MA134 College Algebra Common Final Topic Review December 9, 2010 For the final exam, you should be able to:

1. Solve equations/find zeros of functions

a. Quadratic (You may use your choice of the methods below.)

• By factoring (page 115 example 2 and 3)

• By completing the square (page 118 example 4, page 119 example 5)

• By quadratic formula (page 120 example 9)

b. Polynomial with degree greater than 2

• Use the rational zeros theorem to find zeros of a polynomial function/solve a polynomial equation and write as a

product of linear factors (page 385 example 4, page 386 example 5, except you do not need to use Descarte’s Rule of

Signs and may use synthetic division to simplify steps 7 through 10 to get to the quadratic quotient )

c. Solve exponential and logarithmic equations (See #5d and #5e below)

2. Solve inequalities

a. Polynomial of degree greater than or equal to 2 (page 150 example 6)

b. Rational (page 151 example 7)

3. Graph functions. Accurately plot and label x-intercepts and any horizontal or vertical asymptotes. Label axes appropriately.

a. Quadratic

• Find the vertex by the method of completing the square (page 332 example 4)

• Find the vertex using the formula (page 333 example 5)

b. Use transformations

• Know basic functions listed on page 257

• Be able to apply transformations listed on page 274

• See examples 1 through 6 on page 266-273

c. Polynomial functions of degree greater than 2

d. Rational functions (page 400 example 9)

• Refer to the summary box on page 398. The final will not include a rational function with a slant asymptote.

e. Piece-wise functions (page 255 example 6)

f. Exponential and logarithmic functions (See #5d and #5e below)

4. Work with and understand function notation

a. Add, subtract, multiply and divide functions (page 280 example 1)

b. Composition of functions (page 282 example 3)

c. Find domain (page 236 example 10, page 283 example 4)

d. Find the inverse of a function (summary box on page 296, page 297 example 7)

e. Evaluate a difference quotient (page 235 example 9)

f. Use graphs to find intervals over which a function is increasing or decreasing (page 250 example 2)

g. Find the average rate of change for a given function between two given points (page 251 example 3)

5. Work with and understand exponential and logarithmic functions

a. Graph exponential functions

• Understand properties of exponential functions (page 424)

• Graph an exponential function using transformations (page 427 example 5)

• Find domain, range and equation of the asymptote of an exponential function

b. Graph logarithmic functions

• Understand properties of logarithmic functions (page 447)

• Graph a logarithmic function using transformations (page 442 example 7 and page 443 example 8)

• Find domain, range and equation of the asymptote of a logarithmic function

c. Understand and be able to use properties of logarithmic functions (summary box on page 451, page 452-457 examples 1

through 9)

d. Solve exponential equations (page 462-464 examples 2 through 4)

e. Solve logarithmic equations (page 464 -465 examples 5 through 7)

6. Work with and understand sequences and series

a. Understand notation for the nth term of a sequence (page 706 example 1)

b. Use a recursive formula for a sequence to write several terms (page 710 example 5

c. Use sigma (summation) notation (page 711 example 7)

d. Arithmetic sequences and series

• Find the nth term (page 719 examples 2 and 3)

• Find the sum of the first n terms (page 721 example 4 and 5)

e. Geometric sequences and series

• Find the nth term (page 728 example 2)

• Find the sum of the first n terms (page 730 example 4)

• Find the sum of an infinite geometric series, if it exists (page 731 example 6)

7. Given an equation, identify and graph conics

a. Parabolas (see 3a for alternative methods)

• Graph a parabola (gray box on page 642, page 644 example 2)

b. Circles

• Know and be able to use the standard equation of a circle (box on page 210, page 211-213 examples 1, 3 and 5)

c. Ellipses

• Graph an ellipse centered about the origin (box on page 656, page 657 examples 1 and 2)

d. Hyperbola

• Graph a hyperbola centered about the origin (box on page 668, page 670-671 examples 3 and 4)

8. Solve applied problems involving

a. Linear equations and functions (page 204 example 11)

b. Maxima or minima (page 335 example 7)

c. Exponential growth and decay (page 471-472 examples 1 and 2)

Textbook: Young, Cindy. (2009) College Algebra (2rd ed.), John Wiley & Sons, Inc.

