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Math 2: Unit 3 – Radical and Rational Functions

Review of Exponent Rules

Rule 1: Multiplying Powers With the Same Base – Multiply Coefficients, Add Exponents

1. hhh 53 2. )5)(3( 32 xx 3. )6)(6( 43 4. )4)(5( 3223 nmnm 5. ))(8( 52baab

Rule 2: Dividing Powers With the Same Base – Divide/Simplify Coefficients, Subtract Exponents.

6. 4

6

x

x 7.

2

3

4

4 8.

5

24

xy

yx 9.

73

24

2

8

nm

nm 10.

63

29

16

36

yx

yx

Rule 3: Zero Power Property – A number raised to a power of zero is 1 (exception is zero).

11. 3

3

2

2 12.

5

5

p

p 13. 0745 14. 3014 ba 15.

0

7

x

Rule 4: Negative Exponents – Move the base with the negative exponent across the fraction bar.

16. 310 17. 5x 18. 7

1x

19. 43 rq 20.

952

423

cba

cba

21. 223

353

10

8

zyx

zyx 22.

834

75

10

15

zyx

zxy

23. ))(( 625 yxyx

24. )5)(2( 52 xx 25. 142

125

36

6

qnm

qnm

Rule 5: Raising a Power to a Power – Multiply exponents. Simplify.

26. (36)2 27. (n2)5 28. 534 )( yy 29. (a-4)7 30. T2(T7)-2

Rule 6: Raising a Product or a Quotient to a Power – Multiply each inside exponent by the outside

exponent. Simplify.

31. (5d2)3 32. (x2y)4 33. (4g5)-2 34. (-4pq2r3)2 35. (-2a2b)3(a2b6)4

36.

3

37

3

y

x 37.

4

2

4

n

nn 38.

2

3

62

ba

ab 39.

2

25

2

3

4

ba

ba 40.

0

200

27

576

492

y

x

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Simplifying Radicals & Basic Operations

We are familiar with taking square roots (√ ) or with taking cubed roots (√3

), but you may not be as familiar with the elements of a radical.

An index in a radical tells you how many times you have to multiply the root times itself to get the

radicand.

For example, in √81 = 9, 81 is the radicand, 9 is the root, and the index is 2 because you have to

multiply the root by itself twice to get the radicand (9 ∙ 9 = 92 = 81). When a radical is written without

an index, there is an understood index of 2.

√643

=?

Radicand: Index:

Root is ______ because _____ ∙ _____ ∙ _____ = _____3 = 64

√32𝑥55=?

Radicand: Index:

Root is ______ because _____ ∙ _____ ∙ _____ ∙ _____ ∙ _____ = _____5 = 32𝑥5

To use your calculator:

An index of 2:

Step 1: Press

Step 2: Type in radicand.

An index of 3:

Step 1: Press

Step 2: Choose 4.

Step 3: Type in the radicand.

Any index:

Step 1: Type in the index.

Step 2: Press

Step 3: Choose 5: √𝑥

Step 4: Type in the radicand.

You Try:

√243𝑦55=

√1296𝑚4𝑛84= √144𝑣8=

BUT not every problem will work out that nicely! Use your calculator to find an exact answer for

√243

= ____

The calculator will give us an estimation, but we can’t write down an irrational number like this

exactly – it can’t be written as a fraction and the decimal never repeats or terminates. The best we

can do for an exact answer is use simplest radical form.

Here are some examples of how to write these in simplest radical form. See if you can come up with

a method for doing this. Compare your method with your neighbor’s and be prepared to share it

with the class. (Hint: do you remember how to make a factor tree?)

√𝟏𝟐 = 𝟐√𝟑

√𝟐𝟒𝟑

= 𝟐√𝟑𝟑

√𝟒𝟖𝟒

= 𝟐√𝟑𝟒

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Simplifying Radicals: 1) ________________________________________________________

2) ________________________________________________________

3) ________________________________________________________

Examples:

1. √16𝑥2

2. √8𝑥 3. √15𝑥3

4. √−83

5. √80𝑛53

6. √964

7. √814

8. √4865

9. √−403

10. √18𝑥43

11. √64𝑥34

12. √−32𝑥3𝑦65

13. √81𝑥3𝑦2𝑧43

14. √192𝑥5𝑦7𝑧23

15. √1875𝑥4𝑧24

Simplifying Radicals Homework

Simplify each expression.

