chapter 9: rational equations and functions direct, inverse, & joint variation ex: the variables...

Chapter 9: Rational Equations and Functions

Chapter 9: Rational Equations and Functions Assignment Sheet Date Topic Assignment Completed 9.1: Inverse and Joint Variation pg. 537 # 21 -‐ 43 odd, 54 9.2.1: Graphing Simple Rational

Function pg. 543 # 23-‐29 Be sure to state the domain and range

9.2.2: Graphing Simple Rational Function

pg. 543 # 20-‐22, 32-‐38 Be sure to state the domain and range

9.3.1: Graphing Rational Functions

pg. 550 #12-‐18 (even), 20-‐25, 28, 29, 31, 34

9.3.2: Graphing Rational Functions with Slant Asymptotes

Worksheet

Review 9.1-‐9.3 pg. 576 #1-‐15 pg. 579 #1-‐15

Quiz 9.1-‐9.3 None 9.4: Multiplying and Dividing

Rational Expressions pg. 558 #21, 23, 27, 29-‐49 (odd)

9.5.1: Addition and Subtraction of Rational Expressions pg. 566 #26-‐37

9.5.2: Addition, Subtraction, and Complex Fractions pg. 566 #38-‐40, 42-‐45

9.6: Solving Rational Equations pg. 571 #22-‐30 (even), 31, 34-‐50 (even)

Review 9.4-‐9.6 9.4-‐9.6 Review Review 9.4-‐9.6 9.4-‐9.6 Review Review 9.4-‐9.6 9.4-‐9.6 Review Test 9.4-‐9.6 None

9.1 Direct, Inverse, & Joint Variation

Ex: The variables x and y vary directly and = =12 when 4y x .

a.) Write an equation that relates x and y.

b.) Find the value of y when x = 10.

Ex: Tell whether the data show direct variation. If so, write an equation relating x and y.

Direct variation? _________________

If yes, what’s the equation? _________________ If yes, find the price for a 28 inch gold chain. ___________

Direct variation? _________________

If yes, what’s the equation? _________________

If yes, find the price for .8 carats of loose diamonds._________

Direct Variation

Two variables x and y show direct variation provided y = kx and k ≠ 0. The nonzero constant k is called the constant of variation and y is said to “vary directly with x.” The graph of y = kx is a line through the origin.

Ex: The variables x and y vary inversely, and = =8 when 3y x .

a.) Write an equation that relates x and y.

b.) Find the value of y when x = - 4.

Ex: Tell whether x and y show direct variation, inverse variation, or neither.

Given Equation Rewritten Equation Type of Variation

a) =5y x

b) = + 2y x

c) = 4xy

Word Problems 1. The speed of the current in a whirlpool varies inversely with the distance from the whirlpool’s center. The Lofoten Maelstrom is a whirlpool located off the coast of Norway. At a distance of 3 kilometers (3000 meters) from the center, the speed of the current is about 0.1 meter per second. Describe the change in the speed of the current as you move closer to the whirlpool’s center.

Inverse Variation

Two variables x and y show inverse variation if they are related as follows:

As above, the nonzero constant k is called the constant of variation and y is said to “vary inversely with x.”

2. The table below compares the wing flapping rate r (in beats per second) to the wing length l (in centimeters) for several birds. Do these data show inverse variation? If so, find a model for the relationship between r and l.

Ex: z varies jointly with x and y. Given that = = =3, 8 and 6.x y z a.) Write an equation relating x, y, and z.

b.) Then find z when x = -2 and y = 4.

Ex: Write an equation for the given relationship.

a) x varies inversely with y and directly with z. __________________

b) y varies jointly with z and the square root of x. __________________

c) w varies inversely with x and jointly with y and z. __________________

d) y varies inversely with the square of x. __________________ e) z varies directly with y and inversely with x. __________________

Homework: p. 537 – 538 # 21 - 43 odd, 54

Joint Variation

Joint variation occurs when a quantity varies directly as the product of two or more other quantities. For instance, if z = kxy where k ≠ 0, then z varies

jointly with x and y. Additionally, if , z varies directly with y and

inversely with x.

Inverse variation?

______________

If yes, what’s the equation?

