Activity 7.1 Roots and Radicals Name: In general, we state ...

Activity 7.1 Roots and Radicals Name:__________________

Math 95

25

4 16

1. In general, we state that if nx a , then x is called an ______________ of a.

2. Evaluate, if possible. If the result is not a real number, state: Not a real number.

a. 49 b. 3 125 c. 6 64

d. 25 e. f.

25

g. 3 27 h. 3 27 i.

3 27

j. 4 16 k. l.

4 16

m. 5 32 n. 5 32 o.

5 32

3. The time, in seconds, that it takes for an object to fall, from rest, is given by 1

4t d , in which d

is the distance fallen in feet. Show your work and round to the nearest hundredth second.

a. Find the time required for an object to fall to the ground from a building that is 540 feet high.

Interpret your results with a complete sentence.

b. Find the time required for an object to fall to the ground from a building that is 660 feet high.


Activity 7.1 of 2

c

4

8

10

8

b

c

12

9

17

a

8

4. State the Pythagorean Theorem: __________________________

5. Find the missing length in the following right triangles using the Pythagorean Theorem.

Express your answer in radical form where appropriate.

a. c = ___________________

b.

b = ___________________

c.

a = ___________________

d.

c = ___________________

Activity 7.2 Roots and Radicals Name:_________________

Math 95

3 16 4 16

3 16

2018

40

9

40032

5

7

5

10

5

12

3 32 4 325 32

1. State the Product Property for Radicals: _____________________________

2. Simplify the following, if possible. If the result is not a real number, state: Not a real number.

a. b. c.

d. e. f.

3. State the Quotient Property for Radicals: _____________________________


a. b. c.


a. 16 b. c.

d. 16 e. f. 4 16

g. h. i.

Activity 7.2 Page 2 of 2

c

4

8

4

a

8

10

b

7

6. Find the missing length in the following right triangles using the

Pythagorean Theorem: 2 2 2 a b c . Express your answer in simplest radical form.

a.

a = ___________________

b.

c = ___________________

c.

b = ___________________

Activity 7.5 Rational Exponents Name:______________________

Math 95

1. For any real number b and a positive whole number n > 1, then 1 n

b = _________

If n is even and 0b , then 1 n

b is called the _______________________.

If n is even and 0b , then 1 n

b is ____________________________.

If n is odd then 1 n

b is called the ________________________________.

2. Write each expression in radical form and simplify. If the result is not a real number, state: Not a

real number.

a. 1

2100 b. 1

249 c.

1

327

8

d. 1

481 e. 1

481 f. 1

481

3. For any real number b and a positive integers m and n with n > 1, then m

nb =______ OR_______

4. Write each expression in radical form and simplify.

If the result is not a real number, state: Not a real number

a. 3

216 b. 2

532 c. 3

281

d. 2

327 e. 5

38 f.

3

216

9


5. Review: For any nonzero real numbers a and b and rational numbers m and n,

Product Rule: m na a Product-Power Rule: n

a b

Quotient Rule: m

n

a

a Quotient Rule:

na

b

Product Rule: n

ma Product Rule: na

6. Simplify. If the result is not a real number, state: Not a real number.

a. 1

249

b. 1

327

c. 1

481

d. 2

38

e. 3

416

f. 5

664

7. Use the properties of exponents to simplify each expression. Assume all variables represent

positive real numbers.

a. 5 2

6 3x x b.

7

6

2

3

b

b

c.

43 5

4 2

p p

8. Explain the difference between 1

3x and 3x .

Activity 8.1 Solving Quadratic Equations Name:_________________

Math 95

1. The standard form of a quadratic equation is: ________________

2. State the Zero-Product Principle: ________________

3. Solve the following by factoring.

a. 2 9 0x b. 2 121x

4. State the Square-Root Property: __________________________

5. Solve the following using the Square Root Property and simplify your answers.

a. 2 18 0x b. 2 48 0x

c. 2

5 36 0x d. 2

4 24 0x


-4 -2 2 4

-8

-6

-4

-2

2

4

6

8

x

y

ramp

1

12

6. State the Pythagorean Theorem: __________________________

7. Laws regarding access ramps for the disabled state that a ramp must be in the form of a right

triangle, where every vertical length of 1 foot has a horizontal length of 12 feet. If a ramp has

a 1 foot vertical length and a 12 foot horizontal length, what is the length of the ramp?