1. √24 2. √10003

3. √−1623

4. √512

5. √128𝑛84 6. √98𝑘 7. √224𝑟75

8. √24𝑚33

9. √392𝑥2 10. √512𝑥2 11. √405𝑥3𝑦24 12. √−16𝑎3𝑏83

13. √128𝑥7𝑦74 14. √16𝑥𝑦3

15. √448𝑥7𝑦76 16. √56𝑥5𝑦

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Rational Exponents & Radicals

Simplifying Exponents:

Evaluate each expression.

3

2

1

0

1

2

4 ______

4 ______

4 ______

4 ______

4 ______

4 ______

Where would

1

24 ,

1

24 and

3

24 fall in this list?

Enter

1

24 into the calculator.

1

24 ________

Enter

1

24 into the calculator.

1

24 _______

Without using calculator predict the value of

3

24.

Bases with fraction exponents can be written

as radicals

Radical expressions can be written with

bases with fractions for exponents

n pn

p

bb

n

p

n p bb

We have already done this when simplifying radicals:

43

12

43 12 xxbecausexx and 22

4

24 xxbecausexx

Even if a fraction cannot reduce evenly, we can still write radicals using fraction exponents.

Write using rational (fraction) exponents:

3 2 _______x 4 3 2 _______a b 52 _______t

3

1_______

x

4

3 2 _______x

Write each in Radical Form.

1.)

2

5 _______x 2.)

1

42 _______x 3.)

1

4(2 ) _______x

4.)

2

3(3 ) _______x 5.) 1.5 _______x 6.) 2.8 _______x

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You try: Rewrite each of the following expressions in radical form.

1. 32x

2.

23( 27)

3. 54(16 )x

4. 98y

5. 142a

6. 724

7.

25

5

3

8.

1.2x

Now, reverse the rule you developed to change radical expressions into rational expressions.

1.

5 2

2.

5

3 6

3.

7

5

4.

7

5.

4 39

6.

2

7 3x

Rational Exponents and Radicals Homework:

Write each expression in radical form.

1. 71

2 2. 44

3 3. 25

3 4. 74

3

5. 63

2 6. 21

6 7. (5𝑥)−5

4 8. (5𝑥)−1

2

9. (10𝑛)3

2 10. 𝑎6

5 11. (6𝑣)1.5 12. 𝑚−1

2

Write each expression in exponential form.

13. (√10)3 14. √2

6 15. (√2

4)

5 16. (√5

4)

5

17. √23

18. √106

19. (√𝑚4

)3 20. (√6𝑥

3)

4

21. √𝑣4

22. √6𝑝 23. (√3𝑎3

)4 24.

1

(√3𝑘)5

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Graphing Square Root Functions

Make a table for each function.

f(x) = x2

x f(x)

0

1

2

3

4

5

6

7

8

9

f(x) = √𝑥

x f(x)

0

1

2

3

4

5

6

7

8

9

Ignore the points with decimals. What do you notice about the other points?

__________________________________________________________________________________________________

These functions are _______________ of each other. By definition, this means the _____________ and the

_____________ ______________.

Plot the points from the tables above.

As a result, the graphs have the same numbers in their points but the _____ and the _________

coordinates have ___________ _______________.

This causes the graphs to have the _____________ _______________ but to be __________________ over

the line ____________.

The Square Root Function

Reflect the function f(x) = x2 over the line y = x.

Problems? ______________

We have to define the Square Root ______________ as ________________. This means that we will only

use the _________________ side of the graph.

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The result: f(x) = √𝑥 Characteristics of the graph

Vertex

End Behavior

Domain

Range

Symmetry

Pattern

Transforming the Graphs

Now that we know the shapes we can use what we know about transformations to put that shape

on the coordinate plane.