______________

9.2.1 Graphing Simple Rational Functions in the form: y = ax − h

+ k

xy

1= graphs a hyperbola

x -4 -3 -2 -1 0 1 2 3 4 y

Vertical Asymptote:

Horizontal Asymptote: Domain: Range:

132 −+−=x

y

x -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 y Vertical Asymptote:


213 +−

=x

y

Vertical Asymptote: Horizontal Asymptote:

Domain: Range: Homework: p. 543-544 #23-29 (Be sure to state the domain and range)

Rational Functions in the form khxa

y +−

= have graphs that are hyperbolas

with _____________ asymptotes _________ and ______________ asymptotes ________________

x y

9.2.2 Graphing Rational Functions in the Form y = ax + b

cx + d

421−+=xx

y

x -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 y Vertical Asymptote:


332+−=xx

y Vertical Asymptote:


Rational Functions in the form dcxbax

y++= have graphs that are hyperbolas

with _____________ asymptotes _________________________ and ______________ asymptotes ________________

x y

432

++=

xx

y

Vertical Asymptote:

Horizontal Asymptote: Domain: Range: Homework: p. 543 #20-22 all, 32-38 all (Be sure to state the domain and range)

x y

9.3.1 Graphing General Rational Functions Graphs of Rational Functions: Let p x( ) and q x( ) be polynomials with NO common factors other than 1.

The graph of f x( ) = p x( )q x( ) has the following characteristics:

Example 1: Graph y =2x2

x2 − 9

a) Find the x-‐int (zeros of the numerator) x-‐int: ______________________

b) Find the VERTICAL asymptotes (zeros of the denominator)

Vertical Asymptote(s): ___________

c) Find the HORIZONTAL asymptotes (compare the degree of the numerator and the degree of the denominator)

Horizontal Asymptote(s):_________

d) Use a table and above information to graph

x y

Example 2: Graph y = 4x2 + 2

a) Find the x-‐int (zeros of the numerator) x-‐int: ___________________


Vertical Asymptote(s): ___________ c) Find the HORIZONTAL asymptotes (compare the degree of the numerator and

the degree of the denominator)

Horizontal Asymptote(s):_________ d) Use a table and above information to graph

Example 3: Graph y =x2 + 3x − 4x − 2







x y

x y

Example 4: Graph y = xx2 +1







Example 5: Graph y =2x3

x3 −1



Vertical Asymptote(s): ___________

c) Find the HORIZONTAL asymptotes (compare the degree of the numerator and the degree of the denominator)


x y

x y

Example 6: Graph y =x2 − 3x − 4x − 2



Vertical Asymptote(s): ___________ c) Find the HORIZONTAL (compare the degree of the numerator and the degree of

the denominator)


Homework: Pg 550-551 #12-18 even, 20-25 all, 28, 29, 31, 34

x y

9.3.2: SLANT ASYMPTOTES Use long division to determine the slant asymptote for each problem below. Then graph. Your complete graph should contain: 1) slant asymptote (when degree of numerator > degree of denominator BY 1) 2) vertical asymptote or horizontal asymptote 3) x-‐ and y-‐intercepts listed as coordinate pairs 4) maximum and minimum values where appropriate listed as coordinate pairs 5) a minimum of 3 points per branch of the graph

1) f (x) = x2 − 2x

2) f (x) = x2 − 3x + 5x − 2

3) f (x) = x2 − 6x + 8x + 2

4) f (x) = 3x2 + 2x − 5x − 2

5) f (x) = 3x2 − 2x −17x − 3

6) f (x) = 2x2 + 9x + 22x + 3

7) f (x) = 2x2 + 3x −1x +1

8) f (x) = 2x2 + 3x −1x −1

9.3.2: Graphing Rational Functions Homework Worksheet Graph each of the following. Be sure to include: a) Slant asymptote b) Vertical asymptote or Horizontal asymptote c) x-‐ and y-‐intercepts listed as coordinate pairs d) Maximum and minimum values where appropriate listed as coordinate pairs e) A minimum of 3 points per branch of the graph

1) f (x) = 2x2 − 5x + 5x − 2

2) g(x) = 2x3 − x2 − 2x +1x2 + 3x + 2

3) t(x) = 3x2 +1x2 + x + 9

4) f (x) = 4x − 2( )3

5) f (x) = x2 +1x

9.4 Multiplying and Dividing Rational Expressions Factor the following perfect cube binomials: 1.) 1253 −x 2.) 18 3 +x Which of the following can you simplify and why?

1.) )3(3)3(

2 ++xxx 2.)

