1. Which of the 12 parent functions we know from chapter 1 are power functions? List their equations and names.

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Pre Calculus Worksheet 2.2

2. Analyze each power function using the terminology from lesson 1-2. Think of these as parent functions too.

a) f x( )³ x b) 1₂

( )

f x  x

Why is this a power function? Why is this a power function?

Graph this function (label 5 points) Graph this function (label 2 pts & asymptotes)

Domain: Domain:

Range: Range:

Symmetry: Symmetry:

Boundedness: Boundedness:

Asymptotes: Asymptotes:

Discontinuities: Discontinuities:

Increasing/Decreasing: Increasing/Decreasing:

Extrema: Extrema:

x y

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3. Graphs of Power Functions: The key to power function graphs is the exponent in y



x^a. If a is a positive even number or a positive odd number, you already know what these functions look like from your parent functions.

a) Use your knowledge of parent functions to fill in the 1^st column of the chart below with a basic sketch . 1

a a 0

0  

a

1

b) Use your graphing calculator to graph the following functions and fill in the 2^nd column in your chart.

2 3 4 5

, , , ,...

y



x^ y



x^ y



x^ y



x^

c) Use your graphing calculator to graph the following functions and fill in the 3^nd column in your chart.

1 1

3 5

2

, ,

4

, ,...

y



x y



x y



x y



x

d) Now go back to the functions given in question 2 and rewrite them in y



x^a form.

Which part of the chart above do they match?

4. Match the equations to one of the curves labeled in the figure below

y= -²₃x⁴

1 5

y=2x^-

14

2 y= x

5

y= -x 3

2 2

y= - x^-

23

1.7

y= x

“Even” “Odd”

a b c

d

g

i h j

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5. For mammals and other warm-blooded animal to stay warm requires quite a bit of energy. Temperature loss is related to surface area, which is related to body weight, and temperature gain is related to circulation, which is related to pulse rate. In the final analysis, scientists have concluded that the pulse rate r of mamals is a power function of their body weight w.

a) Find the power regression model.

(No ROUNDING)

b) Use your model from part a to predict the pulse rate for a 450-kg horse.

The remaining questions on this worksheet focus on the application of power functions to direct and inverse variation.

6. Multiple Choice: Which of the following exhibit inverse variation?

A The distance traveled as a function of speed.

B The total cost as a function of the number of items purchased.

C The area of a circular swimming pool as a function of its radius.

D The number of posts in a 20ft fence as a function of distance between posts.

For questions 7 and 8, write a power function to model the situation.

7. The surface area S of a sphere varies directly as the square of the radius r.

8. The period of time T for the full swing of a pendulum varies directly as the square root of the pendulum’s length L.

9. The time it takes for a group of volunteers to build a house varies inversely with the number of volunteers v.

a) Write a power function to model this situation.

b) If 20 volunteers can build a house in 62.5 working hours, how many volunteers would be needed to build a house in 50 working hours. (Hint: Find the value of k first.)

Mammal Body Weight (kg)

Pulse Rate (beats/min)

Rat 0.2 420

Guinea pig 0.3 300

Rabbit 2 205

Small dog 5 120

Large dog 30 85

Sheep 50 70

Human 70 72

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10. The power P (in watts) produced by a windmill is proportional to the cube of the wind speed v (in mph).

a) Write a power function to model this situation.

b) If a wind of 10 mph generates 15 watts of power, how much power is generated by a wind speed of 40mph?

11. Write a power function for the following situation, and then use the equation to solve for the missing information.

The intensity I of light varies inversely as the square of the distance D from the source. If the intensity of illumination on a screen 5 feet from a light is 2 foot-candles, find the intensity on a screen 15 feet from the light.

12. The volume V of a gas varies inversely as the pressure P and directly as the temperature T. A certain gas has a volume of 10 liters (L), a temperature of 300 kelvins (K), and a pressure of 1.5 atmospheres (atm). If the gas is

compressed to a volume of 7.5 L and is heated to 350 K, what will the new pressure be?

13. The power P that must be delivered by a car engine varies directly as the distance d that the car moves and inversely as the time t required to move that distance. To move the car 500 m in 50 s, the engine must deliver 147 kilowatts (kW) of power. How many kilowatts must the engine deliver to move the car 700 m in 30 s?

