cferreras.weebly.com · web viewunit 4– radical and rational functions _____ day 1: exponent...

Honors Math 2 Unit 4 Class Packet Sanderson High School

NOTESUNIT 4– Radical and Rational Functions

_________________________________1

Honors Math 2 Unit 4 Class Packet Sanderson High SchoolDay 1: Exponent Rules and Rational Exponents

Warm-UpQuincy has been confused about simplifying algebraic expressions. His brother Oscar prepared this quiz for him to practice. Look at Quincy’s responses below. If the response is correct, write correct. If it is incorrect, write the correct answer.

1. 4y – 5y -y2. 3x + 2x 5x2

3. 2n2 + 4n2 6n4

4. 2x • 3y 5xy5. 2a – (a – b) a – b

6.n+5n 5

7. 4 – (n + 7) -3 – n

8.2x6

x3 2x2

9. (2xy)2 4x2y2

10. 2a + 3b + 3a + 4b 6a2 + 12b2

2


3


4


Day 2: Graphing Square Root FunctionsWarm-UpElizabeth was going to complete her algebra assignment using her graphing calculator. Her older sister Carolyn decided to change the equations that she had copied into her notebook to tease her. Carolyn tells her that the equations are still equivalent to what her teacher had given her. Help Elizabeth by making each equation below quicker and easier to type into her calculator.

5

Honors Math 2 Unit 4 Class Packet Sanderson High School1.y = 2x-1

2. y = (2x)-1

3.y=7 .2 x

7

3 .6x 4

4. y = (-4)0

5. y = x-4.2x6.2

6. y = 33 • 32

7.y= 15−4

Graphing Square Root Functions

Make a table for each function.f(x) = x2 f(x) = √ x

0 01 12 23 34 45 56 67 78 89 9

6

x f(x)x f(x)x f(x)x f(x)x f(x)x f(x)x f(x)x f(x)x f(x)x f(x)x f(x)x f(x)x f(x)x f(x)x f(x)x f(x)x f(x)x f(x)x f(x)x f(x)x f(x)x f(x)x f(x)x f(x)x f(x)x f(x)x f(x)x f(x) x f(x)x f(x)x f(x)x f(x)x f(x)x f(x)x f(x)x f(x)x f(x)x f(x)x f(x)x f(x)x f(x)x f(x)x f(x)x f(x)x f(x)x f(x)x f(x)x f(x)x f(x)x f(x)x f(x)x f(x)x f(x)x f(x)x f(x)x f(x)


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.

7

Honors Math 2 Unit 4 Class Packet Sanderson High SchoolThe result: f(x) = √ x Characteristics of the graph

Vertex

End Behavior

Domain

Range

Symmetry

Pattern

Transforming the GraphsNow that we know the shapes we can use what we know about transformations to put that shape on the coordinate plane. Remember:Translate Reflect Dilate

1) f(x) = √ x−3

2) f(x) = √ x+4

8


3) f(x) = −√ x

4) f(x) = √−x

5) f(x) = 2√x+3

6) f(x) = 12 √ x

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) = √4 x−12 This is not in graphing form.

9


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

10


Day 3: Solving Square RootsWarm-UpSometimes quadratic equations contain hidden information that is revealed when the equations are written in factored form. Change each quadratic equation below to an equivalent equation in factored form.

1.y = -16t2 – 48t

2. y = 5x2 – 10x

3. y = 3(x – 1) + x(x – 1)

4. y = 52x2 – 13

5. y = x2 + 5x + 4

6. y = x(2x + 5) + x(x – 10)

11


On your own…

1. x=√−45+14 x

2. √ x−1=√x−7

3. (3 x+21 )43+9=90

12


Day 4: Radical ApplicationsWarm-UpFind any roots or zeros of each quadratic function below. Then explain what roots or zeros mean graphically.

1. f(x) = 2x2 – 5x – 3

2. f(x) = x2 + 2x + 1

3. f(x) = x2 + 2x + 3

4. f(x) = 2x2 + 3x - 1

13


Day 4 Classwork/Practice1. 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 d (h )=√2h 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 most common example of a pendulum (this side of Edgar Allen Poe) is the kind found in many clocks. The regular back-and-forth motion of the pendulum is periodic, and one such cycle of motion is called a period. The time, in seconds, that it takes for one period is given by the radical

equation t=2π √ lg 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 0.9 m long.

b. Find the period if the pendulum is 0.049 m long. 14


c. Solve the equation for length l.

d. How long would the pendulum be if the period were exactly 1 s?

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 s ( x )=2√5 x , in which s is skidding distance and x is speed. Calculate the speeding distance for the following speeds.

a. 55 mph

b. 65 mph

c. 75 mph

d. 40 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.

Solve each of the following applications.