PROBLEM SET

Label word problem answers correctly. Show all work for full or partial credit. Give exact, simplified answers

unless otherwise specified (e.g., 2

3 instead of 10

15, and 23 instead of 18 ).

1. Find all solutions for a) 2

0 2 2 5x x= − + b) 3 22 3 0x x x+ − > c)

24 8x x− = d.)

( 3)( 2)0

1

x x

x

− +≤

−

2. Give an equation of the circle with a radius of 5 and center at (3, -2).

3. Give the center and radius for the circle with the given equation: 2 2

8 29 04x y x y+ − − =+

4. Graph each of the following. Give the original function being transformed and list each transformation. Label at least two

points on the final graph. Sketch and label any asymptotes.

a) 3

1( ) 4

3g x x= − + b.) ( ) 6 500h x x= − + c) ( )2

log 4 3y x= + + d) 4

2x

y e−= − −

5. Find the equation of the function that is finally graphed after the following transformations are applied to the graph of

y x= : Reflect about the x-axis, then reflect about the y-axis, then shift up 2 units.

6. For ( ) 2 ( 7) 14g x f x= + − , explain how the graph of ( )g x has been transformed from the graph of ( ).f x

7. Find the domain of

a) 2

5( )

7 18

xf x

x x

−=

− − b) ( ) 3 8 5f x x x= − + c)

3( )

xf x e

−= d) 2

( ) 4f x x x= −

e) ln(7 )y x= −

8. Given the functions 2)( −= xxf and 2

( ) 1g x x= − , find the following:

a.) ( )(3)f g+

b.) ( )g x−

c.) ( ) )(xfg ⋅ d.)

( )( )g f x�

e.) ( )g x−

f.)

( ( ))f g x

g.) ( )( )xgf /

and state the domain

9. Find ( ) ( )f x h f x

h

+ − for a) ( ) 3 5f x x= − b)

2( ) 5f x x x= + −

10. Determine if each of the following has a maximum or minimum value, and find the maximum or minimum value.

a) 2

( ) 3 6 1g x x x= − +

b) 2

( ) 0.002 6 100g x x x= − − +

11. Graph, accurately plotting x-intercepts and any horizontal or vertical asymptotes, and give the location of any holes in the

graph. Label all the asymptotes. For piecewise functions label at least one point on each piece.

a.)

2

2( )

2

2 3

50

xf x

x

x=

+ −−

b.) 2 2 3

2 ( 4) ( 3)y x x x= + −

c.)

2 27 4 28x y+ =

d.)

2

4( )

1f x

x=

+

e.)

2

2

4 21

7 12

x xy

x x

+ −=

− +

f.)

25 10 5y x x= + +

g.)

2 24 16y x− =

h.)

2 12y x=−

i.)

3 0

( )4 0

x if xh x

x if x

+ <=

− ≥ j.)

5 1

( ) 2 1 3

3

if x

f x x if x

x if x

<= + ≤ <− ≥

12. Write 4 3 2

( ) 6 4 8f x x x x x= − − + + as a product of linear factors.

13. Find all zeros of 3 2( ) 13 57 85f x x x x= + + + .

14. Find the inverse of each of the following. Give the domain and range of the function and its inverse.

a) ( ) 4 3g x x= − b)5

2y x= − c) 2

( )3

xf x

x=

− d)

3 5y x= +

15. Write 3 54 xy += as a logarithmic equation.

16. Write 2log ( 3)m p= + as an exponential equation.

17. Write 1

3ln( 2) ln ln( 1)2

x x x+ + − − as a single logarithm.

18. Write ( )3

4log 5x x − as a sum or difference of logarithms. Express powers as factors.

19. Solve each of the following equations. Give both exact answers and approximate answers to 3 decimal places (where

appropriate).

a.)

3log ( 4) 2 7x − + =

b.)

4 4log log ( 3) 1x x+ − =

c.)

