Name: Block:

Unit 7: Rational Functions

Day 1 Multiplying & Dividing Rational Expressions Day 2 Adding & Subtracting Rational Expressions Day 3 Review Days 1 & 2 for Quiz

Day 4 Quiz: Operations with Rational Functions

Day 5 Solving Rational Equations Day 6 Graphing Rational Functions Day 7 Direct & Inverse Variation Day 8 Test Review Day 9 Test: Unit 7 Rational Functions

Tentative Schedule of Upcoming Classes

Day 1 A Monday, March 28 Day 1 Notes: Multiplying & Dividing

Rational Expressions B Tuesday, March 29

Day 2 A Wednesday, March 30 Day 2 Notes: Adding & Subtracting

Rational Expressions B Thursday, March 31

Day 3 A Friday, April 1 Day 3: Review Operations with Rationals; Skills Check B Monday, April 4

Day 4 A Tuesday, April 5 Day 4: Quiz: Operations with

Rationals B Wednesday, April 6

Day 5 A Thursday, April 7 Day 5 Notes: Solving Rational Equations B Friday, April 8

Day 6 A Monday, April 11 Day 6 Notes: Graphing Rational Functions B Tuesday, April 12

Day 7 A Wednesday, April 13 Day 7 Notes: Variation B Thursday, April 14

Day 8 A Friday, April 15 Review for Test: Unit 7 B Monday, April 18

Day 9 A Tuesday, April 19 Test: Unit 7 B Wednesday, April 20

Absent?

See Ms. Huelsman AS SOON AS POSSIBLE to get work and any help you need.

Notes are always posted online on the calendar. (If links are not cooperative, try changing to “list” mode)

Handouts and homework keys are posted under assignments

You may also email Ms. Huelsman at Kelsey.huelsman@lcps.org with any questions!

____

Need Help?

Ms. Huelsman and Mu Alpha Theta are available to help Monday, Tuesday, Thursday, and Friday mornings in L506 starting at 8:10.

Ms. Huelsman is in L402 on Wednesday mornings.

Need to make up a test/quiz?

Math Make Up Room schedule is posted around the math hallway & in Ms. Huelsman’s classroom

Day 1: Multiplying and Dividing Rational Expressions In these notes we will introduce, multiplying and dividing rational expressions So we can eventually GRAPH rational functions. What IS a RATIONAL FUNCTION?

A function of the form f(x)= ( )( )

p xq x

where q(x) ≠ 0 This is just a formality

Both the numerator and the denominator are _________________________________. Simplified form: a rational expression is simplified if its numerator and denominator have NO common factors (other than 1± ).

Simplify Rational Expressions Property ac a c abc b c c

•= =

•

To simplify Rational Functions you need to FACTOR and CANCEL

Examples: 15 3 5 365 13 5 13

•= =

• 2

4 12 4( 3) 42 15 ( 5)( 3) 5

x xx x x x x

+ += =

− − − + −

When can we “cancel”? How do we look for “ones in disguise”? Simplify the following.

1. 63

x + 2. 3 63

x + 3. 2 37x x

x−

4. 34 7x

x− 5. ( 1) ( 1)

( 1)x x x

x+ − +

+ 6. ( 1)

( 1)x x

x− ++

Practice Problems: Simplify each expression.

1. 2

2

7x xx+

2. 2

2

8 162 24

x xx x

− ++ −

3. 2

416

xx+−

Multiplying Rational Expressions Property a c acb d bd• = Simplify if possible.

You can cancel any common factors that are in both the numerator & denominator!

To Multiply Rational Expressions you need to FACTOR each expression & CANCEL common factors

4. 2 3 5

4 4

5 273 15x y xxy x y

•

5. 2

2 2

2 10 325 2

x x xx x− +

•−

6. 2 2

2 2

20 5 3 416

x x x xx x x− + −

•− −

Dividing Rational Expressions Property a c a d adb d b c bc÷ = • = Simplify if possible.

To Divide Rational Expressions you need to Flip (use reciprocal) the second term and multiply the rational expressions.

7. 2 2

2

4 21 3 705 15 100

x x x xx x− − + −

÷+ −

8. 2

22

3 13 10 (3 2 )6

x x x xx

+ −÷ −

9. Complex Fraction

2

2

2

2

6 272 2

14 45

x xx x

x xx

− −+

− +

Day 2: Adding and Subtracting Rational Expressions In these notes we will add and subtract rational expressions (which means we must learn to find common denominators for them)

So we can SOLVE PROBLEMS that have rational functions

Warm-Up – Add or subtract the following.