4. The sum of an integer and its square root is 12. Find the integer.

5. The difference between an integer and its square root is 12. What is the integer?

6. The sum of an integer and twice its square root is 24. What is the integer?

7. The sum of an integer and 3 times its square root is 40. Find the integer.

15

Honors Math 2 Unit 4 Class Packet Sanderson High SchoolDay 5: Direct & Inverse Variation

Warm-UpAn Olympic-size swimming pool measures 25 meters by 50 meters. The Raleigh City Recreation Department is planning to construct a new pool that will be bordered with a walkway of a uniform width w. Find the possible widths of the walkway if the total area of the walkway is to be greater than 76 square meters, but no more than 400 square meters.

16


17


18


Day 6: Solving Rational EquationsWarm-UpGrace has found the following equation in her notebook:

x=−8±√82−4 (1 ) (10 )2 (1 )

She knows that she was using the quadratic formula to solve an equation and is wondering what the original equation might have looked like. Reconstruct her equation. Then find the solution (roots) of the equation. What are the x-intercepts for the graph of the equation?

On your own…

19


20


21


22


Day 7: Graphing Inverse VariationWarm-UpGwen and Rita are discussing the characteristics of certain functions. Rita claims that the equation y = x2 does not model a function because the line y = 5 intersects the graph of the equation in two points. Gwen disagrees. How might she present her case to Rita?

23


24


25


26


27


28


29

Honors Math 2 Unit 4 Class Packet Sanderson High SchoolDay 8: Systems of Equations

Warm-UpLily found the price of table tennis balls listed on the Internet at $4.75 for a package of 6 balls. Shipping and handling was listed at $1.00 per package.

1. Write an equation that represents the total cost for different numbers of packages of table tennis balls.

2. Sketch a graph of this relationship.

3. If you shift your graph up a value of $0.50, does this mean the price per package increased, or the shipping price increased?

4. Write a new equation for the situation in question 3.

30


31


32


Day 9: Unit ReviewA radical expression can be written as a rational exponent, and vice versa.

3. Give the index:

a) b) c) d)

33


4. Graph. State the transformations in order. State the Domain and Range.

a) b) c)

5. Working backwards : Writing the equation when given a translation.

a) The parent function is translated 2 units to the left and one unit down.

b) The parent function is translated 3 units to the right.

c) The parent function is compressed vertically by a factor of and then translated 2 units up.

6. Solve each radical equation.

34

Honors Math 2 Unit 4 Class Packet Sanderson High Schoola) b)

c) d)

e) f)

g) h)

i) j)

35


k)

36

Honors Math 2 Unit 4 Class Packet Sanderson High SchoolWrite an equation of variation to represent the situation and solve for the missing information.

7. 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?

8. 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?

9. 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?

10.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?

11.For a fixed number of miles, the gas mileage of a car (miles/gallon) varies inversely with the number of gallons used. One year an employee driving a truck averaged 24 miles per gallon and used 750 gallons of gas. If the next year, to drive the same number of miles the employee drove a compact car averaging 39 miles per gallon, how many gallons of gas would be used?

37


(The following are extensions and a mixture of the previous types.)

12.To build a sound wall along the highway, the amount of time t needed varies directly with the number of cement blocks c needed and inversely with the number of workers w. A sound wall made of 2400 blocks, using six workers takes 18 hours to complete. How long would it take to build a wall of 4500 blocks with 10 workers?

13.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?

14.The time to prepare a field for planting is inversely proportional to number of people who are working. A large field can be prepared by five workers in 24 days. In order to finish the field sooner, the farmer plans to hire additional workers. How many workers are needed to finish the field in 15 days?

15.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 ½ second for the egg to fall eight feet, how long will it take the egg to fall 200 feet?

16.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?

17.The weight of a person varies inversely as the square of the distance from the center of the earth. If the radius of the earth is 4000 miles, how much would a 180 pound person weigh, 2000 miles above the surface of the earth?

38

Honors Math 2 Unit 4 Class Packet Sanderson High School18.The strength of a rectangular beam varies jointly as its width and the square of it depth.

If the strength of a beam three inches wide by 10 inches deep is 1200 pounds per square inch, what is the strength of a beam four inches wide and six inches deep?

19.The cost c of materials for a deck varies jointly with the width w and the length l. If c = $470.40 when w = 12 and l = 16, find the cost when w = 10 and l = 25.

20.The value of real estate V varies jointly with the neighborhood index N and the square footage of the house S. If V = $376,320 when N = 96 and S = 1600, find the value of a property with N = 83 and S = 2150.

21.The number of gallons g in a circular swimming pool varies jointly with the square of the radius r2 and the depth d. If g = 754 when r = 4 and d = 2, find the number of gallons in the pool when r = 3 and d = 1.5.

Solve the following. Show all work!

22. x+3=10x

23. 0.8 x=20x

24.3n−2

=n5

39


40

cferreras.weebly.com · web viewunit 4– radical and rational functions _____ day 1: exponent...

Documents