11

−+xx 3.)

1+xx

Simplifying, Multiplying, and Dividing Polynomials

• In order to fully simplify a polynomial you must factor every numerator and denominator and then see what you can divide out.

1.) 2

43

32

32

1810

26

yyx

yxyx ⋅ 2.)

165

2

2

−−−

xxx

3.) 4124

2

2

−−−

xxx 4.)

xxx

xxxx

3143

123273 2

2

3 +−⋅−−

−

5.) )1416(1643 2

3 ++⋅−

− xxxx

Remember you NEVER DIVIDE fractions you always INVERT the fraction and MULTIPLY!

1.) 6

384

32

2

−++÷

− xxxx

x

2.) )32(6

376 22

2

xxxxx +÷−+

3.) 5259)53(

5

2

+−÷−⋅

+ xxx

xx

4.) 127

1039

101

2

2

23 +++⋅

+−÷

+ xxx

xx

xx

Homework: pg. 559 # 16-48 even Skydiving

A falling skydiver accelerates until reaching a constant falling speed called the terminal velocity. Because of air resistance, the ratio of the skydiver volume to his or her cross-sectional surface area affects the terminal velocity.

A.) The diagram shows a simplified geometric model of a skydiver with maximum cross sectional surface area. Use the diagram to write a model for the ratio of volume to cross sectional surface area for a skydiver.

Body Part Volume Cross sectional Surface Area

Arm or leg

Head

Trunk

=aSurfaceAre

Volume

B.) Use the result from part A to compare the terminal velocities of two skydivers: One who is 60 inches tall and one who is 72 inches tall.

9.5 Addition and Subtraction of Complex Fractions

1.) =−21

32 2.) =+

xx 35

34 3.) =

+−

+ 34

32

xxx

4.) 56x2

+x

4x2 −12x

5.) x +1

x2 + 4x + 4−

3x2 − 4

6.) 4x3x3

+x

6x3 + 3x2

7.) x +1

x2 + 6x + 9−

1x2 − 9

8.) 2x + 5

x2 + 6x + 9+

xx2 − 9

+1

x − 3

9.) 5

2(x +1)−

12x

−3

2(x +1)2

Homework: Pg. 563 #26-37 Simplify the following:

1.) =

4321

2.) =+31

211

In order to Simplify COMPLEX fractions.

1.) Simplify the numerator and denominator separately. 2.) Simplify the fraction by multiplying by the reciprocal.

1.)

xx

x2

21

22

++

+

2.)

13

41

43

++

−

−

xx

x

3.)

x

x11

11

−

+

4.)

51

51

52

254

2

−+

+

++

−

xx

xx

Using Complex Fractions: Monthly payment for items that are financed such as a car, house, or steno system will vary with the rate of interest, the amount of time, the loan and the length of the loan. The formula for a given monthly payment is the following:

⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

+−

= t

r

r

PM 12

121

11

12 P= r= t=

A.) Find the monthly payment when P= $20, 000, r = 0.75, and t = 1 B.) Find the monthly payment for financing a $12,500 car for 3 years using the

following interest rates.

a.) 8% b.) 12% c.) 18% For each of the interest rates in part b, how much more would you be paying for the car?

Homework: pg. 566 #38-40, 42-45

9.6 Solving Rational Equations Solve the following equation by getting rid of ALL the fractions:

625

3=+ xx

In order to get rid of the fractions you multiplied by the Least Common Denominator. To solve rational equations you want to get RID of the fractions by multiplying all pieces of the equations by the LCD. Equations with one solution:

xx11

254 −=+ LCD: Check:

Equations with an Extraneous Solution:

2107

25

−+=

− xxx

LCD: Check:

Equations with Two Solutions:

31

12114

2 +−

=++

xxx

LCD: Check:

Solving by cross multiplying:

112

2 −=

− xxx LCD: Check:

Solve the following and be sure to check for extraneous solutions:

1.) xxx4

123 =+

− Check:

2.) 44

442

−=

−−

xxx Check:

3.) 254

2102 −

=+− xxxx

Check:

4.) 154

15

+−=

+ xxx Check:

5.) 316

16412

2 +−

=−+

xxx Check:

Homework: Pg. 571-572 #22-30 even, 31, 34-50 even

chapter 9: rational equations and functions direct, inverse, & joint variation ex: the variables...

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