college pre – calculus a - ntschools.org · 11. find the remainder when p x x x( ) 73 2 2 53 21...

College Pre – Calculus A Name _____________________________

Final Exam Review #1 Due: ___________ Period ____________

Show all work for every question in the box. Be sure to include all formulas and substitutions.

1. Find the distance between (-2, 1) and 3 2 6 0x y .

2. Write the coordinates of the relative minimum

for the equation: 3 25 3 1y x x x .

3. Factor completely : 38 1.x

4. Find the coordinates of the x-intercept of

5)2(log4 3 xy

5. If f(x) = 9 32 17x x , find '( )f x .

6. Find the product of )3

2sin

3

2(cos8

i and

)6

3sin

6

3(cos2

i .

7. Find the largest real root for the equation: 4 32 5 1y x x x .

8. If 2( ) 2 1h x x and ( ) 3 2g x x , find

( ( ( 2))g h .

9. Use a sign chart to solve 4

03

x

x

. Write the

answer in interval notation

10. Solve for x: 15 2x x .

11. Find the remainder when 53 21( ) 73 2 2P x x x is divided by x – 1

12. If 5 3 4(3 ) ,y x x find dy

dx .

13. Find the center of the circle :

2 2 6 2 4 0x y x y .

14. If 7 43 5 18y x x x , find ''y .

15. If )18

5sin

18

5(cos2

iz , find 3z .

16. Find the sin 2x if sin x =3

5 and x is in

quadrant II.

17. If 2

( )3

xf x

x

, find 1( )f x .

18. Write the equation of the line that passes through the point (3, 1) and is parallel to the line whose equation is 2 3 5 0x y .

19. Find the 5 2

6

4 7 1lim

8 9x

x x

x

.

20. What is the slope of the tangent line to 32 6 7y x x at x = 2.



1 a) Graph the function:

( )f x =

1;2

1;1

2 xx

xx

b) Use your graph to answer the

following questions:

1) 1

lim ( )x

f x

__________

2) 1

lim ( )x

f x

__________

3) 1

lim ( )x

f x

_________

4) 2

lim ( )x

f x

_________

5) lim ( )x

f x

= __________ 6) Domain of f(x): ______________ 7) Range of f(x): _______________

2. a) Solve 2

13cos x for b) Prove the identity:

all x in the interval )2,0( . 2 2 2cos tan cos 1x x x

.

3. Given: 2 216 9 32 36 164x y x y Identify the Conic:___________

Standard Form:

__________________________

Center: __________

Vertices: ___________________

Covertices: _________________

Foci: ______________________

Lines of Symmetry:

_________________________

Equations of Asymptotes:

__________________________

4. List all possible rational zeros and 5. ) Solve for x:

then find all real and complex zeros of

15115)( 23 xxxxf 2

5 5log ( 36) log ( 6) 2x x

All possible rational zeros:

Find all zeros:




1 If 532)( 2 xxxf find the difference

quotient ( ) ( )

, 0f x h f x

hh

.

2. The following three transformations are applied

(in the given order) to the graph of 3y x .

I Vertical stretch by a factor of 2

II Reflection in the x-axis

III Vertical shift down 3 units

Which of the following is the equation that is the

result of applying these transformations?

a) 32( 3)y x

b) 32( 3)y x

c) 32 3y x

d) 32 3y x

3. Solve for x: ln ln( 2) ln3x x

4. Find the exact value of x: 1

3

log 9 .x

5. Evaluate: 1 1cos( ( )).

2Sin

6. Simplify : 2 2

2

1 1

5 5

x y

y xy

7. Find the range for the function:

( ) 2 3f x x .

8. Express the complex number 2 + 2i in trig form

9. What is the 3rd term in (5x2 – 2y4)5?

10. Given the parabola 2( 2) 3y x ,

find the vertex

11. Find dxxx )736( 2

12. Find the domain of 7

12)(

x

xxf

13. Complete the square to write the equation of

6105)( 2 xxxg in standard form of a

quadratic equation for graphing

14. Find the distance between the two parallel lines

5x – 3y = -5 and 5x – 3y – 6 = 0

15. An object 175 feet above the ground is shot straight up with an initial velocity of 52 ft/sec.

a. Write the general formula for projectile motion ______________________________

b. Write the equation that would be used to

represent this problem ______________________________ c.

