one or more fractions in the numerator, denominator, or both.

–  Example

4.3 Add & Subtract Rational Expressions•  Adding Expressions with Like Denominators

–  With like denominators, when adding fractions, combine the numerators. Leave the denominator the same.

–  Examples:

4.3 Add & Subtract Rational Expressions•  Subtracting Expressions with Like Denominators

•  With like denominators, when adding fractions, combine the numerators. Leave the denominator the same.

–  Then simplify. –  Examples

4.3 Add & Subtract Rational Expressions•  Adding Expressions with Different Denominators

–  With different denominators, you will need to find the least common denominator (LCD) of the two fractions (this is the least common multiple (LCM) of the two denominators)

–  Example

4.3 Add & Subtract Rational Expressions•  Subtracting Expressions with different denominators

–  Find the LCD and rewrite each fraction –  Simplify each numerator –  Subtract the numerators –  Simplify the fraction –  Example:

4.3 Add & Subtract Rational Expressions•  Using Rational Expressions

–  A certain truck gets 25% better gas mileage when it holds no cargo than when it is fully loaded. Let m be the number of miles per gallon of gasoline the truck gets when it is fully loaded. The truck drops off a full load and returns empty. What is an expression for the number of gallons of gasoline the truck uses?

4.4 Inverse Variation•  Identifying Direct and Inverse Variations

–  Direct Variation •  y = kx, where k ≠ 0 •  As x increases, y increases proportionally •  The graph is linear

–  Inverse Variation •  xy = k, y = k/x, or x = k/y ,where k ≠ 0 •  As x increases, y decreases proportionally •  The graph is hyperbolic

–  k is the constant of variation –  Graphs give rough information, but are not enough to identify

an inverse or direct variation by themselves.

4.4 Inverse Variation–  Examples: Is the relationship between the

variables direct variation, an inverse variation, or neither? Write the model that corresponds to the appropriate variation.

x y

2 15

4 7.5

10 3

15 2Asxincreases,ydecreases,andaplotverifiesthatthisislikelyaninverserela+onship.

(2)(15)=30(10)(3)=30(4)(7.5)=30(15)(2)=30Theconstantofvaria+onis30,sothemodelforthisinversevaria+onisxy=30.

x y

0.2 8

0.5 20

1.0 40

1.5 60

Directy=40x

x2.5

x2x1.5

x2.5

x2x1.5

4040

40

40

y/x x y

0.2 40

0.5 16

1.0 8

2.0 4

x2.5

x2x2

x0.4

x0.5x0.5

88

8

8

xyInverseY=8/x

4.4 Inverse Variation•  Determining an Inverse Variation

–  Suppose x and y vary inversely, and x = 4 when y = 12. What function models the inverse variation? What does the graph look like? What is y when x = 10?

xy=k(4)(12)=k

48=kxy=48ory=48/x

y=48/xy=48/10y=4.8

4.4 Inverse Variation•  Modeling an Inverse Variation

–  Your math class has decided to pick up litter each weekend in a local park. Each week there is approximately the same amount of litter. The table shows the number of students who worked each of the first four weeks of the project and the time needed for the pickup. What function models the data? How many students should there be to complete the project in at most 30 minutes each week?

n t

3 85

5 51

12 21

17 15

Numberofstudents=n+meinminutes=t

(3)(85)=255(12)(21)=252(5)(51)=255(17)(15)=255

(n)(t)isalmostalways255.Themodelisnt=255

nt=255;t=30n=?

n(30)=255

n=255/30

n=8.5

Thereshouldbeatleast9studentstocompletethejobinatmost30min.

4.4 Inverse Variation•  Using Combined Variation

–  Combined variation: when one quantity varies with respect to at least two others.

–  Joint variation: when one quantities varies directly with at least two others.

–  Equations: •  z varies jointly with x and y •  z varies jointly with x and y and inversely with w

•  z varies directly with x and inversely with the product wy

4.4 Inverse Variation–  Example: The number of bags of grass seed n needed to

reseed a yard varies directly with the area a to be seeded and inversely with the weight of w of a bag of seed. If it takes two 3-lb bags to seed an area of 3600ft2, how many 3-lb bags will seed 9000ft2?

4.4 Inverse Variation•  Applying Combined Variation

–  Gravitational potential energy PE is a measure of energy. PE varies directly with an object’s mass and its height in meters above the ground. Physicists use g to represent the constant of variation, which is gravity. A skateboarder has a mass of 58kg and a potential energy of 2273.6J at the top of a 4m halfpipe. What is the gravitational potential energy of a 65kg skateboarder on the same halfpipe?