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Before we can use what we know about polynomials, we need to check that we do have a polynomial function. Thinking about what a polynomial looks like, we need identify functions as polynomials or not.

1. Which of the following functions are polynomial functions? For those that are, state the degree and leading coefficient. For those that are not polynomials state why not.

a) ^{f x}^{( )}^{ }⁹ ²^x⁵^ ²

 

^x² ^b)^{g x}^{( )}^³^x^⁴ ^¹⁷ ^c)^{h x}^{( )}^⁵ ^d)^{k x}^{( )}^{ }⁹^x³

2. Which of the polynomial functions above are also power functions (if any). Explain.

3. Using function notation, if f (a) = b, then the function contains the point _____________ .

The next few problems focus on Linear Functions…

4. The average rate of change of a function between two points (a, f (a)) and (b, f (b)) is given by:

5. Write the equation of the linear function, f, if you know the given information.

a) f (2) = –7 and f (–1) = 5 b) f (–5) = –1 and f (2) = 4

6. The table shows the average weekly earnings of construction workers for several years. Let x = the number of years since 1970.

a) Find the linear regression model for the data.

b) What does the slope of the regression represent in terms of this situation?

c) Use the linear regression model to predict the construction worker average salary in 2005.

DO NOT ROUND your regression model to determine your answer.

Year 1970 1980 1990 1995 2000

Weekly Earnings 195 368 526 587 702

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Now to focus on Quadratic Functions…

We have covered Quadratic functions extensively in Algebra 2, F.S.T. and in our study of parent functions in chapter 1.

In order to put a standard form quadratic function into vertex form, we use a process called Completing the Square. This method works well when we have an a value of 1 and/or a b value that is even.

7. Complete the square to rewrite each quadratic function in vertex form, describe the transformation, and then graph the function. A sample is provided for your reference.

a) f x( )x²4x 6 b) f x( )2x²

 6

x7

8. Use x = -₂^b_a to find the vertex of the function in question 7a and 7b. What do you notice?

9. Find the equation of a quadratic function of the form ^{f x}

 

^^{a x}



^^h



²^^k with the given information:

a) passes through (2, –13) and has vertex (–1, 5) b)

10. Among all the rectangles whose perimeters are 100 ft, find the dimensions of the one with the maximum area.

11. A large painting is 3 feet longer than it is wide. If the wooden frame is 12 inches wide and the area of the painting is 208 ft , find the dimensions of the painting. ²

Sample:

2 2 2

2

3 12 7 3( 4 ) 7

3( 4 ) 7

3( 2) 5

y x x

y x x + 4 12

y x

  

   

  

x y

(1,3) (3, 11)

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12. Jack was named manager of the month at the Sylvania Wire Company due to his hiring study. The study showed that each of the 30 salespersons he supervised averaged $50,000 in sales each month, and that for each additional sales person he would hire, the average sales would decrease by $1,000 per salesperson.

a) Write a function for Jack’s study showing the total sales as a function of the number of additional salespersons Jack hires.

b) Use your function to determine how many salespersons Jack should hire to maximize the income from sales. What is this maximum sales Jack’s company can anticipate according to the model?

Projectile Motion

All projectile motion can be represented by a quadratic function. Most quadratics are written in standard form based on a formula discovered by Sir Isaac Newton in the 17^th century. Newton discovered that the height h on an object at time t after it has been thrown upward with an initial velocity vo from an initial height ho satisfies the formula:

1 2

( ) 2 _o _o

h t  gt v th where g is the acceleration due to gravity (–9.8 meters per second² or –32 feet per second²) 13. What is the leading coefficient for every projectile motion function whose height is measured in feet? What is the leading coefficient for every projectile motion function whose height is measured in meters?

14. A baseball throwing machine is use to train little league players to catch pop-ups. The machine throws baseballs straight upward with an initial velocity of 48 ft/sec from a height of 3.5 feet.

a) Find an equation that models the height of the ball as a function of the ball t seconds after its thrown.

b) What is the maximum height the ball will reach? How many seconds will it take to reach this height?

c) If the player misses the catch, how long will it take the ball to hit the ground?