Give an exact answer and an approximation to two decimal places.

8. The two equal sides of an isosceles right triangle are of length s.

Find a formula for the length of the hypotenuse and simplify your results.

h

s

s

9. Given the function 2 2 8f x x x , determine the following.

a. Solve the equation 20 2 8x x

b. Coordinates of the x-intercept(s) __________________

__________________

c. Coordinates of the y-intercept __________________

d. Sketch the graph on the given axes and label at least 4 points.

Your graph should be consistent with your answers above.

Activity 8.2 Quadratic Formula Name:_________________

Math 95

1. Solve the following by factoring.

a. 2 24 10x x b. 22 7 15x x

2. State the Quadratic Formula: _______________________

3. Solve the following by using the Quadratic Formula.

a. 2 24 10x x b. 22 7 15x x

4. Solve each equation using the Quadratic Formula. Leave answers in simplest radical form.

a. 2 3 5 0x x b. 4 2 x x


-4 -2 2 4

-8

-6

-4

-2

2

4

6

8

x

y

5. When using the Quadratic Formula to solve a quadratic equation:

Step 1: Write the equation in standard form: __________________

Step 2: Identify the coefficients ____________________

Step 3: Substitute these values in the ___________________.

Step 4: _______________ your answers.

6. Complete the table.



b. Coordinates of the x-intercept(s)_________________

c. Coordinates of the y-intercept ___________________

d. Sketch the graph on the given axes. Your graph

should be consistent with your answers above.

Equation Evaluate the discriminant 2 4b ac

Number and

type of solution(s)

24 10 x x

3 0 x x

2 14 49x x

2 10 0 x x

2 7 5 0 x x

22 5 12 0 x x

Activity 8.3 Graphing Parabolas Name:_______________ Math 95

-4 -2 2 4

-10

-5

5

x

y

-10 -8 -6 -4 -2 2 4 6 8

-25

-20

-15

-10

-5

5

10

15

x

y



b. Coordinates of the x-intercept(s): __________________

c. Coordinates of the y-intercept: ___________________

d. Axis of Symmetry: Coordinates of the vertex:

e. Sketch the graph on the given axes and label at least 4 points.

f. State the domain: ________________ State the range: _________________

2. Answer the following for the function 2 16f x x .

a. Solve 20 16x .

b. Coordinates of the x-intercept(s):

c. Coordinates of the y-intercept : ___________________

g. Axis of Symmetry: Coordinates of the vertex:

d. Sketch the graph on the given axes and label at least 4 point.

e. State the domain: ________________ State the range: _________________


-8 -6 -4 -2 2 4 6 8

-12

-10

-8

-6

-4

-2

2

4

6

8

x

y

3. Given the equation 2 8 12 f x x x , determine the following.

a. Coordinates of the x-intercept(s) __________________

b. Coordinates of the y-intercept _________________

c. State the coordinates of the vertex. ______________

d. Sketch the graph on the given axes and label at least 4 point.

e. State the domain: ________________ State the range: _________________

4. A patient’s body temperature, T °F, after x hours after taking a fever reducing drug can be

approximated by 2

0.4 2.8 103 T x x x . Interpret your results with a complete

sentence.

a. When will the patient’s temperature be a minimum?

b. Determine the patient’s minimum temperature. Round to nearest tenth.

Activity 8.4 Application Problems Name:_______________

Math 95

-4 -3 -2 -1 1 2 3 4-10

10

20

30

40

50

60

70

80

90

100

110

x

y

1. A company’s monthly profit P is related to the number of items sold by the function

20.2 50 800.P x x x Algebraically determine the number of items sold each month to

achieve the maximum profit and determine the maximum profit. Write your answer in a complete

sentence.

2. A ball is thrown upward from the roof of a 100-ft building with an initial velocity of 20 ft/sec. The

equation 216 20 100f t t t represents this situation.