1. 7

1073+ 2.

83

41+ 3.

72

35+

4. 58

54− 5.

56

38− 6.

83

29+

−

7. Describe how you found least common denominators (LCD). List any factors that can cancel. If nothing can cancel, write “N/A”

Expression Which factors cancel?

Expression Which factors cancel?

1 4

2 5

3 6

x(x+5)(x+5)

x+55

16x4

(x+3)(x+5)(x+3)

3+(x+2)(x+2)

4ab4a

Now let’s apply this to RATIONAL EXPRESSIONS. Adding and Subtracting with LIKE Denominators – Life is GOOD! Does the denominator change when you add or subtract?

1. 12 25 5x x

− = 2. 3 1

2 5 2 5x

x x+ =

+ +

3. 2 22 2

x xx x

+− =

− − 4.

4 2 73 3

x xx x

+− =

+ +

Careful with the minus! Adding and Subtracting w ith UNLIKE Denominators – ONE MORE STEP!

1. Find a LCD – Factor each denominator (One of EACH factor to the highest degree that it occurs)

2. Multiply to create the common denominator 3. Add or Subtract numerators 4. Simplify if possible. (Factor and Cancel)

1. 4

2 5x

x x+

+ +

LCD: ________________ Hint: Factor first!

2. 2

4 2 73 9

x xx x

+−

+ −

LCD: ________________

Properties

Addition: a b ad bc ad bcc d cd dc cd

++ = + =

Subtraction:

a b ad bc ad bcc d cd dc cd

−− = − =

Hint: Factor first!

3. 2 2

5 73x x x

+−

LCD: ________________ Hint: Factor first!

4. 2

2 2

2 21 3 3

xx x

++ +

LCD: ________________ Hint: Factor first!

5. 2

2 13 15 4 5

x xx x x

− −−

− − −

LCD: ________________ Hint: Factor first!

6. 2

5 4 98 5 24

x xx x x

−+

+ + −

LCD: ________________

Review: Operations with Rational Expressions - Classwork Simplify each expression: (Remember to FACTOR completely!) 1. 324 81−x

2. 3 3125 729+x y

3. 3729 − x

4. 6 10225 81−x y

5. 23 9 54+ −x y xy y

6. 2 216 25−x y

Assuming no denominator equals zero, perform the indicated operation. Show all work!! Completely simplify each expression.

1. 2

2

8 15 9 451 22 5 3

+ + − −÷

−+ −x x x

xx x

2. 2

2 2

4 15 9 2 18 10 3 4

+ + +•

+ + +x x xx x x x

3. 2

2

5 245 4 352 32 11 21

− +−

+− −x x

xx x

4. 2 2

2 320 7 12

+− − + +x x x x

Assuming no denominator equals zero, perform the indicated operation. Show all work!! Completely simplify each expression.

1. 2

2 2

6 34 3 2

+ −•

+ + − −x x xx x x x

2. 2 2

2 2

16 9 6 3 312 10 8 3 2 8

− + −•

+ − + −x x x

x x x x

3. 2 1 2 1

1 1+ −

−− +

x xx x

4. 2

2

12 3 3 436

− +−

−− −x x

xx x

5. 2 215 12 15 3 12

36 45 24 30− + −

÷+ +

x x x xx x

6. 2

6

542

57

−

−

mm

mm

7. 4 2

3 2

4 47 8−

− −d d

d d d

8. 2

16 2416

++− xx

Operations with Rational Functions: Quiz Review Homework Perform the indicated operation. Show all work!! Put your answer into simplest form. 1.

4x6x2

15x520x5

−+

•+−

2.

12x3

3x12

−•

−

3.

5xy36

yx48 22

÷

4.

2x3

21+

5.

1xx3

1xx

2 −+

−

6.

2x6

3xx

+−

+

7.

x2x8

2xx2

2 +−

+

8.

1xx3

1xx2

2

+

−

9.

)1x(26x5x2

−++

•3x1x2

+−

10.

2

7 2 12 2

xx x x

+−

+ − +

11.

2

6 123 5 6

x xx x x

+−

+ + +

12.

1432

12413

22 +++

•−−

+xx

xxx

x

13. 2 2

2

2 3 21

x x x xx x

− − +•

−

14. 2

22

13 40 ( 5 24)2 15

x x x xx x− +

− −− −

÷

15. 2 23 2

5 5x x x xx x− +

−+ +

16. 2 2

2 3

6 2 88 12 8

x x x xx x x

+ + −•

+ + −

Day 5: Solving Equations with Rational Functions In these notes we will get rid of denominators and solve rational equations So that we can find where two rational functions intersect Steps: 1) Get rid of all denominators.