PE=gmh

2273.6=g(58)(4)

2273.6=232g

9.8=g

PE=9.8mh

PE=9.8mh

PE=(9.8)(65)(4)

PE=2548J

4.4 Inverse Variation–  TheformulafortheIdealGasLawisPV=nRTwherePisthepressurein

kilopascals(kPA),Visthevolumeinliters(L),TisthetemperatureinKelvin(K),nisthenumberofmolesofgas,andR=8.314intheuniversalgasconstant.

•  Writeanequa+ontofindthevolumeintermsofP,n,r,andT.•  Whatvolumeisneededtostore5molesofheliumgasat350Kunderapressureof190kPA?

•  A10Lcylinderisfilledwithhydrogengastoapressureof5000kPA.Thetemperatureofgasis300K.Howmanymolesofhydrogengasareinthecylinder?

4.5 The Reciprocal Function Family•  Graphing an Inverse Variation Function

–  Reciprocal functions: •  Parent function: f(x) = 1/x, where x ≠ 0 •  General form:

•  Asymptotes are lines the graph approaches but does not touch.

– The vertical asymptote for a reciprocal function is x = h. – The horizontal asymptote for a reciprocal function is y =

k. –  Inverse variation function: f(x) = a/x, where x ≠ 0.

•  a determines stretches, compressions, and reflections on x-axis.

4.5 The Reciprocal Function Family–  Example: What is the graph of y = 8/x, x ≠ 0? Identify the x- and

y- intercepts and the asymptotes of the graph. Also state domain and range.

Therearenox-ory-interceptsastheasymptotesofthegrapharey=0andx=0.Thedomainisallrealnumbersexceptx=0andtherangeisallrealnumbersexcepty=0.

4.5 The Reciprocal Function Family•  Identifying Reciprocal Function Transformations

–  Branches •  Each part of the graph of a reciprocal function •  In Quadrants I and III when a is positive. •  In Quadrants II and IV when a is negative. •  All stretches (|a|>1)/shrinks(|a|<1) remain in the same

Quadrants. –  Example: For each given value of a, how do the graphs of

y = 1/x and y = a/x compare? What is the effect of a on the graph? a=6

a=0.25a=–6

4.5 The Reciprocal Function Family•  Graphing a Translation

–  Start by graphing the asymptotes. –  Then translate the graph. –  Draw the branches through these points.

4.5 The Reciprocal Function Family–  Example: What is the graph of ? Identify the

domain and range.

h=4k=6

Ver+calasymptote:x=4Horizontalasymptote:y=6

Usethepoints(1,1)and(–1,–1)fromtheparentfunc+ontoselectpointson

thisgraph.(1,1)+(4,6)(5,7)

(–1,–1)+(4,6)(3,5)

Drawthebranchesthroughthesepoints.

Thegraphismoved4unitsrightand6unitsup.

4.5 The Reciprocal Function Family•  Writing the Equation of a Transformation

–  If you know the asymptotes and the value of a of a reciprocal function, you can write the equation of the function.

–  Use the intersection of the new asymptotes to determine how the graph was translated (h and k).

–  Example: The graph below is a translation of y = 2/x. What is an equation for the function?

•  Theasymptotescrossat(–3,4).•  Thismeansthath=–3andk=4aretheasymptotes.

•  Therefore,theequa+onis

4.5 The Reciprocal Function Family•  Using a Reciprocal Function

–  The rowing club is renting a 57-passenerg bus for a day trip. The cost of the bus is $750. Five passengers will be chaperones. If the students who attend share the bus cost equally, what function models the cost per student C with respect to the number of students n who attend? What is the domain of the function? How many students must ride the bus to make the cost per student no more than 20?

Tosharethecostequally,divide$750byn.

Withacapacityfor57peopleand5chaperones,thereareamaximumof52students.Domainis1≤x≤52.

Graph byplokngthefunc+onandy=20.Determinewheretheyintersect.

Atleast38studentsmustridethebus.

4.6 Graphing Rational Expressions•  Finding Points of Discontinuity

–  Rational function: a function containing a rational expression. –  If a function has a polynomial in its denominator, its graph has a

gap at each zero of the polynomial. •  One-point hole •  Vertical asymptote

–  The domain is all real numbers except the zeros of the denominator.

–  These graphs are discontinuous graphs (graphs with a break in the domain)

–  If there are no values that make the denominator zero, the graph is a continuous graph.

4.6 Graphing Rational Expressions•  Point of Discontinuity

–  If a is a real number of which the denominator of a rational function f(x) is zero, then a is not the domain of f(x). The graph of f(x) is not continuous at x = a and the function has a point of discontinuity at x = a.