Draw the graph that represents the problem situation:

d. When will the object reach is maximum height?

_____________________________

e. What is the maximum height?

_____________________________

f. When will the projectile hit the ground?

___________ __________________




1. In ABC if c = 12 miles, the m = 49o

and m = 27o, find b.

2. Factor completely : 3 24 8 2x x x

3. If 4 2 6(2 3 )y x x , finddy

dx.

4. Simplify: 2 2

1 13 3

x y

x y

5. Write the equation of the line that passes through

the point (-2, 3) and is perpendicular to the line

whose equation is x – 2y +1 = 0.

6. Find the quotient of )12

7sin

12

7(cos8

i

and )4

sin4

(cos2

i .

7. Use a sign chart to solve: 08

1032

x

xx

8. Find 4z if )16

3sin

16

3(cos2

iz

9. Solve the equation: 2sin 2sin 3x x on the

interval 0,2 .

10. What is the slope of the tangent line to 53 2 16y x x at x = -1.

11. Find the area of a triangle with sides of 11

inches, 16 inches and an included angle of 49o.

12. Find the 7 4

7

5 8 1

6 3lim

x

x x

x

.

13. Evaluate: 1 3tan( )

2Cos

14. Write the equation of the resulting function

when the following four transformations are

applied to the graph of y=sin x.

I Vertical stretch by a factor of 3

II Reflection in the x – axis

III Horizontal shift 1 unit left

IV Vertical shift 2 units down

15. A box without a top is made from cutting equal squares from the corners of a piece of a

rectangular piece of cardboard that measures 12 inches by 18 inches.

a. Find all possible lengths of the sides of the squares if the volume of the box is 125 in3

b. Find the maximum volume that could be formed with the given conditions.

Draw picture here: Draw a graph of the problem situation here:

Identify variables and write equation here: Completely answer the problem here:

. a. Possible lengths of the sides of the square cut:

b. Maximum possible volume




1. Find dx

dy of

3

1;

13

24

x

x

xy

2. Find the first derivative of 52 )37(4)( xxg

3. Find the largest real root for the equation:

4595)( 325 xxxxf

4. Factor: 405 9 a

5. Find the distance between (-2,4) and 3x-2y+2=0

6. Solve for x: 4242 xx

7. Find the quotient and remainder when

325)( 3 xxxf is divided by x – 2

8. Find the domain of 82

4)(

2

2

xx

xxf

9. Find a polynomial of degree 3 whose zeros are

i32 and -5

10. Solve using sign patterns:

32

12

x

xx

11. If $2500 is invested at 6.75% APR

compounded quarterly, find the value of the

investment after 12 years.

12. If Jasper invests $500 at 7% APR compounded

continuously, how much money will he have saved

when he starts college in 13 years?

13. Write the equation of the parabola that has a directrix of x = -1 and focus (3, 2). Find the coordinates of the

vertex, the equation of the line of symmetry, the length of the latus rectum and graph the parabola.

Equation of the parabola: __________________

Vertex: ____________

Line of Symmetry: _______________

Length of the latus rectum: ______________

14. Given: 131243)( 234 xxxxf

a. Draw a complete graph of f(x): b. Find all real zeros of )(xf

c. Find all local extrema of )(xf

Local Max: ______________________

Local Min: _______________________

d. Find all intervals where )(xf is

increasing and decreasing

Increasing: _________________ Decreasing: _________________

15. Given: 3

2;

23

56)(

x

x

xxf b. Is f(x) a one – to – one function? Why?

a. Draw a complete graph of f(x): c. Find the domain and range of f(x)

d. Find )(1 xf



1. Find the coordinates of the focus of the parabola

with vertex (5, -2) and Directrix x = 8.

2. Find the remainder when

61 50( ) 47 8 7 2P x x x x is divided by x + 1.