A) If you have one rational expression on each side: cross-multiply. B) If you have more than one rational expression on each side then find LCD of all rational expressions and multiply all terms by LCD.

2) Solve for x. 3) Check for extraneous solutions! (denominator cannot equal 0)

Solve by cross multiplying – only if you see FRACTION = FRACTION Before we start, what x-values will make these functions undefined (denominator = 0)? That means that if we get these x-values as solutions, they are “extraneous”

7 11

2 2 10x x=

− −

Before we start, what x-values will make these functions undefined (denominator = 0)? That means that if we get these x-values as solutions, they are “extraneous”

4x1

x4x3

2 +=

+

Before we start, what x-values will make these functions undefined (denominator = 0)? That means that if we get these x-values as solutions, they are “extraneous”

1

2 5 11 8x

x x=

+ +

Solve by using the LCD. Multiply each term by the LCD to eliminate denominators. Examples:

43

325

21

+=+− xx

Find LCD = _____________________

xx12

213=− Find LCD = _____________________

1x54

1xx5

+−=

+ Find LCD = _____________________

Check the denominator for an extraneous solution!! Remember, the denominator cannot equal 0.

8 315x x

− =−

Find LCD = _____________________

14x

62x2x3

2 +−

=−−

Find LCD, factor both denominators

LCD = _____________________

2

2

6 8 43 9 3

x xx x x

= −− − + Find LCD, factor all denominators

LCD = _____________________

Day 6: Investigating Rational Functions In these notes we will define and investigate horizontal and vertical asymptotes So that we can graph rational functions Domain Restrictions based on an equation: 1. Dividing by zero is undefined: a denominator can NEVER be equal to zero. 2. The square root of a negative number does not exist . . . we NEVER put a negative number under a square root (unless we are dealing in complex numbers). – This was in Unit 6!

Find the domain of 1( )

2f x

x=

+ Find the domain of 2

1( )4 5

f xx x

=+ −

Think: what would make the denominator = 0?

Having a DENOMINATOR with x and a function that is a RATIO introduces new twists….. We will have ASYMPTOTES.

The graph of a rational function is called a HYPERBOLA. This graph is made of symmetrical parts called branches. Investigation: Match the function to its graph. Remember that numerators and denominators should have parentheses around them. Discuss how the function definitions are different. How do these differences affect their graphs?

_____1. 1( )f xx

=

_____2. 1( )

2f x

x=

−

_____3. 1( ) 3

2f x

x= +

−

−4 −3 −2 −1 1 2 3 4

−4

−3

−2

−1

1

2

3

4

x

y

−4 −3 −2 −1 1 2 3 4

−4

−3

−2

−1

1

2

3

4

x

y

−4 −3 −2 −1 1 2 3 4 5 6 7

4

−3

−2

−1

1

2

3

4

5

6

x

y

A.

B.

C.

Some rational functions vocabulary:

Rational Function – written in the form f(x) = ( )( )

p xq x

where p(x) and q(x) are polynomials

Asymptote – a line that a graph approaches more and more closely.

Graphed as a dashed line.

1) Vertical Asymptote – occurs when the denominator equals zero

x = # (there can be more than one VA)

2) Horizontal Asymptote – for now we will use our calculator to help – but it’s good to know these! a) If the degree of the numerator < the degree of the denominator,

ayx

=

khx

ay +−

=

y = 0 or y = k is the horizontal asymptote.

b) If the degree of the numerator = the degree of the denominator,

y = lead coefficient of the numeratorlead coefficient of the denominator

is the horizontal asymptote.

c) If the degree of the numerator > the degree of the denominator,

you will NOT have a Horizontal asymptote … you will look at this more in future math courses.

3) Removable Discontinuity (Hole in the graph) – occurs when the numerator and denominator have a common factor that cancels. Your table will read “ERROR” as the y-value, but you can find the y-value by plugging in the x-value of the factor that cancels.

Steps to Graphing a Rational Function:

1. Identify and draw the asymptotes (as dashed lines)

2. Plot 2 points on the left and right side of the vertical asymptotes (use easy numbers).

3. Draw the branches of the hyperbola through the plotted points.

Use your calculator to graph the function. Find integer ordered pairs off the table on your calculator.

1. Graph 4yx

= 2. Graph 3 21

yx−

= +−

V.A. ____________ table will say “undefined” V.A.____________ table will say “undefined” H.A. ____________ H.A. ____________ Domain: Domain: Range: Range:

5. 9

2)( 2

2

−=

xxxf 6.