15. Consider the function f x

( )

=2x²-3x.

a) Find the average rate of change between the points on the function when x = 1 and x = 2.

b) On a graph, a line that connects any two points is called a secant line. Find the equation of the secant line that connects the points on the function when x = 1 and x = 2.

c) In calculus, we find the slope between a point on the graph when x = a and another point that is a small distance away. We use x = a + h for the second point. Find the average rate of change between the points x = a and x = a + h.

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In lesson 2.3 we are putting together lots of skills from Algebra 2. If you are having difficulty with any of these individual skills, you need to check out a textbook and do the OPTIONAL problems for your particular area of difficulty listed at the bottom of this worksheet BEFORE you attempt to put all the skills together.

For each of the functions below…a) State the degree of the polynomial, b) list the zeros of the polynomial function, c) state the multiplicity of each zero and whether the graph crosses the x-axis or bounces off the x-axis at the corresponding x-intercept, and d) list the end behavior using limit notation and e) using all this information, sketch the function without a calculator.

1. ^{k x}

 

^⁽²^x^¹⁾⁽^x^⁴⁾³ ^2.^{h x}

 

^^x³⁽²^^x⁾²

3. ^f

 

^x ^⁶⁽⁴^x^^{3) (}² ^x^⁸⁾⁴ ^4.^{h x}

 

^^{(10 5 )(9}^ ^x ^x²^{ }³^x ²⁾

5. f x( )4x³20x²9x45 6. g x( ) 3x⁴

 5

x³

 2

x²

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7. What is the y-intercept of the graph of ^{f x}^{( )}^²



^x^¹



³^⁵? (NO, its not 5 and NO using your calculator!!)

8. Explain why the function g x

( ) 

x⁹

 

x

50

has at least one real zero.

9. Using only algebra, find a cubic function in standard form with zeros 3,



¹₄ and 6. Check with a graphing calculator.

10. Many students begin to question why we spend all this time graphing by hand when we have a graphing calculator?

To answer this question…graph the function f x

( ) 10 

x⁴

 19

x³

 121

x²

 143

x

 51

using the given window and sketch the graph, then answer the questions that follow.

Window A: 0.5 x 1.5,  1 f x( )1 Window B:   6 x 4, 2000 f x( )2000

a) Which window matches what you know about the end behavior of the function? Why?

b) What is missing from Window B that shows up in Window A?

c) Why can’t we just rely on “what the calculator says?”

Next let’s do some numerical analysis of polynomial functions. Given each function and corresponding table of values below, find the difference in the y values of the function. Then, find the difference in the results. Continue this process until you get a constant. How many “levels” of differences did you take? What can you tell about function according to the number of “levels” of differences?

11. f x( )x³5x12 12. f x( )x⁴ 7x³2x x –3 –2 –1 0 1

f(x) 0 14 16 12 8

x –1 0 1 2 3 4

f(x) 6 0 –4 –36 –102 –184

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13. Use the given table of values to determine the degree of the polynomial function whose curve will “fit” the data.

Then, find the equation of that polynomial function using the regression capabilities of your graphing calculator.

14. Sometimes the numerical analysis does not work and the differences are never constant. In this situation, you can graph a scatterplot, as well as trying the possible regression equations (linear, quadratic, cubic or quartic) to determine which function best fits the given data. Using the data below, determine the best model and explain why you chose this model. A graphing calculator should be used!

Now for an application…

15. A state highway patrol safety division collected data on stopping distances according to certain speeds of travel. This data is summarized in the table below.

a) Find the quadratic regression model.

b) Use the regression model to find the stopping distance for a vehicle traveling at 25 mph. Do NOT round your model, but round your answer to the thousandths.

c) After an accident, a policeman measured the stopping distance (skid mark) of one car to be 300 feet. The driver claimed the car was traveling at the 55 mph speed limit when the brakes were applied. Use the regression model (and the graphing capabilities of your calculator) to predict the speed of a car when the stopping distance was 300 feet.

Was the driver telling the truth?

OPTIONAL EXTRA PRACTICE…see your teacher to borrow a textbook.