Algebraically determine the following and give all results to three decimal places.

a. When will the ball reach its maximum height and what is the maximum height?

Answer the question with a complete sentence and label the results on the graph.

b. Find the time the ball hits the ground. Answer the question with a complete sentence and

label the results on the graph.


3. Each table shows a relationship between speed (miles per hour) and gasoline consumption (miles

per gallon, MPG). In each case, use quadratic regression to find a quadratic function that best fits

the data and answer the following.

#1 Car: Oldsmobile

Speed 5 15 20 25 30 50 60 70 75

MPG 5.1 11.4 12.5 15.6 19.0 27.3 28.2 22.9 21.6

a. Find the Oldsmobile quadratic function

(in the form 2 y ax bx cand round to 3 decimal places): __________________________

b. Use the quadratic function from part a. to estimate the speed

(in miles per hour) the Oldsmobile should be driven to give the best

gas mileage (round answer to one decimal place and include units). ___________________

c. State the Oldsmobile’s best gas mileage

(round answer to one decimal place and include units). ___________________

#2 Car: Honda

a. Find the Oldsmobile quadratic function

(in the form 2 y ax bx cand round to 3 decimal places): __________________________

b. Use the quadratic function from part a. to estimate the speed

(in miles per hour) the Oldsmobile should be driven to give the best

gas mileage (round answer to one decimal place and include units). ___________________

c. State the Honda’s best gas mileage

(round answer to one decimal place and include units). _______________________

Speed 10 15 20 25 30 45 60 70 75

MPG 15.9 21.2 25.1 27.3 28.0 30.1 28.3 23.8 22.9

Activity 9.1 Simplifying Rational Expressions Name:________________

1. A rational expression is an expression of the form _________

where P and Q are _________________________________ and ____________.

2. A company has setup costs of $4,500 for the production of Wibble-Wobbles. The cost to produce

a single Wibble-Wobble is $7.75.

a. Write an expression that gives the total cost of producing x Wibble-Wobbles.

b. Write a rational equation that gives the average cost per Wibble-Wobble when x Wibble-

Wobbles are produced.

c. Use your rational equation to find the average cost when 80 Wibble-Wobbles are produced.

Answer with a complete sentence.

d. Use your rational equation to find the average cost when 1000 Wibble-Wobbles are produced.

Answer with a complete sentence.

3. Simplify2

2

12 5 2

3 5 2

x x

x x

.

Here is a student solution: 2 2

2

12 5 2 12

3 5 2

x x x

x x

5x 2

23x 5x 2

4 1 12

1 1 1

Show the student how to correctly simplify the expression:


4. For what value(s) of x is the following rational expression undefined?

a. 16

x

x b.

2 1

x

x c.

16

x

5. Recall:

6. Simplify. Assume the denominator is not 0 and express all answers with positive exponents.

a. b.

c. d.

e. f.

7. Polynomial opposites:

8. Perform the indicated operation and express as a simplified expression using positive exponents.

a. b.

9 2

2 9

21

12

x y

x y

2

2

2 35

11 30

x x

x x

9 2

2 9

21

12

x y

x y

4

2

81

5 6

x

x x

2 2

2 2

9

8 15

x y

x xy y

__________

a b

b a

2

2 2

4

x

x

2

5 20

16

x

x

2

2

11 24

64

x x

x

0 n

m n m na a a a a

Activity 9.2 Simplifying Rational Expressions Name: ________________

Math 95

1. Recall:

2. Multiplying Rational Expressions:

____________ P R

Q S where Q and S are not equal to ___________.


a. b.

4. Dividing Rational Expressions:

____________ P R

Q S where Q, R and S are not equal to __________.


a. b.

0 n

m n m na a a a a

9 2 6 3

2 5 9

15 18

27 45

x y x y

x x y

9 2 6 3

2 5 9

15 18

27 45

x y x y

x x y

2 23 18 9

9 54 12 36

x x x

x x

2 23 18 9

9 54 12 36

x x x

x x


6. Let 2 3 4

5

x xf x

x

and

2 20

4

x xg x

x

. Answer the following and show all work.

a. State the domain of f: _______________ State the domain of g: _______________

b. Find 0f .

c. Find 0g .

d. Find 0 0f g .

e. Find f x g x .

f. Use your results from part e. to evaluate 0 0f g . How does this compare to your result

in part d.?