3 52 6

x x− = d.) 72 4 −= −xe

e.) 5 5log (2 3) 2 log 3x + =

f.) 3 3log (4 2) log ( ) 2x x+ − =

20. An object is thrown vertically up and its height in feet after t seconds is given by the formula 2

( ) 96 16h t t t= − .

a) Find the maximum height attained by the object.

b) After how many seconds does it attain the maximum height?

c) After how many seconds does it return to its starting position?

21. Give both exact answers and approximate answers to 3 decimal places, where appropriate. (Exception, give a whole

number of bacteria.)

a) Suppose the half-life of Girardium is 20 minutes. Find how much will be left in 105 minutes if you start with 100 g.

b) A culture of bacteria obeys the law of exponential growth. If 500 bacteria are present initially, and there are 800 after

1 hour, how many will be present in the culture after 5 hours?

22. Joe has available 200 meters of fencing and wishes to enclose a rectangular field along a bluff. He does not need to fence

the edge along the bluff. What are the dimensions of the field with the largest possible area?

23. The average weight of a baby born in 1900 was 6.25 pounds. In 2000, the average weight of a newborn was 6.625

pounds. We will assume for our purposes that the relationship is linear. Find an equation that relates the year to the

average weight of a newborn. Using that equation, predict the average weight of a newborn in 2035.

24. Find the first 5 terms for each: a) ( ) ( )12 4

n

na n+

= − + b) 1 1

3 2 4;n na a a−= − =

25. Series practice:

a) Find the nth term of 10, 6, 2, -2, -6, …

b) Find the 100th term and the sum of the first 100 terms of 7, 10, 13, 16 . . .

c) Find the 20th term of 2, 6, 18, 54, …

d) Evaluate

100

0

(4 3).k

k=

+∑

e) Evaluate

6

3

( 2).k

k k=

+∑

f) Find the sum of the first 10 terms of 2 1 1 1

...3 3 6 12+ + + +

g) Find the sum: 2 1 1 1

...3 3 6 12+ + + +

h) Evaluate 1

6(0.4)k

k

∞

=∑ .

26. Use the graph of the function ( )f x to answer questions a – h:

a) (2)f = ___

b) For what value(s) of x is ( ) 3f x = ?

c) The domain of f is ___

d) The range of f is ___

e) For what interval(s) is ( )f x decreasing?

f) For what interval(s) is ( )f x increasing?

g) There is a relative maximum of _____ at _______.

h) There is a relative minimum of _____ at _______.

i) Use the graph to solve f(x) < 0.

j) Find the real zeroes of the function.

k) f(0) = _____

Solutions

1a.) 1 3

2 2i± b.) ( 3,0) (1, )− ∪ ∞ c.) 2 2 3± d.) ( , 2] (1,3]−∞ − ∪ 2.

2 2( 3) ( 2) 5x y− + + =

3. ( , ) ( 2, 4) 7h k r= − = 4a.) 3 1 1 11

, up 4, reflect over x-axis, mult. y by shrink vertically by , points (0, 4), 1,3 3 3

x

4b.) ( ), reflect over y-axis, up 500, mult. y by 6 stretch vertically by 6 , points (0,500), ( 1,506)x −

4c.) 2log , up 3, left 4, points ( 3,3), ( 2, 4), : 4x VA x− − = −

4d.) , right 4, reflect over x-axis (make y neg),down 2, points (4, 3),(5, 2), : 2xe e HA y− − − = −

5.) 2y x= − − + 6.) ( )left 7, mult. y by 2 stretch vertically by 2 ,down 14 7a.) { }9, 2x x x≠ ≠ −

b.) 3

8x x

≤

c.) all reals d.) all reals e.) { }7x x < 8a.) 9 b.)

2 1x − c.) ( )2 1 2x x− −

d.) 3x − e.) 2 1x− + f.)