Using limit notation to describe end behavior of polynomials – page 203: #25-28 Find zeros of polynomial by factoring – page: 203: #33-38

Use end behavior and multiplicity of zeros to sketch polynomial by hand – page: 203: #39-42, 73, 74

x –3 –2 –1 0 1 2

f(x) –191 –84 –27 –2 9 24

x 3 4 5 6 8

f(x) –7 –4 –11 8 3

Speed

(mph) 10 20 30 40 50

Stopping

Distance (ft) 15.1 39.9 75.2 120.5 175.9

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A summary statement is another way to write the answer to a division problem. For example, look back at example 1 from notes 2.4. We found that

4 3

3 2

2 3 5 1

2

2 2 9 17

2

x x x

x x x x

x

  

     

 . Suppose we multiplied BOTH sides of this equation by x – 2. The fractions would cancel to leave ²^x⁴^³^x³^⁵^x^{ }¹



^x^^{2 2}

 

^x³^^x²^²^x^{ }⁹



¹⁷^.

In other words, ²^x⁴^³^x³^⁵^x^{ }¹ ^{p x}^{( )}^



^x^^{2 2}

 

^x³^^x²^²^x^{ }⁹



¹⁷

1. Use long division to divide. Write a summary statement in polynomial form.

a)

2

x

 1 4

x³10x²

 6

b)

4 3 2

2

2 3 4 6

2 1

x x x x

x x

   

 

2. Explain why the following problems can be done using synthetic division; then use synthetic division to divide. Write a summary statement in polynomial form.

a)

3 5 2 3 2

1

x x x

x

  

 b)



⁵^x³^⁷^x^{ }³

 

^x^⁴



When we are only concerned with the remainder from division, we can use the remainder theorem to evaluate a

polynomial when given a value or a factor. Find the remainder when f x

( ) 

x³



x²

 5

x

 5

is divided by x k



for the given value of k.

3. a) k

  3

b) k

 1

4. What do you notice about question 3b above? Use this information to factor f (x).

This is the summary statement.

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5. Find g( 3) when g w

( )   2

w⁵

 8

w³

 25

w

 4

using two different algebraic strategies from this lesson.

Sometimes we can give you a graph to help you factor a polynomial.

6. For each given function, use the graph to guess and synthetic division to verify possible linear factor(s). Then, write the polynomial in factored form.

a) f x

( )  5

x³

 7

x²

 49

x

 51

b) g x

( )  5

x³

 12

x²

 23

x

 42

7. Use the factor theorem to decide if (x – 2) is a factor of x³3x4. Explain how your work relates to the factor theorem.

8. Find the polynomial function in factored form: y = a(x – b)(x – c)(x – d) for the given data.

x 1 3 –4 –1 y 0 0 0 2

9. Find the polynomial function in standard form with leading coefficient 2 that has the given degree and zeros.

a) Degree 3, with –1, 0, and 4 as zeros b) Degree 3, with –2, ¹₂, and ⁴₃as zeros.

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10. What does the Rational Zeros Theorem tell you?

11. Multiple Choice: Let f x( )2x³7x²2x . Which of the following is NOT a possible rational root of f ? 3 A. –3 B. –1 C. 1 D. 1/2 E. 2/3

12. Multiple Choice: Let f be a polynomial function with (3)f  . Which of the following statements is NOT true? 0 A. x + 3 is a factor of f(x).

B. x – 3 is a factor of f(x).

C. x = 3 is a zero of f(x).

D. 3 is an x intercept of f(x).

E. The remainder when f(x) is divided by x – 3 is zero.

13. List all possible rational zeros for the given function. Then, determine which, if any, possible zeros are actually zeros. NO CALCULATOR!!

3 2

( ) 2 9 14 5

f x



x



x



x



14. For the function below, list the possible rational zeros. Then, use your calculator to determine which of these possible zeros are zeros and verify them algebraically (don’t just say the calculator said so!!). Finally, rewrite the function in factored form.

4 3 2

( ) 6 73 227 224 64

h x



x



x



x



x



15. The Sunspot Small Appliance Company determines that the supply function for their EverCurl hair dryer is

( ) 6 0.001

3

S p

 

p and the its demand function is D p

( )  80 0.02 

p², where p is the price of the hair dryer.

Determine the price for which the supply is equal to the demand and the number of hair dryers corresponding to this equilibrium price.