Remember:

Translate Reflect Vertical Change

Domain: Range:

1) f(x) = √𝑥 − 3

Transformations:

Domain:

Range:

As :x

2) f(x) = √𝑥 + 4

Transformations:

Domain:

Range:

As :x

3) f(x) = −√𝑥

Transformations:

Domain:

Range:

As :x

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4) f(x) = √−𝑥

Transformations:

Domain:

Range:

As :x

5) f(x) = 2√𝑥 + 3

Transformations:

Domain:

Range:

As :x

6) f(x) = 1

2√𝑥

Transformations:

Domain:

Range:

As :x

(HONORS) Sometimes the functions are not in graphing form. We may have to use some of our

algebra skills to transform the equations into something we can use.

Ex: f(x) = √4𝑥 − 12 This is not in graphing form.

Transformations:

Domain:

Range:

As :x

Ex: f(x) = √9𝑥 + 36 − 5 This is not in graphing

form.

Transformations:

Domain:

Range:

As :x

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Classwork:

Graph each function. Then state the domain and range.

1. 𝑓(𝑥) = √𝑥 + 4 − 2 2. 𝑓(𝑥) = −√𝑥 + 3

3. 𝑓(𝑥) = (𝑥 − 2)1

2 + 3 4. 𝑓(𝑥) = −√𝑥 + 2 + 2

Write an equation for each graph. Then state the domain and range.

Describe the transformations in the graphs of each equation. Then state the domain and range.

8. 14)( xxf 9. 2

1

4)( xxf

10. 23)( 2

1

xxf 11. 2

1

45)( xxf

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1. 2. 3.

D: ___________ D: _____________ D: ___________

R: ___________ R: _____________ R: ___________

4. 5. 6.

D: ___________ D: _____________ D: ___________

R: ___________ R: _____________ R: ___________

7. 8. 9.

D: ___________ D: _____________ D: ___________

R: ___________ R: _____________ R: ___________

10. 11. 12.

D: ___________ D: _____________ D: ___________

R: ___________ R: _____________ R: ___________

13. 14. 15.

D: ___________ D: _____________ D: ___________

R: ___________ R: _____________ R: ___________

xy

xy

xy

xy

xy

xy

xy

xy

xy

xy

xy

xy

xy

xy

xy

)15

2

1)14

13

2)13

12

1)12

32)11

2)10

3)9

2

1)8

31)7

24

1)6

21)5

212)4

1)3

13)2

5)1

Homework. Graph each. Write the Domain and Range.

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Solving Radical Equations Notes

Extraneous Solutions:

1. 522 x 2. 312 x

3. 592 x 4. 416 xx

5. xx 24 6. xx 23

7. The number of people, y, involved in recycling in a community is modeled by the function

400390 xy , where x is the number of months the recycling plant has been open.

(a) Find the number of people involved in recycling exactly 3 months after the plant opened.

(b) After how many months will 940 people be involved in recycling?

8. The period of a pendulum (T), in seconds, is the length of time it takes for the pendulum to make

one complete swing back and forth. The formula 32

2L

T gives the period T for a pendulum

of length L in feet. If you want to build a grandfather clock with a pendulum that swings back and

forth once every 3 seconds, how long, to the nearest tenth of a foot, would you make the

pendulum?

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Solving Radical Equations Classwork

Solve each equation.

1. 823 xx 2. 33 4294 xx

3. 723 xx 4. 722 xx

5. xx 24 6. 223 2 xx

7. xx 5132 (Calculator) 8. 1216 xx

9. 44 7233 xx 10. xx 3515 (Calculator)

11. The velocity of a free falling object is given by ghV 2 where 𝑉 is the velocity (in meters per

second), 𝑔 is acceleration due to gravity (in meters per second square), and ℎ is the distance

(in meters) the object has fallen. The value 𝑔 depends on which body/planet is attracting the

object. If the object hits the surface with a velocity of 30 meters per second, from what height

was it dropped in each of the following situations? a. You are on Earth where 𝑔 = 9.81 𝑚/𝑠2?

b. You are on the moon where 𝑔 = 1.57 𝑚/𝑠2?

c. You are on Mars where 𝑔 = 3.72 𝑚/𝑠2?

