f ( x )= 4 + 1 Chapter 3 - Exponential and Logarithmic Functions

Algebra 3 with Trigonometry Name: _______________________________

Semester 2 Final Exam Review Date: ________________ Hour: ________

Chapter 3 - Exponential and Logarithmic Functions

Section 3.1

For problems 1-6, match the function with its graph.

1) f (x )=4 ^x _{________ 2)} f (x )=4

^x _{________ 3)} f (x )=−4 ^x _{______}

4) f (x )=4 ^x +1 _{________ 5)} f (x )=4

^x −1 _{________ 6)}

f (x )=4

_{______}

Sketch each graph by hand. Use the RST method.

7) f (x )=6 ^x ₈₎ f (x )=0. 3 ^x ₉₎ f (x )=6

***You need to know the compound interest formulas!***

 Compounding n times per year  Compounding continuously

10) Complete the table to determine the amount of money P that should be invested at rate r to produce a final balance of

$200,000 in t years.

r = 8%, compounded continuously

Algebra 3 with Trigonometry Semester 2 Exam Review Page 1 of 18

Paren t

R S T Paren

t

R S T Paren

t

R S T

Domain: ________________

Range: ________________

HA: ________________

Domain: ________________

Range: ________________

HA: ________________

Domain: ________________

Range: ________________

HA: ________________

Domain: ________________

Range: ________________

HA: ________________

Domain: ________________

Range: ________________

HA: ________________

Domain: ________________

Range: ________________

HA: _______________

Y-int: ________________

Additional Pt: _____________

Domain: ________________

Range: ________________

HA: _______________

Y-int: ________________

Additional Pt: _____________

Domain: ________________

Range: ________________

HA: _______________

Y-int: ________________

Additional Pt: _____________

t 1 10 20 30 40 50

P

11) A deposit of $10,000 is made in a savings account for which the interest is compounded continuously. The balance will double in 12 years.

a) What is the annual interest rate for this account? b) Find the balance after 1 year.

12) Determine the balance of an account in which $250 was initially invested at the rate of 10% compounded quarterly for 8 years.

Section 3.2

Write the exponential equation in logarithmic form. YOU ARE NOT SOLVING!!!!

1) 4

=64 ₂₎ 3

=243 ₃₎ 25

3

2

=125 4)

12

= 1 12

Evaluate the expression. DO NOT USE A CALCULATOR!

5) log

8

₆₎ log

1 ₇₎ log

216 ₈₎

log

1024

9) log

1

6 ₁₀₎ ^log

0.001

Sketch each graph by hand. Use the RST method. State the domain and range using interval notation.

11) f (x )=−log

x+5 ₁₂₎ f (x )=2 log

( x−3 ) 13) f (x )=log

( x−1)+ 6

Paren t

R S T Paren

t

R S T Paren

t

R S T

Domain: ________________

Range: ________________

Domain: ________________

Range: ________________

Domain: ________________

Range: ________________

VA: _______________

X-int: ________________

Domain: ________________

Range: ________________

VA: _______________

X-int: ________________

Domain: ________________

Range: ________________

VA: _______________

X-int: ________________

14) f (x )=log

(x +2)−3 15) f (x )=ln x+3

Evaluate the expression. DO NOT USE A CALCULATOR!

16) ln e

₁₇₎ ln 1 ₁₈₎ 6ln e

₁₉₎ − 3

2 ln e

Section 3.3

Evaluate the logarithm using the change of base formula. Round to the nearest thousandth.

1) log

9 2) log

5

Use the properties of logarithms to write the expression as a sum, difference, and /or constant multiple of logarithms…..expand!

3) log

5 x

₄₎ log

√ ^x

4 ₅₎ log _m 5 √ ^y

x

6) ln| x−1

x+1 |

7) ln [ ^{( x}

^{+1 ) ( x−1)} ] ₈₎ ^ln

√

^{4 x 4 x}

^{−1 + 1}

Algebra 3 with Trigonometry Semester 2 Exam Review Page 3 of 18

Domain: ________________

Range: ________________

HA: ________________

Domain: ________________

Range: ________________

HA: ________________

Paren t

R S T Paren

t

R S T

Domain: ________________

Range: ________________

VA: _______________

X-int: ________________

Additional Pt: _____________

Domain: ________________

Range: ________________

VA: _______________

X-int: ________________

Additional Pt: _____________

Write the expression as the logarithm of a single quantity…..condense.

9) log

5+log

x ₁₀₎ log

y−2 log

z ₁₁₎

1

2 ln|2 x−1|−2ln|x+1|

12) 5ln|x−2|−ln|x+2|−3 ln|x| ₁₃₎ ^{ln 3+ 1} 3 ln(4−x

)−ln x

14) 3[ ln x−2 ln( x

+1)]+2 ln 5

Section 3.4

Solve each equation for “x”.