–  The graph of has a removable discontinuity at x = -2. The hole in the graph is called a removable discontinuity because you could make the function continuous by redefining it at x = -2 so that f(-2) = 1.

–  The graph of has a non-removable discontinuity at x = 2. There is no way to redefine the function at 2 to make the function continuous.

4.6 Graphing Rational Expressions–  When you are looking for discontinuities, remember to factor

the denominator. –  The discontinuity caused by (x – a)n is removable if the

numerator also has (x – a)n as a factor. –  Example: What are the domain and points of discontinuity of

each rational function? Are the points of discontinuity removable or non-removable? What are the x- and y- intercepts?

Domain:allrealnumbersexceptx=1,3Thepointsofdiscon+nuityarenon-removablex-intercept:(-3,0)y-intercept:(0,1)

Domain:allrealnumbersexceptx=4Thepointofdiscon+nuityisremovablex-intercept:(-1,0)y-intercept:(0,1)

4.6 Graphing Rational Expressions•  Finding Vertical Asymptotes

–  If a rational function has a non-removable discontinuity at x = a, the graph of the rational function will have a vertical asymptote at x = a.

–  The graph of the rational function f(x) = P(x) / Q(x) has a vertical asymptote at each real zero of Q(x) if P(x) and Q(x) have no common zeros. If P(x) and Q(x) have (x – a)m and (x – a)n as factors, respectively and m < n, then f(x) also has a vertical asymptote at x = a.

–  Example: What are the vertical asymptotes for the graph of Since2and3arethezerosofthedenominatorandneitherisazeroofthenumerator,thelinesx=2andx=3arever+calasymptotes.

Since–3iszerosofthedenominatorwithnomatchinthenumerator,x=–3isaver+calasymptote.2isazeroofbothnumeratoranddenominator,soitisahole.

4.6 Graphing Rational Expressions•  Finding Horizontal Asymptotes

–  To find the horizontal asymptotes of the graph of a rational function, compare the degree of the numerator m to the degree of the denominator n.

•  If m < n, the graph has a horizontal asymptote y = 0 (x-axis) •  If m > n, the graph has no horizontal asymptote •  If m = n, the graph has horizontal asymptote y = a/b where a

is the coefficient of the greatest degree in the numerator and b is the coefficient of the greatest degree in the denominator.

–  Examples: What is the horizontal asymptote for the rational function?

m=n;y=2/1ày=2 m<n;y=0 m>n;none

4.6 Graphing Rational Expressions•  Graphing a Rational Function

–  By finding all intercepts and asymptotes, you can get a reasonable graph.

–  Sometimes, a few extra points will be necessary. –  Example: What is the graph of the rational function

Horizontalasymptote:m=n;y=1/1ày=1

Ver+calasymptotes:x=2,x=–2

x-intercepts(rootsofnumerator):(3,0)&(–4,0)

y-intercept(x=0):(0,3)

x y

–3 –6/5

–1 4

1 10/3

4 2/3

Morepoints:

4.6 Graphing Rational Expressions•  Using a Rational Function

–  You work in a pharmacy that mixes different concentrations of saline solutions for its customers. The pharmacy has a supply of two concentrations, 0.5% and 2%. The function below gives the concentration of the saline solution after adding x mL of the 0.5% solution to 100mL of the 2% solution. How many mL of the 0.5% solution must you add for the combined solution to have a concentration of 0.9%?

Graphthefunc+onunderY1and0.009underY2onthecalculator.Findthepointofintersec+on.Itwilltake275mLofthe0.5%solu+on.

4.7 Solving Rational Equations•  Solving a Rational Equation

–  A rational equation contains at least one rational expression •  Find the Least Common Denominator (LCD) •  Multiply both sides by the LCD •  Simplify and Solve •  Check for extraneous solutions •  Write the solutions

–  Example:

4.7 Solving Rational Equations•  Using Rational Equations

–  A flight across the US takes longer east to west than it does west to east. Assume that winds are constant in the eastward direction. When flying westward, the headwind decreases the airplane’s speed. When flying eastward, the tailwind increases its speed. The time for a round trip between Chicago and San Francisco (1850mi one way) is 7.75hr. If the airplane cruises at 480 mi/h, what is the speed of the wind?

Letx=windspeedRate*+me=distance,so+me=distance/rate

Dist Rate Time

WesttoEast 1850 480+x

Easttowest 1850 480–x

Speedisposi+ve.West-to-eastwindspeedisabout35mi/h

4.7 Solving Rational Equations•  Using a Graphing Calculator to Solve a Rational Equation

–  To solve using the graphing calculator, plot each side of the equation and see where they intersect.

–  Example:

Thesolu+onisx=0