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Solving Radical Equations Homework

Solve each equation.

1. 123 x 2. 142

1

x

3. 623 x 4. 647 x

5. 213 x 6. 112 xx

7. xx 2518 8. 212 2 xx

9. 2329 xx 10. 3 23 2 642 xxx

11. The speed that a tsunami (tidal wave) can travel is modeled by the equation

dS 356

where S is the speed in kilometers per hour and d is the average depth of the water in kilometers.

a.) What is the speed of the tsunami when the average water depth is 0.512 kilometers?

(round to nearest tenth)

b.) Solve the equation for d.

c.) A tsunami is found to be traveling at 120 kilometers per hour. What is the average depth of the

water? (round to three decimal places)

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Radical Applications

Warm Up

Solve the following equations.

1. 92 x 2. x5 3. x221220

Solving Radical Equations Application Worksheet

1. A pendulum can be measured with the equation 𝑇 = 2𝜋√𝐿

𝐺, where T is the time in seconds, G is the

force in gravity (10m/s2) and L is the length of the pendulum.

a) find the period (to the nearest hundredth of a second) if a pendulum is 0.9m long

b) find the period if the pendulum is 0.049 m long.

c) solve the equation for length L.

d) how long would the pendulum be if the period were exactly 1 s?

Solve the following applications.

2. The difference between an integer and its

square root is 12. What is the integer?

3. The sum of an integer and twice its square root

is 24. What is the integer?

4. The sum of an integer and three times its square root is 40. Find the integer.

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Use ( ) 2d h h , where d is the distance to the horizon in miles from a given height h in feet for #5 & 6.

5. If a plane flies at a height 30,000 ft, how far

away is the horizon?

6. Janine was looking out across the ocean from

her hotel room on the beach. Her eyes were 250

ft above the ground. She saw a ship on the

horizon. Approximately how far was the ship from

her?

When a car comes to a sudden stop, you can determine the skidding distance (in feet) for a given

speed (in miles per hour) using the formula 𝒔(𝒙) = 𝟐√𝟓𝒙 where s is skidding distance and x is speed.

Calculate the skidding distance for the following speeds (round to the nearest tenth).

7. 55mi/h

8. 65 mi/h 9. 75 mi/h 10. 40 mi/h

Radical Applications Homework

1. Did you ever stand on a beach and wonder how far out into the ocean you could see? Or have

you wondered how close a ship has to be to spot land? In either case, the function hhd 2

can be used to estimate the distance to the horizon (in miles) from a given height (in feet).

a. Cordelia stood on a cliff gazing out at the ocean. Her eyes were 100 ft

above the ocean. She saw a ship on the horizon. Approximately how

far was she from that ship?

b. From a plane flying at 35,000 ft, how far away is the horizon?

c. Given a distance, d, to the horizon, what altitude would allow you to see that far?

2. A weight suspended on the end of a string is a pendulum. The time, in seconds, that it takes for

one period is given by the radical equation g

lt 2 in which g is the force of gravity (10 m/s2)

and l is the length of the pendulum.

a. Find the period (to the nearest hundredth of

a second) if the pendulum is 1.2 m long.

b. Find the period if the pendulum is 0.045 m

long.

c. Solve the equation for length l.

d. How long would the pendulum be if the

period were exactly 2 seconds?

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3. When a car comes to a sudden stop, you can determine the skidding distance (in feet) for a

given speed (in miles per hour) using the formula xxs 52 , in which s is skidding distance and

x is speed. Calculate the skidding distance for the following speeds (round to the nearest tenth).

a. 45 mph b. 70 mph c. 90 mph

d. 5 mph

e. Given the skidding distance s, what formula would allow you to calculate the speed in miles

per hour?

f. Use the formula obtained in (e) to determine the speed of a car in miles per hour if the skid

marks were 35 ft long.