1) 8

=512 ₂₎ 3

=729 ₃₎ 6

= 1

216

4) 6

=1296 ₅₎ ^log

x=4 ₆₎ log

243=5

Solve each exponential equation for “x”. Round to the nearest thousandth.

7) e

=12 ₈₎ e

=25 ₉₎ 3e

=132

10) 14e

=560 ₁₁₎ e

+13=35 ₁₂₎ e

−28=−8

13) −4 ( 5

) =−68 ₁₄₎ 2 ( 12

) =190

15) e

−7 e

+10=0 ₁₆₎ e

−6 e

+8=0

Solve each logarithmic equation for “x”. Round to the nearest thousandth.

17) ln 3 x=8.2 ₁₈₎ ln 5 x=7.2 ₁₉₎ 2ln 4 x=15

20) 4 ln 3x=15 ₂₁₎ ln x−ln 3=2 ₂₂₎ ln √ ^x+8=3

23) ln √ ^x+1=2 24) ln x−ln 5=4

25) log ( x−1)=log( x−2)−log ( x+2) ₂₆₎ log ( x+2)−log x=log( x+5)

Algebra 3 with Trigonometry Semester 2 Exam Review Page 5 of 18

27) log (1−x)=−1

Chapter 4 – Trigonometric Functions

Section 4.1

(a) Sketch the angle in standard position, (b) determine the quadrant in which the angle lies, and (c) list one positive and one negative co-terminal angle.

1)

π

16 ₂₎

40 π

47 ₃₎ − 9π

15 ₄₎ − 11π

3

Find the complement and supplement (if possible) of each angle.

5)

π

8 ₆₎

13π

18 ₇₎

3 π

10 ₈₎

2 π 21

Convert the measure from radians to degrees. Round to the nearest hundredth.

9)

5 π

7 ₁₀₎ − 3 π

5 ₁₁₎ −3.5 ₁₂₎ 1.55

(a) Sketch the angle in standard position, (b) determine the quadrant in which the angle lies, and (c) list one positive and one negative co-terminal angle.

13) 40 ^∘ ₁₄₎ 190 ^∘ ₁₅₎ −110 ^∘ ₁₆₎

−405 ^∘

Quad: ________

Pos:___________

Neg:___________

Quad: ________

Pos:___________

Neg:___________

Quad: ________

Pos:___________

Neg:___________

Quad: ________

Pos:___________

Neg:___________

Comp:_________

Supp:__________

Comp:_________

Supp:__________

Comp:_________

Supp:__________

Comp:_________

Supp:__________

_ Leave answer in terms of π. Do not covert to degrees!

Leave answer in terms of π. Do not covert to degrees!

Find the complement and supplement (if possible) of each angle.

17) 8 ^∘ ₁₈₎ 94 ^∘ ₁₉₎ 171 ^∘ ₂₀₎ 49 ^∘

Convert from DMS to degrees.

21) 135 16’ 65” 22) -234 40” 23) 5 22’ 53” 24) 280 8’ 50”

Convert from degrees to DMS.

25) 135.29 26) 25.8 27) -85.36 28) -327.93

Convert from degrees to radians.

29) 480 30) -16.5 31) -33 45’ 32) 84 15’

33) Find the radian measure of the central angle of a circle with a radius of 12 feet that intercepts an arc of length 25 feet.

Section 4.2

Fill out the unit circle below.

Algebra 3 with Trigonometry Semester 2 Exam Review Page 7 of 18

Quad: ________

Pos:___________

Neg:___________

Quad: ________

Pos:___________

Neg:___________

Quad: ________

Pos:___________

Neg:___________

Quad: ________

Pos:___________

Neg:___________

Comp:_________

Supp:__________

Comp:_________

Supp:__________

Comp:_________

Supp:__________

Comp:_________

Supp:__________

_

Leave answer in terms of π.

Find the point (x, y) on the unit circle that corresponds to the real number, θ. No decimals.

1) θ= 2π

3 ₂₎ θ= 5 π

6

Evaluate, if possible, the six trigonometric function of the real number. No decimals.

3) θ= 7 π

6 _{4 )} θ=− 2 π

3

Evaluate the trigonometric function using its period as aid.

5) sin

11π

4 _{6) sin} ( ^{− 17 π} 6 )

Evaluate each trigonometric function using a calculator. Round to two decimal places.

7 ) cot 2.3 8) cos

5 π 3 Section 4.3

Find the exact values of the six trigonometric function of the angle θ, in the given figure. No decimals.

1)

sin θ = csc θ =

cos θ = sec θ =

tan θ = cot θ =

sin θ = csc θ =

cos θ = sec θ =

tan θ = cot θ =

sin θ = csc θ =

cos θ = sec θ =

tan θ = cot θ =

Leave answer in terms of π. Do not covert to degrees!