3. Evaluate:

2

1

2 )2( dxxx

4. Solve for x: 2ln( 36) ln( 6) ln5x x .

5. Find the exact value of x: 2

1log

16x

6. Find dx

dy if

21;

12

1

x

x

xy

7. Find

4

2

4

lim

x

x

x

8. Factor completely : 96 54128 ba

9. Find the range of |102|5)( xxg

10. Complete the square to write the equation of

6123)( 2 xxxg in standard form of a

quadratic equation for graphing

11. Algebraically find the range of

23;

32

4)(

x

xxf

12. Factor completely: 32128 23 xxx

13. Find the difference quotient for

5147)( 2 xxxf

14. If the 17

8cos

x and

x

2, find sin2x in

fractional form.

15. Determine an equation, the line of symmetry, the directrix, length of the latus rectum and graph the

parabola whose focus is (1, 2) and vertex is (1, 4).

Equation:

Line of Symmetry: ___________

Directrix: __________________

Length of the Latus Rectum: ___

16. Algebraically find all intervals where xxx

xf 623

)(23

is increasing and decreasing and all local

extrema in fractional form.

Increasing: ____________ Decreasing: ___________

Show work here to algebraically find the coordinates of the extrema in

fractional form:

Local Max: ______________ Local Min: ____________

17. For the piecewise function, find (x,y) values, graph and find domain, range and find the indicated

limits if

1,

1,22)(

2

xx

xxxxf

Domain: ___________________________

Range: ____________________________

Find:

1

lim

x = _______

1

lim

x= _______

1

lim

x= ________ )1(f _________

x

lim= _______

0

lim

x= ________



1. Factor: 2189 23 xxx 2. Find the domain and range of

52)( xxg

3. Write the equation of the line parallel to

3y – x = 5 and passing through the point (6,-1)

in general form.

4. Complete the square and find the vertical shift

of the function 684)( 2 xxxf

5. Find the distance between the parallel lines

054 yx and 114 yx

6. Find the center and the radius of the circle

01581222 yxyx

7. Find the inverse, f -1 (x), if

5

23)(

x

xxf , 5x

8. Solve using sign patterns and write answer in

interval notation: 05

32

xx

x

9. Write the equation of the graph when the

following transformation are applied to 2xy : a

vertical stretch of 3, followed by a reflection over

the x-axis, with a horizontal shift right of 2 and a

vertical shift down of 4.

10. Find a polynomial of degree 3 that has a zeros

of 4 and i43


1726 3 xx is divided by x – 1

12. If )12

sin12

(cos3

iz , find z3

13. Find 2

1

z

z if z1= )

9

20sin

9

20(cos24

i

and )9

5sin

9

5(cos42

iz .

14. Find the longer diagonal of a parallelogram

with sides of 12.5 in, 14.8 in and an included angle

of 51o.

15. A candy box (probably to hold some chocolate for Mrs. Wiech) is to be made out of a piece of

cardboard that measures 16 inches by 24 inches. Squares of length x will be cut from each

corner and then the ends will be folded up to create the box. Find the height of the box that will

allow the maximum volume and the maximum volume of the candy box created ( holding

the maximum amount of chocolate)

Draw picture here: Draw a graph of the problem situation here:

Identify variables and write equation here: Completely answer the problem here:

. What is the height that will allow the

maximum volume?

What is the maximum volume?

Review #7 continued on next page

Final Exam Review #7 Continued…..

16. An object 30 feet above the ground is shot straight up with an initial velocity of 250 feet per second.

Find when the maximum height will occur and the maximum height and for what period of time will

the object be at least 550 feet the ground?

a. Write the general formula for projectile motion ______________________________

b. Write the equation that would be used to

represent this problem ______________________________ c.

Draw the graph that represents the problem situation:

d. When will the object reach is maximum height?

_____________________________

e. What is the maximum height?

_____________________________

f. When will the object be at least 550 feet above the

ground?

___________ __________________

17.

a. Solve for x: 4)2(log3

12 x

b. Solve for x:

1)2(log)42(log 1010 xx

c. Solve for x:

4)12(log2)12(log4 33 xx



1. Solve using a sign chart: 365

3

x

x

2. Solve from ]2,0[ xx cos1cos2 2

3. Find the difference quotient of

653)( 2 xxxf

4. Complete the square and identify the horizontal

shift of 5124)( 2 xxxf .