.

g. Use your results from part e. to evaluate 10 10f g

h. Are you able to use your results from part e. to evaluate 4 4f g ? CAREFUL!!!

Activity 9.5 Graphing Rational Functions Name:________ _________

Math 95

1. Find the vertical asymptote of the following functions.

a. 1

4f x

x

has vertical asymptote at ____________________________

b. 1

5

xg x

x


c. 2 5

xh x

x


d. 15

5 10

xk x

x


2. Find the horizontal asymptote of the following functions.

a. 1

4f x

x

has horizontal asymptote at ____________________________

b. 1

5

xg x

x


c. 2 5

xh x

x


d. 15

5 10

xk x

x



-10 -8 -6 -4 -2 2 4 6

-6

-4

-2

2

4

6

8

10

x

y

-10 -8 -6 -4 -2 2 4 6

-6

-4

-2

2

4

6

8

10

x

y

3. Given the function3

( )3

xf x

x , determine the following and graph the function.

a. Domain: _______________

b. State the x-intercept: ____________

c. State the y-intercept: ____________

d. State the equation of the vertical asymptote

e. State the equation of the horizontal asymptote

f. Graph the function. Plot and label at least 4 points with their coordinates and draw smooth

curves with arrows on the ends. Draw and label the vertical and horizontal asymptotes.

4. Given the function 3

3

h x

x , determine the following and graph the function.

a. Domain: _______________

b. State the x-intercept: ____________

c. State the y-intercept: ____________

d. State the equation of the vertical asymptote

e. State the equation of the horizontal asymptote

f. Graph the function. Plot and label at least 4 points with their coordinates and draw smooth

curves with arrows on the ends. Draw and label the vertical and horizontal asymptotes.

Activity 9.6 Solving Rational Equations Name:_________________

Math 95

40

4

x

x

7 2

9 9

x

x x

16 3 3 3 2

6 5 15

x x x

x

1 1 1

6 8 t

1. Algebraically solve the following and check for extraneous solutions.

a. b.

c. d.


2 2 2

2 3 2

4 3 9 2 3x x x x x

2. Algebraically solve

3. In sections 9.1 and 9.2, we simplified expressions. In section 9.6, we solved equations.

Categorize each of the following by writing either “Expression” or “Equation”.

a. 2

2 35

9x x b.

2 4

2

x

x x

c. 2 1

13

xx

d.

2 11

3

xx

e. 4 3 10

4x x f.

4 3 10

4x x

Activity 10.1 Algebra of Functions Name:_________________

Math 95

-4 -3 -2 -1 1 2 3 4

-4

-3

-2

-1

1

2

3

4

x

y y p x

y h x

1. Algebra of Functions:

( ) ( )f g x f g x

( ( ) )f g x f g x

2. Use the information from the tables to answer the

following. Show all work.

a. (2)f b. (2)g c. (2)g f

d. (3)f g e. ( 3)f g f. (3)g f

h. ( 3)f g i. ( 3)f g j. ( 3)g f ?

3. Use the information from the graphs to answer the following. Show all work.

a. (1)p b. ( 4)p c. ( 4)h

d. ( 2)p h e. ( 2)p h

f. ( 2)p h g. ( 2)p h h. ( 2)h p

x f(x) x g(x)

-3 -1 -3 20

-1 7 0 30

2 6 2 40

3 -3

3 50


Math 95

4. Let ( ) 2 7f x x and ( ) 5 15g x x . Answer the following. Show all work

a. (2)f b. ( )f y

c. (2)g d. ( )g

e. (2)f g f. (5)f g

g. (3)f g h. ( )f g x

i. ( )f g x j. What is domain of ( )f g x ?