2 3x − g.) { }2

2, domain 2

1

xx x

x

−≥

− 9a.) 3 b.) 2 1x h+ +

10a.) min (1, 2)− b.) max ( 1500, 4600)−

11. See graphs below. 11a.)Intercepts3

( 3,0), (1,0), (0, )50

− ; Asym: 1

, 5, 52

y x x= = = −

11b.) Int ( 4,0), (0,0), (3, 0)− 11c.) Int (2,0), ( 2,0), (0, 7 ), (0, 7 )− − 11d.) Int (0,4) Asym 0y =

11e.) Hole (3,-10); Int (-7,0) Asym 1, 4y x= = 11f.) Int (-1,0), (0,5) 11g.) Int (0,4), (0,-4); Asym y = 2x, y = -2x

11g.) Int (0,0) 11h,i.) Just need one labeled point on each piece. Need not be the same as on the key.

12.) 2( 2) ( 2)( 1)x x x− + + 13.) 5, 4 , 4i i− − + − −

14a.) 1 4( ) domain and range of both are all reals

3

xg x− − +

=

b.) 1 5( ) x+2 domain and range of both are all realsf x− = c.) { }1 3

( ) domain of ( ) is 3 2

xg x f x x x

x

− = ≠−

{ } { } { }1range is 2 ,domain of ( ) is 2 range is 3y y f x x x y y−≠ ≠ ≠

d.) 1 3( ) 5, domain and range of both are all realsf x x− = − 15.) 4log 3 5y x= + 16.) 2 3m p= +

17.)

3( 2)ln

1

x x

x

+− 18.) 4 4

13log log ( 5)

2x x+ − 19a.) 247x = b.) 4x = c. )

5ln 2, 12.047

3ln 2 ln 6x x= ≈

−

d.) 4 ln 9, 6.197x x= + ≈ e.) 3x = f.) 2

5x = 20a.) 144 ft b.) 3 sec c.) 6 sec 21a.) 2.628 g

b.) 5243 22.) 50 ,100m m

23.) ( ) .00375 0.875, (2035) 6.756f x x f lb= − = 24a.) 20, 48,112, 256,576− −

b.) 4, 5,13, 23, 49− − 25a.) 4 14na n= − + b.) 100 100304, 15550a S= = c.) 20 2324522934a = d.) 20503

e.) 122 f.) 341

256 g.)

4

3 h.) 4 26a.) (2) 8f = − b.) 3, 5, 4.5− − c.) ( 7, 5]− d.) [ 9, 7]−

e.) ( 4,1)− f.) ( 7, 4) (1,5)− − ∪ g.) max of 4 at - 4 h.) min of -9 at 1 i.) ( 7, 6) ( 2, 4)− − ∪ −

j.) 6, 2, 4− − k.) (0) 8f = −

Graphs See the key above for the two labeled points. Be sure to use exact coordinates.

4a.) b.) c.)

d.) 11a.) b.)

11c.) 11d.) 11e.)

11f.) g.) h.)

11i.) (-1,2), (0,4) j.) (0,5), (1,3), (3,-3)

Formulas

The following formulas will be available on the Final. You should know very well how to use them.

2 1

2 1

y ym

x x

−=

−

y mx b= + 1 1( )y y m x x− = −

2 4

2

b b ac

a

− ± −

, 2 2

b bf

a a

− −

loglog

log

ba

b

xx

a=

1 ( 1)na a n d= + −

1( )2

n n

ns a a= +

rtA Pe=

1

1

n

na a r −=

1(1 )

1

n

n

a rS

r

−=

−

1

1

aS

r=

−

Below is a list of most (not necessarily all) of the formulas that might be needed on the final that will not be

supplied – if you need any formula besides those above, you must know it.

log y

a x y a x= ⇔ = log M

a a M=

log log loga a aMN M N= + log log loga a a

MM N

N= −

log logr

a aM r M= log loga aM N M N= ⇔ =

2 2 2( ) ( )x h y k r− + − = 2( )y a x h k= − +

2( )x a y k h= − +

2 2

2 21

x y

a b+ =

2 2

2 21

y x

a b+ =

2 2 2c a b= −

2 2

2 21

x y

a b− =

2 2

2 21

y x

a b− =

2 2 2c a b= +