444)( 2

2

−−

=xxxf

V.A. _________ H.A. _________ V.A._________ H.A.__________ Domain: Domain: Range: Range:

−10 −8 −6 −4 −2 2 4 6 8 10

−10

−8

−6

−4

−2

2

4

6

8

10

x

y

−10 −8 −6 −4 −2 2 4 6 8 10

−10

−8

−6

−4

−2

2

4

6

8

10

x

y

−10 −8 −6 −4 −2 2 4 6 8 10

−10

−8

−6

−4

−2

2

4

6

8

10

x

y

−10 −8 −6 −4 −2 2 4 6 8 10

−10

−8

−6

−4

−2

2

4

6

8

10

x

y

Remember rules for Horizontal Asymptotes – even though we can use the calculator, let’s look at this rule:

If the degree of the numerator = the degree of the denominator,

y = lead coefficient of the numeratorlead coefficient of the denominator

is the horizontal asymptote.

Example 5: 9

2)( 2

2

−=

xxxf

What was the horizontal asymptote (see last page)? Can you see why, looking at the equation?

Example 6: 444)( 2

2

−−

=xxxf

What was the horizontal asymptote (see last page)? Can you see why, looking at the equation?

PRACTICE: Can you identify asymptotes? Use your calculator to graph each function

1. Which are asymptotes for the function 3yx−

= ?

x = -3 y = -3 x = 0 y = 0 no asymptotes

2. Which are asymptotes for the function 1

3y

x=

− ?

x = 3 y = 3 x = 0 y = 0 no asymptotes 3. Which are asymptotes for the function 3y x= − ? x = 3 y = -3 x = 0 y = 0 no asymptotes

4. Which are asymptotes for the function 1 2

3y

x= +

− ?

x = 3 y = 3 y = 0 y = 2 no asymptotes

5. Which are asymptotes for the function 3 1

4xy

x−

=+ ?

x = - 4 x = 4 y = 0 y = 3 no asymptotes

−10 −8 −6 −4 −2 2 4 6 8 10

−10

−8

−6

−4

−2

2

4

6

8

10

x

y

−10 −8 −6 −4 −2 2 4 6 8 10

−10

−8

−6

−4

−2

2

4

6

8

10

x

y

PAY ATTENTION….THESE ARE REALLY UNIQUE!!!!

7. 234)( 2

2

−−++

=xxxxxf Look at the graph…

How many vertical asymptotes do you see? Look at the table… “ERROR” is y-value for two different x-values: -1 and 2. Are there two vertical asymptotes? NO! There is ONE vertical asymptote… Which x-value is the vertical asymptote? What is causing the other “ERROR” in the table? How can we find the actual y-value when the table says error? What is the domain? What is the range? Include the asymptote and the hole Include the asymptote and the hole

10. 2

2

9( )8 15

xf xx x

−=

+ + Look at the graph…

How many vertical asymptotes do you see? Look at the table… “ERROR” is y-value for two different x-values: -3 and 5. Are there two vertical asymptotes? NO! There is ONE vertical asymptote… Which x-value is the vertical asymptote? What is causing the other “ERROR” in the table? How can we find the actual y-value when the table says error? What is the domain? What is the range? Include the asymptote and the hole Include the asymptote and the hole

Day 7: Types of Variation

In these notes we will define and investigate types of variation. So that we can apply the process to real-world problems. How to Solve a VARIATION Problem

1. Determine which formula to use. 2. Substitute values. Solve for a (the constant of variation) 3. Find an equation that relates to the variables by substituting a into the formula

from Step 1 Understanding VARIATION

Direct Variation and Cookies Inverse Variation and Cookies You are bringing cookies for your class. You want each person to have 2 cookies (the constant of variation). The number of cookies you bring will depend on the number of people!! y = 2x where x = # of people and y = # of cookies you will bring More people? More cookies!! Y will always be 2 times x!!

You only have 2 cookies to eat (the constant of variation). You are a very generous person. If someone asks you to share, you do. If more people ask you to share, you do. The amount of cookie you will have left is inversely proportional to the number of people sharing the cookie. 2 total: you each get 1 cookie

4 total: you each get 21

cookie

y = x2

where x = # of people

and y = the amount of cookie for each person More people? Less cookie per person!! The product of x and y will always be 2!

Direct Variation Inverse Variation Joint Variation

y ax= ayx

= y axz=

Direct Variation: (This type is from Algebra 1) A direct variation is a function that can be represented in the form y = kx

• Say: “y varies directly as x”. • The graph passes through the origin.