4. Solve each of the following applications.

a. The sum of an integer and its square root is

56. Find the integer.

b. The difference between an integer and its

square root is 56. What is the integer?

c. The sum of an integer and twice its square

root is 35. What is the integer?

d. The sum of an integer and 3 times its square

root is 130. Find the integer.

Notes: Direct and Inverse Variation

I. Direct Variation: y = kx

Stated: y varies directly as x or y is directly proportional to x where k is the constant of proportionality

or constant of variation.

Key Idea: As x gets larger, y gets larger or as x gets smaller, y gets smaller.

A line with a y-intercept of 0 is a direct variation.

Real world examples of direct variation include: fruit sold by the pound, distance traveled by a car

over time, characters printed from a computer per second, circumference of a circle varies directly

as the diameter, and wages varying directly to hours worked. Can you think of others?

Example 1: If you buy three pounds of grapes at $2.99 per pound, how much would you pay for the

grapes? What are the two variables and what is the constant of variation?

How can you write this as a linear equation? _______________________________________

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Data that represents direct variation:

What is the constant of proportionality? ___________

Example 2: If y varies directly as x and x = 10 when y=9, then what is y when x = 4?

Example 3: When a bicycle is pedaled in a certain gear, the distance traveled varies directly to the

number of pedal revolutions. The bicycle travels 16 meters for every 3 pedal revolutions. How many

revolutions would be needed to travel 600 meters?

Example 4: A refund r you get varies directly as the number of cans you recycle. If you get a $3.75

refund for 75 cans, how much should you receive for 500 cans?

II. Inverse Variation: y = k

x or xy = k

Stated: y varies inversely as x or y is inversely proportional to x where k is the constant of

proportionality or constant of variation

Key Idea: As x gets larger, y gets smaller or as x gets smaller, y gets larger.

Real world examples of indirect variation include: Boyles’ Law of Gases is a real world example of

inverse variation. Likewise, for a trip to Myrtle Beach, the greater your car speed, the less time it would

take you to get there. If a rectangle has an area of 15 square units, then as the length increases the

width decreases.

Data that represents inverse variation:

What is the constant of variation? _________

Example1: If y varies inversely as x and x = 3 when y = 9, then what is x when y = 27?

Example 2: If y varies inversely as the square of x and y = 20 when x =4, find y when x =5.

x y

-1 3

0 0

-2 6

4 -12

x y

3 4

2 6

9 43

10 65

1 12

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Example 3: Find x when y = 3, if y varies inversely as x and x = 4 when y = 16.

Example 4: The amount of resistance in an electrical circuit required to produce a given amount of

power varies inversely with the square of the current. If a current of .8amps requires a resistance of 50

ohms, what resistance will be required by a current of .5 amps?

(HONORS) Combined Variation: describes a situation where a variable depends on two (or more)

other variables, and varies directly with some of them and varies inversely with others (when the rest

of the variables are held constant).

x

y kz

Example:

The number of hours needed to assemble computers varies directly as the number of computers and

inversely as the number of workers. If 4 workers can assemble 12 computers in 9 hours, how many

workers are needed to assemble 48 computers in 8 hours?

For each problem: a) write a function of variation to represent the situation and

b) solve for the indicated information.

1. The number of gallons g of fuel used on a trip

varies directly with the number of miles m

traveled. If a trip of 270 miles required 12 gallons

of fuel, how many gallons are required for a trip

of 400 miles?

2. Karen earns $28.50 for working six hours. If the

amount m she earns varies directly with h the

number of hours she works, how much will she

earn for working 10 hours?

3. A bottle of 150 vitamins costs $5.25. If the cost

varies directly with the number of vitamins in the

bottle, what should a bottle with 250 vitamins

cost?

4. Wei received $55.35 in interest on the $1230 in

her credit union account. If the interest varies

directly with the amount deposited, how much

would Wei receive for the same amount of time if

she had $2000 in the account?

5. The volume V of a gas kept at a constant

temperature varies inversely as the pressure p. If

the pressure is 24 pounds per square inch, the

volume is 15 cubic feet. What will be the volume

when the pressure is 30 pounds per square inch?

