Algebra 2 and Trigonometry

Unit 4: Radicals, Circles, and Variation

Name _____________________________

Exponents and Radicals IntroDo Now:

For all values of b for which the expressions are defined, is equivalent to

1) 2) 3) 4)

1)

2)

3)

4)

5)

6)6)6)6)

7)

8)

9)

10)

Example What the rule is…Multiplying

=Dividing

=Power to a Power

=Product to a Power

=

Quotient to a Power

=Zero Exponent

=Negative Exponents

=Fractional Exponents

=

11) The expression , is equivalent to1)

2)

3)

4)

12Simplify:

13 When simplified, the expression is equivalent to1)

2)

3)

4)

14 The expression is equivalent to1)

2)

3)

4)

15 If x is a positive integer, is equivalent to1)

2)3)4)

HW - Exponents and Radicals Intro

1) Which expression is equivalent to ?

(1)

(2) (3) (4) 0

2) The expression is equivalent to

(1) (2) (3) (4)

3) Rewrite the following expression using only positive exponents:

4) The expression is equivalent to

(1)

(2)

(3)

(4)

5) The expression is equivalent to

(1) (2) (3)

(4)

6) Simplify the expression and write the answer using only positive exponents

7) Simplify:

8) Simplify the expression and write your answer using a positive exponent.

9) What is the value of when ?

10) Express as a fraction in simplest form:

11) Find the discriminate and describe the nature of the roots for the following equation:

Simplifying/Adding/Subtracting Radicals

Do Now:

Express in simplest form.

*To simplify a radical by adding or subtracting you must have like terms.

*Like terms are when the powers AND radicand are the same.

1) 2)

3) 4)

5) 6)

7) 8)

9) 10)

11) 12)

HW - Simplifying/Adding/Subtracting Radicals

1) If , then equals

1) 3)2) 4)

3) Simplify: 4) Simplify:

5) Simplify:

6) The expression written in simplest radical form is

(1) (2) (3) (4)

7) What is the sum of and ?

(1) (2) (3) (4)

8) The expression written in simplest radical form is

(1) (2) (3) (4)

9) Express in simplest radical form

10) Simplify:

2) Simplify:

11) Factor:

12) Express in simplest form:

CW on Multiplying & Dividing Radicals!

Do Now:

1. Find the multiplicative inverse 2. Simplify: of .

Multiplying & Dividing!

1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

13. Express the reciprocal of in simplest radical form with a rational denominator.

HW on Multiplying & Dividing Radicals!

Perform the indicated operation and express your answer in simplest radical form. (If your answer is a fraction, make sure the denominator is a rational expression)

1. 2.

3. 4.

5. 6.

7.

8. The expression can be rewritten as where a and b are integers. Which represents a?

(1) 16 (2) 12 (3) 6 (4) 4

9. Factor completely:

10. Write a quadratic equation if one of the roots is .

CW on Solving Radical Equations!

Do Now:

1. Simplify and express with a positive exponent: 2. Express with a fractional exponent:

Solving Radical Equations:

Solve each equation for x. YOU MUST CHECK YOUR ANSWERS!

1.

2.

3.

4.

5.

6.

7.

8.

9. Which statement is true about the roots of the equation ?

1) The only root is 1.2) The only root is 4.3) Both 1 and 4 are roots.4) Neither 1 nor 4 is a root.

10. The solution set of the equation is

(1) (2) (3) (4)

11. What is the solution set of the equation ?

(1) (2) (3) (4) HW on Solving Radical Equations!

Solve each equation for x.

1. 2.

3. 4.

5. What is the solution set of the equation ?

(1) (2) (3) (4)

6. The expression is equivalent to

(1) (2) (3) (4)

7. Simplify and express as a radical: 8. Simplify:

CW on Equations of Circles

Do Now:

1. Find the reciprocal of 2. Simplify:

CIRCLES!

In general, the equation of a circle is , where the center is (h, k) and the radius is r.

Write the equation of a circle given the following information.