1. Determine whether equation equation below represents a direct variation. If so, identify the constant of variation.

a. 2 3 0x y− = b. 5x y+ = − c. 5xy =

2. Given that y varies directly as x, write a direct variation equation to find the missing value. If x = 24 and y = 64, find y when x = 15. Find Equation Find y when x = 15

Inverse Variation

An inverse variation is a function that can be represented in the form:kyx

=

• Say: “y varies inversely with x” • The graph will be a rational function.

4. Determine whether the variables show direct variation, inverse variation, or neither.

a. 3y x= b. 0.25xy = c. y = x – 5 d. 9rs

=

1. The variables x and y vary inversely, and y = 15 when x = 13

. Write an equation that relates x

and y. Then find y when x = -10. Find Equation Find y when x = -10

2. The number of songs that can be stored on an MP3 player varies inversely with the average size of a song. The MP3 player can store 2500 songs when the average size of a song is 4 megabytes (MB).

a. Write an equation that gives the number n of songs that will fit on the MP3 player as a function of the average song size s (in megabytes).

b. Create a table of values for the following size of songs:

Average Size of Song (MB) 2 2.5 3 5 Number of Songs

Joint Variation – join together Joint Variation occurs when a quantity varies directly with the product of two or more other quantities. In the equations below, a is a nonzero constant. Example: z = axy z varies jointly with x and y p = aqrs p varies jointly with q, r, and s 3. The variable z varies jointly with x and y. Also, z = 60 when x = -4 and y = 5. Find z when x = 7 and y = 2. Find Equation Find z when x = 7 and y = 2

Translations & Applications:

Directions: Write the equation to represent the given relationship. _____________y varies inversely with x. _____________y varies inversely with the square of x. _____________z varies directly with y and inversely with x. _____________R is inversely proportional to s and directly proportional to the square of t. _____________M is directly proportional to n and inversely proportional to s cubed. _____________The power (P) generated for a translational motion varies jointly with the acting force (F) over a distance (d) achieved and inversely with the time (t) taken to perform this motion. Directions: Write an equation for and solve each of the following problems. An equation shows m is directly proportional to n and inversely proportional to s cubed. When m = 5, then n = 160 and s = 2. What is the constant of proportionality? Write your answer as a fraction. The force needed to keep a car from skidding on a curve varies directly as the weight of the car and the square of the speed and inversely as the radius of the curve. Suppose a 3,960 lb. force is required to keep a 2,200 lb. car traveling at 30 mph from skidding on a curve of radius 500 ft. How much force is required to keep a 3,000 lb. car traveling at 45 mph from skidding on a curve of radius 400 ft.?

Day 8: Test Review

Understanding connections between graphing & solving rational functions Remember when solving a “system of equations” we are finding where the two functions cross?

Solve algebraically: 4x

1x4x

32 +

=+

What answers would be “extraneous”? Are both of our solutions extraneous? Now graph each of these functions and see where they cross: Notice that they have a common asymptote AND a common point. The common asymptote is the extraneous solution – since both graphs are “UNDEFINED” at that x-value, we don’t consider it a solution. Only the common point is a solution.

1. What is the domain of 5( )2

f xx

=−

2. What is the domain of 2

( )1

xf xx

=+

3. What is the domain of 7 1( )( 3)( 3)

xf xx x

+=

+ − 4. Factor 3x2 – 3 completely

5. Factor 3x3 – 3 completely

6. Simplify 3

3 2

3 7510 25a a

a a a−

+ +

7. Simplify 36 122

a aa a+

+

8. Solve 5 1 2 9

2 2x x

x x x+ +

+ =+ +

9. Solve 2

4 3 2 516 4 4

xx x x

++ =

− − +

Variation Practice: 1. Variable M varies directly with p. If M = 75 when p = 10, find M when p = 16. 2. R varies inversely with variable T. If R is 168 when T = 24, find R when T = 30. 3. Variable Y varies jointly with P and Q. If Y = 144 when P = 12 and Q = 8, find Y when P = 15 and Q = 25. 4. The volume, V, of a gas varies inversely as the pressure, p, in a container. If the volume of a gas is 200cc when the pressure is 1.6 liters per square centimeter, find the volume (to the nearest tenth) when the pressure is 2.8 liters per sq centimeter. 5. In science, one theory of life expectancy states that the lifespan of mammals varies inversely to the number of heartbeats per minute of the animal. If a gerbil's heart beats 360 times per minute and lives an average of 3.5 years, what would be the life expectancy of a human with an average of 72 beats per minute? Does this theory appear to hold for humans?