Use trigonometric identities to transform one side of the equation into the other side.

2) csc θ tan θ= sec θ 3)

cot θ+tanθ

cot θ =sec

θ

Section 4.4

Find the exact values of the six trigonometric function of the angle θ, in standard position whose terminal side passes through the given point. No decimals.

1) (12, 16)

Find the remaining five trigonometric functions of θ satisfying the given conditions. No decimals.

2) sec θ = 6

5 _{tan θ < 0}

Evaluate the trigonometric function without a calculator. No decimals.

3) tan

π

3 _{4 ) sin} ( ^{− 5 π} 3 )

Find the reference angle for the given angle.

5) 264 6) − 6π

5

Algebra 3 with Trigonometry Semester 2 Exam Review Page 9 of 18

sin θ = csc θ =

cos θ = sec θ =

tan θ = cot θ =

sin θ =

cos θ =

tan θ =

sin θ =

cos θ =

tan θ =

Leave answer in terms of π. Do not covert to degrees!

Evaluate the sine, cosine and tangent of the angle without using a calculator. No decimals.

7) 240 8) -210

9) − 9π

4 ₁₀₎ − π

2

Find the point (x, y) on the unit circle that corresponds to the real number, θ. Then use the result to evaluate the sine, cosine, and tangent without using a calculator. No decimals!

11) θ= 2π

3 ₁₂₎ θ= 7 π

6

Section 4.5/4.6 Graph each function.

1) f(x) = 3cos (2 π x )

Transformations: a= b=

c = d =

Amplitude: Period: Scale:

Minimum = Maximum = Axis (midline):

sin θ =

cos θ =

tan θ =

sin θ =

cos θ =

tan θ =

sin θ =

cos θ =

tan θ =

sin θ =

cos θ =

tan θ =

Paren t

R S T

2) f(x) = 5sin 2 x

5

Transformations: a= b=

c = d =

Amplitude: Period: Scale:

Minimum = Maximum = Axis (midline):

3) Graph f(x) = - tan π x

4

Transformations: a= b=

c = d =

Amplitude: Period: Scale:

Minimum = Maximum = Vertical Asymptote:

4) Graph f(x) = 3cot

x 2

Transformations: a= b=

c = d =

Algebra 3 with Trigonometry Semester 2 Exam Review Page 11 of 18

Paren t

R S T

Paren t

R S T

Paren t

R S T

Amplitude: Period: Scale:

Minimum = Maximum = Vertical Asymptote:

5) Graph f(x) = 1

4 _{csc 2x}

Transformations: a= b=

c = d =

Amplitude: Period: Scale:

Minimum = Maximum = Vertical Asymptote:

6) Graph f(x) = sec ( ^{x− π} 4 )

Transformations: a= b=

c = d =

Amplitude: Period: Scale:

Minimum = Maximum = Vertical Asymptote:

Section 4.7

Find the exact value, if possible, otherwise approximate with a calculator.

1) arcsin 1 2) arcsin 4 3) arccos

√ ²

2 _{4) arccos}

( ^{− √} 2 ³ )

Paren t

R S T

Paren t

R S T

Leave answer in terms of π.

5) Use an inverse trig function to express θ in terms of x.

Write an algebraic expression that is equivalent to the expression. You do not need to rationalize your answer.

6) sec[ arcsin (x – 1) ]

Section 4.8

1) A train travels 3.5 km on a straight track with a grade of 1 10’. What is the vertical rise of the train in that distance?

Round to two decimals places.

2) A passenger in an airplane flying at an altitude of 37,000 feet sees two towns directly to the left of the airplane. The angle of depression to the towns are 32 and 76. How far apart are the towns? Round to two decimals places.

Algebra 3 with Trigonometry Semester 2 Exam Review Page 13 of 18

True or False? SHOW WORK to justify your answer.

3)

sin 60 ^∘ sin 30 ^∘ =sin 2 ^∘

4) tan { ( 0.8 )

} =tan

( 0.8 )

5) y=sin θ is not a function because sin 30 ^∘ = sin150 ^∘ _.

6)

Chapter 6 – Additional Topics in Trigonometry Section 6.1 Law of Sines

Using the Law of Sines, solve each triangle. (i.e., find all of the missing info.) If two triangles exist give both sets of answers.

Round to two decimals places.

1)A = 12  , a = 8, B = 58  Case: AAS ASA SSA SSS SAS 2) A = 75  , a = 2.5, b = 16.5 Case: AAS ASA SSA SSS SAS

2) B = 115  , a = 9, b = 14.5 Case: AAS ASA SSA SSS SAS 4) A = 15  , a = 5, b = 10 Case: AAS ASA SSA SSS SAS

5) Using the Law of Sines, find the area of the triangle: A = 27  , b = 5, c = 8

6) From a certain distance the angle of elevation to the top of a building is 17  . At a point 50 meters closer to the building the angle of elevation is 31  . Approximate the height of the building to the nearest tenth.