1. Center = (4, 5); radius = 6 2. Center = (-1,0); diameter = 10

3. Center is @ the origin; radius = 4. Center = (3, -7); diameter = 15

5. 6.

7. Endpoints of the diameter are (4, 2) 8. Center = (-2, -5) and a point on the circle has and (1, 6). coordinates (1, -4).

Find the center, radius, diameter, area and circumference of each circle.

9. 10.

11. 12.

HW on Equations of Circles

Write the equation of a circle given the following information.

1. Center = (0, 4); radius = 9 2. Center = (-3, 8); diameter = 20

3. Center = (-20, 30); radius = 4. Center = (0,0) and contains the point (1, -3)

5.

Find the center and radius of each circle.

6. 7.

8. Solve for x: 9. Factor:

10. Simplify: 11. Simplify:

Using “Completing the Square” to determine the center and radius of a circle!

Do Now:

Write an equation of a circle whose center is at (3, 6) and passes through the point (9, -2).

Transform each equation into standard form and state the center and radius of the circle.

1. 2.

3. 4.

5. 6.

HW on Using “Completing the Square” to determine the center and radius of a circle!

Transform each equation into standard form and state the center and radius of the circle.

1. 2.

3.

4. Write an equation of a circle whose diameter has endpoints (5, -4) and (0, 8).

5. Solve algebraically for x:

a) b)

6. Find the product of and its conjugate.

7. A rectangular backyard has a length that measures and a width that measures .

a) How much fencing would a person need to enclose the entire backyard? Place your answer in simplest radical form.

b) If I wanted to put fake grass over the entire back yard, how many square units would I need? Place your answer in simplest radical form.

8. Simplify: 9. Evaluate

Direct and Inverse Variation

I. Direct Variation: Both variables change in the same manner!- A direct variation between 2 variables, x and y, is a relationship that is expressed as: _______________, where

k is the constant of proportionality

- To solve a direct variation problem, set up a ________________

- Graphically, direct variation is represented by a _______________. The following graph represents direct

variation:

Examples:

1) Variable M varies directly with p. If M = 75 when p = 10, find M when p = 16.

2) Driving along I-95 at a constant rate, Edgar drove 165 miles in 3 hours. If he continued to travel at the same rate, how long would it take him to travel another 143 miles?

3) Does this table show direct variation? If so, write the formula for the linear function that expresses this relationship.

A 2 4 5 7B 6 12 15 21

II. Inverse Variation: Variables change in the opposite manner!

- An inverse variation between 2 variables, x and y, is a relationship that is expressed as:

_______________, where k is the constant of proportionality

- To solve an inverse variation problem, we set it up as: _______________

- Graphically, inverse variation is represented by a _______________. The following graph represents

inverse variation:

Examples:

1) If y varies inversely as x, and y = 24 when x = 6, what is the value of y when x is 12?

2) The rate at which Jacob travels from home to his college varies inversely as the time it takes to make the trip. If Jacob can make the trip in 4 hours at 45 miles per hour, how many miles per hour must he travel to make the trip in 3 hours?

3) 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?

Direct and Inverse Variation Practice Questions

1. The speed of a laundry truck varies inversely with the time it takes to reach its destination. If the truck takes 3 hours to reach its destination traveling at a constant speed of 50 miles per hour, how long will it take to reach the same location when it travels at a constant speed of 60 miles per hour?

(1) hours (2) hours (3) hours (4) hours

2. According to Boyle's Law, the pressure, p, of a compressed gas is inversely proportional to the volume, v. If a pressure of 20 pounds per square inch exists when the volume of the gas is 500 cubic inches, what is the pressure when the gas is compressed to 400 cubic inches?

(1) 16 lb/in2 (2) 25 lb/in2 (3) 40 lb/in2 (4) 50 lb/in2

3. To balance a seesaw, the distance, in feet, a person is from the fulcrum is inversely proportional to the person’s weight, in pounds. Bill, who weighs 150 pounds, is sitting 4 feet away from the fulcrum. If Dan weighs 120 pounds, how far from the fulcrum should he sit to balance the seesaw?