Activity 10.2 Composition of Functions Name:_________________

Math 95

-4 -3 -2 -1 1 2 3 4

-4

-3

-2

-1

1

2

3

4

x

y y p x

y h x

1. The composition of functions f and g is the function f g x ____________________.

2. Use the table to evaluate the following.

a. 3f g b. 3g f

c. 1g f d. 0f f

d. 2f f e. 0g g

3. Use the graph to find the value of each composition.

a. 1h p b. 2p h

c. 2h p d. 4p h

e. 4h h f. 4p p

x f x g x

1 0 2

0 2 1

1 1 3

2 3 2

3 1 0


4. Given 2 3f x x and 2g x x find the following.

a. 3f g

b. 3f g

c. 3g f d. 0f g

e. f g x

f. g f x

5. Charlie averages 62 mi/hr traveling on the highway. His vehicle averages 26 mi/gal and his tank

holds 15 gallons of gas. Assume the tank is full when Charlie starts his trip.

a. Express the number of miles driven as a function of t, time on the road.

b. Express the number of gallons used, N, as a function of M, miles driven.

c. Express the number of gallons used as a function of t, time on the road. Simplify your answer.

d. If Charlie is on the road for 1 hour, how much fuel has he used? Round to the nearest tenth and

answer the question with a complete sentence.

Activity 10.3 Inverse Functions Name:________________

Math 95

-8 -6 -4 -2 2 4 6 8

-8

-6

-4

-2

2

4

6

8

x

y

-8 -6 -4 -2 2 4 6 8

-8

-6

-4

-2

2

4

6

8

x

y

-8 -6 -4 -2 2 4 6 8

-8

-6

-4

-2

2

4

6

8

x

y

1. Functions f and g are inverse functions if f g x for all x the domain of f and g f x

for all x in the domain of g.

2. To find the Inverse of a Function:

Step 1: If given a function f x , write ________ in place of f x

Step 2: Switch _____and _____

Step 3: Solve for _______

Step 4: Rename y as _____

Step 5: Check that 1 1 and f f x f f x

3. Find the inverse functions for the following.

a. 6 3

5

xf x

b. 4 8f x x

4. A function is one-to-one if no two distinct domain elements are paired with the same _____ element.

Horizontal Line Test: A function is one-to-one if no ________________________ passes through

_______ or more points on its graph.

The ________ of a function is also a function if the graph of f passes the ______________Line Test.

5. Determine whether the following functions are one-to-one and whether the inverse is also a function.

a. b. c.


-8 -6 -4 -2 2 4 6 8

-8

-6

-4

-2

2

4

6

8

x

y

6. Consider the function 2 4f x x

a. Graph the function and its inverse on the same axes.

b. Find the inverse function of.

c. Verify that f and 1f are inverse functions.

Activity 10.4 Exponential Functions Name:_______________

Math 95

1. Let 2

1

4

x

g x

. Express your result as a simplified, improper fraction if necessary.

a. Find 2g .

b. Find 0g .

c. Find 2g .

d. Find 3g .

e. Find 4g . f. Find 5g .

2. Use property of Exponential Equations to solve the following (do not use logarithms, not yet ).

a. 2 13 243x b.

4 25 125x

c. 1

381

x d. 4 12

16

x


-5 -4 -3 -2 -1 1 2 3 4

5

10

15

x

y

-4 -2 2 4 6 8 10 12 14 16 18

5000

10000

15000

20000

25000

3. Consider the function 1

3

x

f x

.

a. Is this function exponential decay or growth?

b. State the x-intercept: _______________

c. State the y-intercept: _______________

d. State the horizontal asymptote: _______________

e. Graph the function on the grid. Plot and label three points.

f. State the domain: ____________________ State the range: ____________________

4. Matt bought a new car for $25,000. If the car depreciates approximately 15% of its value each

year, then the value of car after x years is given by 25000 0.85x

V x .

a. Label the axes with the appropriate units.

b. Is this function exponential decay or growth?

c. Sketch the graph of the function.

d. Use this function to find the value of the car in 5 years.

Plot and label this point on the graph.


e. Use this function to find the value of the car in 15 years.

Plot and label this point on the graph.


f. Use the “Intersect” feature of your calculator to graphically determine when the car will be worth

$5000. Plot and label this point on the graph. Interpret your results with a complete sentence.