6. The time to complete a project varies inversely

with the number of employees. If 3 people can

complete the project in 7 days, how long will it

take 5 people?

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7. The time needed to travel a certain distance

varies inversely with the rate of speed. If it takes 8

hours to travel a certain distance at 36 miles per

hour, how long will it take to travel the same

distance at 60 miles per hour?

8. The number of revolutions made by a tire

traveling over a fixed distance varies inversely to

the radius of the tire. A 12-inch radius tire makes

100 revolutions to travel a certain distance. How

many revolutions would a 16-inch radius tire

require to travel the same distance?

9. (Challenge) An egg is dropped from the roof

of a building. The distance it falls varies directly

with the square of the time it falls. If it takes 12

seconds for the egg to fall eight feet, how long

will it take the egg to fall 200 feet?

10. (Challenge) The time needed to paint a

fence varies directly with the length of the fence

and inversely with the number of painters. If it

takes five hours to paint 200 feet of fence with

three painters, how long will it take five painters

to paint 500 feet of fence?

Direct and Inverse Variation Worksheet

Find the Missing Variable:

1) y varies directly with x. If y = -4 when x = 2, find

y when x = -6.

2) y varies inversely with x. If y = 40 when x = 16,

find x when y = -5.

3) y varies inversely with x. If y = 7 when x = -4, find

y when x = 5.

4) y varies directly with x. If y = 15 when x = -18,

find y when x = 1.6.

5) y varies directly with x. If y = 75 when x =25, find x when y = 25.

Classify the following as: a) Direct b) Inverse c) Neither

6) m = -5p 7) r =

t

9

8) d = 4t 9) c =

4

e

10) n = ½ f 11) z =

t

2.

12) c = 3v 13) u =

18

i

What is the constant of variation for the following?

14) y = 4x 15) y =

x

2

16) x = ½ y 17) x =

y

9

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Answer the following questions.

18) If x and y vary directly, as x decreases, what

happens to the value of y?

19) If x and y vary inversely, as y increases, what

happens to the value of x?

20) If x and y vary directly, as y increases, what

happens to the value of x?

21) If x and y vary inversely, as x decreases, what

happens to the value of y?

Classify the following graphs as a) Direct b) Inverse c) Neither

22) 23) 24) 25)

Answer the following questions:

26) The electric current I, is amperes, in a circuit varies directly as the voltage V. When 12 volts are

applied, the current is 4 amperes. What is the current when 18 volts are applied?

27) The volume V of gas varies inversely to the pressure P. The volume of a gas is 200 cm3 under

pressure of 32 kg/cm2. What will be its volume under pressure of 40 kg/cm2?

28) The number of kilograms of water in a person’s body varies directly as the person’s mass. A person

with a mass of 90 kg contains 60 kg of water. How many kilograms of water are in a person with a

mass of

50 kg?

29) On a map, distance in km and distance in cm varies directly, and 25 km are represented by 2cm.

If two cities are 7cm apart on the map, what is the actual distance between them?

30) The time it takes to fly from Los Angeles to New York varies inversely as the speed of the plane. If

the trip takes 6 hours at 900 km/h, how long would it take at 800 km/h?

Solving Rational Equations

Rational Equations

Restricted/Excluded Values

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Steps for solving rational equations.

Solve each equation. State any restricted/excluded values.

1. 3

22

x

x 2.

xx 7

2

5

1

3. 2

1

3

2

6

32

aa 4.

14

3

27

32

bbb

5. 12

7

5

3

xx 6. 5

2

2

5

kk

k

7. 11

5

1

mm

m 8. 2

23

2

23

4

x

x

x

x

9. 255

5 2

p

p

p 10.

3

122

3

32

aa

a

11. 2

32

2

52

bb

b 12.

6

1

2128

4

2

kk

k

kk

Proportions

1. Set the cross products equal.

2. Solve the equation.

Rational Equations with unlike denominators

1. Find the Least Common Denominator (LCD)

2. Multiply each term by the LCD.

3. Simplify each term.

4. Solve the equation.

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Solving Rational Equations Practice

Solve each equation. State any restricted/excluded values.