(1) 4.5 ft (2) 3.5 ft (3) 3 ft (4) 5 ft

4. The manager of Stuart Siding Company found that the number of workers used to side a house varies inversely with the number of hours needed to finish the job. If four workers can side the house in 48 hours, how many hours will it take six workers working at the same speed to do the same job?

(1) 32 (2) 36 (3) 42 (4) 72

5. Carol notices that the number of customers who visit her coffee shop varies inversely with the average daily temperature. Yesterday, the average temperature was 40° and she had 160 customers. If today’s average temperature is 25°, how many customers should she expect?

(1) 100 (2) 145 (3) 256 (4) 1,000

6. When air is pumped into an automobile tire, the pressure is inversely proportional to the volume. If the pressure is 35 pounds when the volume is 120 cubic inches, what is the pressure, in pounds, when the volume is 140 cubic inches?

7. A pulley that has a diameter of 8 inches is belted to a pulley that has a diameter of 12 inches. The 8-inch-diameter pulley is running at 1,548 revolutions per minute. If the speeds of the pulleys vary inversely to their diameters, how many revolutions per minute does the larger pulley make?

8. The amount of money each member of a band earns playing at a graduation party varies inversely as the number of members in the band. If the band has five members, each member earns $70. Write an equation that models the relationship between the number of members in a band, n, and the amount each member earns, d. Use the equation to calculate the amount each member earns if there are four members in the band.

9. The price per person to rent a limousine for a prom varies inversely as the number of passengers. If five people rent the limousine, the cost is $70 each. How many people are renting the limousine when the cost per couple is $87.50?

10. Julio's wages vary directly as the number of hours that he works. If his wages for 5 hours are $29.75, how much will he earn for 30 hours?

11. If a machine that prints designs on T-shirts prints 500 shirts in 3 hours, how many hours will it take to print designs on 1,800 shirts?

12. A cake recipe calls for 1.5 cups of milk and 3 cups of flour. Seth made a mistake and used 5 cups of flour. How many cups of milk should he use to keep the proportions correct?

13. If x varies directly as y, and when what is the value of x when

(1) 1 (2) 2 (3) 3 (4) 4

14. A group of friends rent a house at the beach for spring break. If 9 of them share the house, it costs $150 each. Is the cost to each person directly or inversely proportional to the number of people sharing the house? Explain your reasoning and write a formula. How many people are needed to share the house if each student wants to pay a maximum of $100 each?

15. Driving at 55 mph, it takes approximately 3.5 hours to drive from Long Island to Albany. Is the time the drive takes directly or inversely proportional to the speed? Explain your reasoning and write a formula. TO get to Albany in 3 hours, how fast would you have to drive?

16. The price per person to rent an apartment varies inversely as the number of people renting it. If 5 people rent the apartment, the cost is $200 per person each month. How much will it cost each person, per month, if 3 couples rent the apartment, to the nearest dollar?

17. If x varies inversely with y, then when x is

a) increased by 2, y is decreased by 2 c) multiplied by 2, y is multiplied by 2b) increased by 2, y is increased by 2 d) multiplied by 2, y is divided by 2

18. The frequency of a radio wave is inversely proportional to the wavelength. If a wave of 300 meters has a frequency of 1,500 kilocycles per second, write an equation to represent the relationship between the frequency, f, and the wavelength, w. What is the length, in meters, of a wave with a frequency of 1,000 kilocycles per second?

19. The volume of a balloon is inversely proportional to the air pressure. If a balloon has a volume of 1.6 cubic meters when the air pressure is 14.7 pounds per square inch, how much greater will the volume, in cubic meters, be of the same balloon if it is raised to an altitude where the air pressure is only 12 pounds per square inch?

20. For a specified distance, speed varies inversely with time. If a car goes a certain distance in one-half hour at 60 mph, what speed, in mph, is needed to go the same distance in 45 minutes.

21. The acceleration due to Earth’s gravitational attraction varies inversely as the square of the distance from

the center of the Earth. The acceleration is 32 at 4000 miles from the center. What is the acceleration,

in , of a satellite 8000 miles from the center of the Earth?