5. Solve for x: 1 3 3x x


2sin

3

2(cos3

i

and )12

sin12

(cos2

i .

7. Write i44 as the trig form of a complex

number. 8. Solve xx cos1cos2 2 for all values of x in

]2,0(

9. In triangle ABC, side BC = 18.75 in,

64Am and 49Cm . Find side AB.

10. Find 52

235lim2

23

x

xx

x

11. Simplify: 2 1

2 2

( )b ab

b a

12. If2 3

( ) , 0x

f x xx

, find 1f x .

13. Evaluate: dxx )1(

4

0

2

14. Find dx

dy if

2

1;

12

23

x

x

xy

15. List all possible rational zeros and then find all real and complex zeros of 1332)( 23 xxxxf

16.

a) Solve 2

33sin x for all x in the

interval )2,0( .

b) Prove the identity: x

xxxx

sin

tancosseccot

.


Final Exam Review #8 Continued…

17. List all transformations and graph one curve of )2sin(23 xy

Vertical Stretch = amplitude = ______________

Reflect over x – axis? _____________

Frequency = ___________

Period = _____________

Horizontal Stretch or Shrink = ______________________

Vertical Shift = __________________

Horizontal Shift (phase shift) = ____________________

18. Given the function 3 2( ) 2 5 7f x x x

a. Find '( )f x .

b. Find the slope of tangent line when x = 1.

c. Find the coordinates of the point of f(x) when x = 1.

d. Find the equation of the tangent line of f(x) at the point found in part c.



1. Find a polynomial of degree 3 whose roots are 3

and i26

2. Write the equation of the line tangent to the

curve 532)( 2 xxxf at the point (-3,4)

3. Evaluate: dxxxx )4324( 65

2

0

4

4. Solve for x: ln(x + 3) + ln x = ln4

5. Find the distance between the parallel lines

x + y – 5 = 0 and x + y + 9 = 0

6. Solve algebraically from )2,0[ xx coscos2

7. What is the 4th term in the expansion of

(3x3 – 2y2)5?

8. Algebraically find how long it will take an

initial deposit of $1250 to triple in value at 7%

interest compounded monthly.

9. Write the equation of the function when the

following transformations are applied to y = |x|:

I. Vertical stretch of 2

II Reflection over x – axis

III Horizontal shift left 3

IV Vertical shift down 4

10. Solve for x: 15

1 x

x

11. Solve for all values of x in the interval ]2,0[ : 33cos2 x

12. Given: 24)( 234 xxxxf

a. Draw a complete graph of f(x): b. Find all real zeros of )(xf

c. Find all local extrema of )(xf

d. Find all intervals where )(xf is

increasing and decreasing

13. Given: 5;5

32)(

x

x

xxf b. Is f(x) a one – to – one function? Why?

a. Draw a complete graph of f(x): c. Find the domain and range of f(x)

d. Find )(1 xf


Final Exam Review #9 Continued… 14. List all transformations and graph one curve of )22cos(32 xy

Vertical Stretch = ______________ Reflect over x – axis? _____________

Frequency = ___________ Period = _____________

Horizontal Stretch or Shrink = __________ Vertical Shift = __________________

Horizontal Shift (phase shift) = _______

15. Given the function: 323314

4

1)( xxxxf

a. Use the first derivative and a sign chart to find interval(s) where f(x) is increasing and decreasing.

Be sure to clearly identify your answers.

b. Clearly identify interval(s) where f(x) is increasing and decreasing:

Increasing: _______________________ Decreasing: ____________________________

c. Using the sign chart from above, algebraically find all local extrema in fractional form of f(x)



1. Solve xx cos1sin2 2 in ]2,0[

2. Find 21 zz if z1 = 6(cos 18

5 + isin

18

5) and

z2 = 8(cos 9

8+ isin

9

8) in simplest a + bi form

3. If Abby invests $750 at 12.5% interest

compounded continually, how much money will

she have from this investment when she starts

college in 14 years?

4. What is the slope of the tangent line to

7352)( 23 xxxxf when x = 2.