1. 7

2

3

x

x 2.

xx

x 2

5

42

3. xx

41

1

2

3 4.

xx

x 12

1

1

5. 3426

1

x

x

x 6.

3

24

3

6

xx

7. 45

2

2

xx 8. 2

34

5

bb

9. yy

41

9

1 10.

8

15

4

13

nn

11. 3

22

3

2

3

4

x

x

xx 12.

2x 2( 3)2

3 1 3 1

x

x x

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Graphing Inverse Variation

Sketch a graph of 10

yx

. Make a table of values that include positive and negative values of x.

x -10 -8 -5 -4 -2 -1 0 1 2 4 5 8 10

y

Graph the points and connect them with a smooth curve. The graph has two parts- each part is called a

branch.

The x-axis is called the _____________________

asymptote.

The y-axis is called the _____________________

Asymptote.

The parent graph of a rational function is 1

yx

.

Like all parent graphs, it passes starts at the

point (_____,_____). It has a horizontal asymptote

at ________________ and a vertical asymptote at

________________. The graph approaches but

does not cross these lines. The parent graph of a

rational function has a domain of

________________ and a range of

_________________. The parent graph of a

rational function is always decreasing.

The Parent Graph of 1

yx

:

Transforming Inverse Variation Functions:

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Steps to Graph Inverse (Rational) Functions: 1

yx

and

ay k

x h

Center of Asymptotes at:

(______, ______) or (______, ______)

(Starting Point)

If “a” is positive, graph ________________

If “a” is negative, graph _______________

Domain: ______________ Range: _______________

To graph:

1. Find C.O.A. ,draw a vertical and horizontal

line thru point

2. Use pattern

Over Up

a 1

1 a

-a -1

-1 -a

3. Connect points with a smooth curve

approaching asymptotes

Examples: For each rational function, state the Center of Asymptotes, Domain and Range

1.

1

4y

x 2.

21

3y

x

Graph each function. State the Domain and Range.

3. Function:

32

1y

x

D: ___________ R: ___________

4. Function:

2

1y

x

D: ___________ R: ___________

5. Function:

42

2y

x

D: ___________ R: ___________

6. Function: 1

3yx

D: ___________ R: ___________

7. Function:

12

1y

x

D: ___________ R: ___________

8. Function:

31

3y

x

D: ___________ R: ___________

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Homework: Graph

each function.

State the domain

and range.

xy

xy

xy

xy

xy

xy

xy

xy

xy

xy

xy

xy

xy

xy

xy

1)15

1

1)14

3

22)13

3

4)12

3

1)11

2

1

)10

31

1)9

2

4)8

2

11)7

5

3)6

21

2)5

1

4)4

13

1)3

2

2)2

51

)1

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Solving Systems involving Radical and Inverse Functions

Algebraically

1. Use subsitution to combine the equations.

2. Solve the equation.

3. Plug variable back into one of the original

equations to find 2nd variable.

4. Write solutions as ordered pairs.

Graphically

1. Type 1st equation (or left side) into y1.

2. Type 2nd equation (or right side) into y2.

3. Find the intersection(s) of the two

functions.

(2nd Trace, 5: Intersection, Enter, Enter, Enter)

Solving Radical Equations/Systems with Radical Equations Notes

Solve each equation algebraically, then check graphically.

1. 324 x 2. 1632 xx

3. 24 xx 4. 51 xx

5.

yx

xy

6

7 6.

yx

xy

4

5

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Solving Systems Homework

Solve each system symbolically. Then represent the solution on a graph. Label any key points.

1.

623

5

9

yx

xy (Calculator)

2.

4

2

yx

xy

3.

6yx

xy 4.

1

994

xy

yx (Calculator)

5.

1

322

xy

yx (Calculator) 6.

xy

xy 2

Applications.

7. Find the lengths of the legs of a right triangle whose hypotenuse is 15 feet and whose area is 3

square feet.

8. A small television is advertised to have a picture with a diagonal

measure of 5 inches and a viewing area of 12 square inches. What are

the length and width of the screen?