Activity 10.5 Logarithmic Functions Name:________________

Math 95

5 10 15 20 25

-1

1

2

3

4

x

y

1. Definition of Logarithms: For 0 and 1, if and only if b b

2. Answer the following using the definition of logarithms.

a. 2log 8 ___________ b. 3

1log

3

_______

c. 7log 1 ___________ d. 3log 3 __________

e. 9log 27 ________ f. 1

4

log 16 ________

g. 3

1log

27

________ h.

3

1log

27

____________

3. Solve for y:

a.

9log 27 y

b. 1

4

log 16 y

c. 3log 0 y d. 5log 2 y

4. Let 3logy x .

a. Find 5 ordered pairs.

b. Graph the function.

Plot and label 3 points.

Graph and label any asymptotes.

c. State any asymptotes.

d. State the domain.

e. State the range:

x 3logy x

http://www.youtube.com/watch?v=TOoXH_IwUx0

http://www.youtube.com/watch?v=TOoXH_IwUx0


-4 -2 2 4 6 8 10 12 14 16 18

5000

10000

15000

20000

25000

5. Find the inverse for each function.

a. 10x

f x b. 3 2xf x

6. Matt bought a new car for $25,000. If the car depreciates approximately 15% of its value each

year, then the value of car after x years is given by 25000 0.85x

V x .

a. Sketch a graph of the function. Label the axes with the appropriate units.

b. Use the function to algebraically determine when the car is worth

$12500. Round the nearest tenth of a year.

Interpret your result with a complete sentence.

Plot and label this information on the graph.

c. Use the function to algebraically determine when the car is worth $5000. Round the nearest

tenth of a year. Interpret your result with a complete sentence. Plot and label this information

on the graph.

Activity 10.6 Properties of Logarithms Name:________________

Math 95

1. Circle either True or False.

True or False: log log logx y x y

True or False: log log logx y x y

2. Evaluate the following without a calculator.

a. 4log 1 b.

4log 2 c. 4log 64 d. 4

1log

2

3. Write 2log 64 6 in exponential form.

4. Write1 1

33

in logarithmic form.

5. What is the value of x for 2log 5x ?

6. Write in expanded form.

a. 2 1/2

5ln

x y

z

b. 6

loga b

c

http://www.regentsprep.org/Regents/math/algtrig/ATE9/logs.htm


7. Write log 4log 5logx y z as a single logarithm.

8. Write the following in expanded form.

a. 2log 2x b. 2

lnx

y

c. 3 5log

x y

z

9. Write the following as a single logarithm.

a. 2 2log logx z b. 3ln 2ln 4lnx y z c. 2 2

13log 4log

5x y

10. Write 3 15ln 13ln 5 ln 1

2x x x as a single logarithm.

http://www.youtube.com/watch?v=n4duiJyCLEM

http://www.youtube.com/watch?v=4HftR1d8laU

http://www.youtube.com/watch?v=4HftR1d8laU

Activity 10.7 Solving Exponential and Logarithmic Equations Name:________________

Math 95

1. Algebraically solve for x.

a. 54 30x

b. 2 5 20x

2. If $1,500 is invested in an account that earns an annual interest rate of 3% and is modeled by

1500 1.03t

A t , algebraically determine the number of years it will take the money to triple.

Round results to the nearest tenth and interpret with a complete sentence.


3. The radioactive element strontium 90 has a half-life of about 28 years. Initially, there is 10grams

of Strontium 90 and the amount remaining after t years is given by /28

110

2

t

A t

.


a. Algebraically determine how much Strontium 90 there will be in 50 years.

b. Algebraically determine how long it will take for the 10 gram sample to decay to 7 grams.

4. Ubertown’s population is about 32,000. If there is a projected growth rate of 2% per year, t years

from now the population can be modeled by 0.0232000 tP t e . Algebraically determine the

number of years it will take for the town’s population double.


Activity 7.1 Roots and Radicals Name: In general, we state ...

Documents

Transcript of Activity 7.1 Roots and Radicals Name: In general, we state ...