5. Find dx

dy if 524 )72(3 xxy

6. If 3 sides of a triangle are 7ft, 10ft and 13ft, find

the measure of the smallest angle of the triangle.

7. Find the 5th term of (-4a5 – 7b3)6

8. Algebraically find the range of

4;4

32)(

x

x

xxg

9. Find z4, if )3

sin3

(cos2

iz in simplest

bia form.

10. Find 6

25

5

732lim

xx

xx

x

11. Given: 1

2sin 12

y x

a.. Find the amplitude, period and

list all transformations.

b. Graph one cycle of the function.

12. Prove the identity: 2

2 2

2

1 tansec csc .

1 cos

xx x

x


Final Exam Review #10 Continued…

13. On graph paper, graph the following function:

2:12

2;4)(

2

xx

xxxf

a) Find the domain and range of the function

D: ________________________

R: ________________________

b) Use the graph to find the following limits:

1) 2

lim ( )x

f x

3) 2

lim ( )x

f x

2) 2

lim ( )x

f x

4) lim ( )x

f x

14. Given the function: 244)( 23 xxxxf

a) List all possible rational zeros

b) Find the exact values of all the real and complex zeros


Final Exam Review Practice Final Exam Due: ___________ Period ____________

June 2018 Final Exam Part I 1. Find the distance between the two parallel lines

1045 yx and 5x – 4y +11 = 0

2. Find the center and the radius of the circle

02101822 yxyx in simplest radical

form.

3. Find the shorter diagonal of a parallelogram with

sides of 18 inches, 26 inches and an included

angle of 39o.

4. Express the complex number i232 in

trigonometric form.

5. Find the domain and range for the function:

72)( xxf

6. If the 13

5sin and

2, find 2sin in

fractional form

7. Find the area of triangle ABC if side BC is 12

feet, side AC is 17.5 feet the measure of

angle C is 52o.

8. Solve the equation: xx cos1cos2 2 on the

interval 0,2 .

Practice Final Exam continued on next page

9. If52

73)(

x

xxf ,

25x , find )(1 xf

10. Factor completely: 2481 12 x

11. If 645 )2(3 xxy , finddy

dx.

12. In triangle ABC, find b if c = 12 miles, the

measure of angle C is 96o and the measure of

angle B is 41.5o.

13. Find the domain of the function:

5143

942)(

2

2

xx

xxxf


2sin

3

2(cos4

i and

)12

sin12

(cos2

i .

15. What is the slope of the tangent line to

1654)( 23 xxxf at 3x ?

16. Find the quotient of )3

7sin

3

7(cos241

iz

and )6

4sin

6

4(cos22

iz , in simplest bia form.


17. Find the remainder when

157234)( 2151835 xxxxxf is divided by

(x + 1).

18. If Jasper invests $725 at 3.25% interest

Compounded quarterly, find the amount of

interest earned in 9 years when he starts college.

19. Use a sign chart to solve: 0592

42

xx

x.

Write the answer in interval notation.

20. Find dx

dy if

2

3;

32

53)(

2

x

x

xxxf

21. If )24

11sin

24

11(cos2

iz , find 4z .

22. Find 64

64

64lim

x

x

x

23. Evaluate: dxxxx )163014(

3

0

356

24. Simplify: 2 1

2 2

( )b ab

b a


25. What is the equation of the line, in general

form, that is perpendicular to the line

0523 yx and passes through the

point (-6, 7).

26. Find the 4th term in the expansion of (4x3 – 3y2)7

27. Solve for x: 24loglog)2(log 777 xx 28. Write the equation of the function when

the following transformations are applied y = |x|

I. Vertical stretch of 4

II. Reflection over the x – axis

III. Horizontal shift right 3

IV. Vertical shift down 6


27543 24 xxx is divided by x – 2.

30. Find a polynomial of degree 3 that has zeros

of 5 and i34

CONGRATUALTIONS!!!! WE DID IT

ANOTHER SUCCESSFUL YEAR OF

PRE-CALCULUS IS DONE

college pre – calculus a - ntschools.org · 11. find the remainder when p x x x( ) 73 2 2 53 21...

Documents