Case: AAS ASA SSA SSS SAS

7) Find the height of a tree that stands on a hillside of slope 28 (from the horizontal) if, from a point 75 feet down the hill, the angle of elevation to the top of the tree is 45. Round to two decimals places.

Case: AAS ASA SSA SSS SAS

Section 6.2 Law of Cosines

Using the Law of Cosines, solve the triangle. (i.e., find all of the missing info.) Round to two decimals places.

1) a = 5, b = 8, c = 10 Case: AAS ASA SSA SSS SAS 2) a = 6 , b = 9, C = 45  Case: AAS ASA SSA SSS SAS

Algebra 3 with Trigonometry Semester 2 Exam Review Page 15 of 18

3) a = 4 , c = 4, B = 110  Case: AAS ASA SSA SSS SAS

4) To approximate the length of a marsh, a surveyor walks 425 meters from point A to point B. Then the surveyor turns 65  and walks 300 meters to point C. Approximate the length of AC of the marsh. Case: AAS ASA SSA SSS SAS

Round to two decimals places.

5) Using Heron’s Area Formula, find the area of a triangle when a = 4, b = 5, and c = 7. Round to two decimals places.

Chapter 9 – Topics in Analytical Geometry Section 9.1

Find the standard form of the equation of the circle.

1) Center at the origin Point on the circle: (-3, -4) 2) Endpoints of the diameter (-1, 2) (5, 6)

3) 16x

+ 16y

-16x + 16y – 3 = 0

Draw a sketch here! 

Find the standard form of the equation of the parabola.

4) Vertex (4, 2) Focus (4, 0) 5) Vertex (0, 2) Directrix x = -3

6) A parabolic archway is 12 meters high at the vertex. At a height of 10 meters, the width of the archway is 8 meters. How wide is the archway at ground level?

Section 9.2

Find the standard form of the equation of the ellipse.

7) Vertices (-3, 0) (7, 0) Foci (0, 0) (4, 0) 8) Vertices (0,1) (4, 1) Endpoints of the minor axis:

(2, 0) (2, 2)

Find the center, vertices, foci, and eccentricity of the ellipse.

9) 16x

+ 9y

– 32x + 72y + 16 = 0 10)

( x+2)

81 + ( y−1 )

100 =1

Algebra 3 with Trigonometry Semester 2 Exam Review Page 17 of 18

4p = h =

k =

4p = h =

k =

4p = h =

k =

a = h =

(Dist. fm. ctr. to vertex.)

b = k =

(Dist. fm. ctr. to co-vertex.) c =

(Dist. fm. ctr. to focus.)

a = h =

(Dist. fm. ctr. to vertex.)

b = k =

(Dist. fm. ctr. to co-vertex.) c =

(Dist. fm. ctr. to focus.)

a = h =

(Dist. fm. ctr. to vertex.)

b = k =

(Dist. fm. ctr. to co-vertex.) c =

(Dist. fm. ctr. to focus.)

a = h =

(Dist. fm. ctr. to vertex.)

b = k =

(Dist. fm. ctr. to co-vertex.) c =

(Dist. fm. ctr. to focus.)

Draw a sketch here!  Draw a sketch here! 

Draw a sketch here! 

Draw a sketch here! 

Section 9.3

Find the standard form of the equation of the hyperbola.

11) Vertices (-10, 3) (6, 3) Foci (-12, 3) (8, 3) 12) Foci (0, 0) (8, 0) Asymptotes

y=±2( x−4)

Find the center, vertices, foci, and the equation of the asymptotes of the hyperbola. Then sketch the graph.

13) 9x

– 16y

– 18x – 32y – 151 = 0 14)

( x−3)

16 − ( y+5)

4 =1

Classify the conic from its general equation. Show work!

15) 3x

+ 2y

– 12x + 12y + 29 = 0 16) 4x

– 4y

– 4x + 8y – 11 = 0

Identify each conic. Then sketch the graph.

17)

( y +3)

16 − ( x−5)

121 =1

18)

( x−2)

4 + ( y +1 )

9 =1

a = h =

(Dist. fm. ctr. to vertex.)

b = k =

c =

(Dist. fm. ctr. to focus.)

a = h =

(Dist. fm. ctr. to vertex.)

b = k =

c =

(Dist. fm. ctr. to focus.)

a = h =

(Dist. fm. ctr. to vertex.)

b = k =

c =

(Dist. fm. ctr. to focus.)

a = h =

(Dist. fm. ctr. to vertex.)

b = k =

c =

(Dist. fm. ctr. to focus.)

Find the standard form of the equation of the graph of each conic.

19) 20) 21)

Algebra 3 with Trigonometry Semester 2 Exam Review Page 19 of 18