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Pracalculus Honors - FINAL REVIEW-2015-2016:
The exam will cover the following chapters and concepts:
Chapter 1
Chapter 2
1.1 Functions
2.1 Power and Radical Functions
1.2 Analyzing Graphs of Functions and Relations
2.2 Polynomial Functions
1.3 Continuity, End Behavior, and Limits
2.3 The Remainder and Factor Theorems
1.4 Extrema and Average Rates of Change
2.4 Zeros of Polynomial Functions
1.5 Parent Functions and Transformations
2.5 Rational Functions
1.6 Function Operations and Composition of
2.6 Nonlinear Inequalities
Functions
1.7 Inverse Relations and Functions
Chapter 3
3.1 Exponential Functions
3.2 Logarithmic Functions
3.3 Properties of Logarithms
3.4 Exponential and Logarithmic Equations
Chapter 4
4.1 Right Triangle Trigonometry
4.2 Degrees and Radians
4.3 Trigonometry Functions on the Unit Circle
4.4 Graphing Sine and Cosine Functions
4.5 Graphing Other Trigonometric Functions
4.6 Inverse Trigonometric Functions
4.7 The Law of Sines and the Law of Cosines
Chapter 5
5.1 Trigonometric Identities
5.2 Verifying Trigonometric Identities
5.3 Solving Trigonometric Equations
5.4 Sum and Difference Identities
5.5 Multiple-Angle and Product-to-Sum Identities
Chapter 6
6.1 Multivariable Linear Systems and Row
Operations
6.2 Matrix Multiplication, Inverses, and
Determinants
6.3 Solving Linear Systems Using Inverses and
Cramer’s Rule
Chapter 7
7.1 Parabolas
7.2 Ellipses and Circles
7.3 Hyperbolas
7.4 Rotations of Conic Sections
Chapter 10
10.1 Sequences, Series, and Sigma Notation
10.2 Arithmetic Sequences and Series
10.3 Geometric Sequences and Series
10.4 Mathematical Induction
10.5 The Binomial Theorem
Helpful Formulas that you will receive:
Sum and Difference
Power Reducing/Half-Angle
sin(𝑢 ± 𝑣) = sin 𝑢 cos 𝑣 ± cos 𝑢 sin 𝑣
sin2 𝑢 =
1−cos(2𝑢)
cos2 𝑢 =
2
cos(𝑢 ± 𝑣) = cos 𝑢 cos 𝑣 ∓ sin 𝑢 sin 𝑣
tan2 𝑢 =
tan 𝑢 ± tan 𝑣
1 ∓ tan 𝑢 tan 𝑣
tan(𝑢 ± 𝑣) =
sin 2𝑢 = 2 sin 𝑢 cos 𝑢
cos 2𝑢 = cos2 u − sin2 𝑢
tan 2𝑢 =
2 tan 𝑢
1 − tan2 𝑢
Product – to – Sum
1
sin 𝑢 sin 𝑣 = 2 [cos(𝑢 − 𝑣) − cos(𝑢 + 𝑣)]
1
[cos(𝑢 − 𝑣) + cos(𝑢 + 𝑣)]
2
Binomial Theorem: where n Cr 
 a  b
n
2
1 − cos(2𝑢)
1 + cos(2𝑢)
Sum – to - Product
Double Angle
cos 𝑢 cos 𝑣 =
1+cos(2𝑢)
𝑢+𝑣
𝑢−𝑣
) cos (
)
2
2
𝑢+𝑣
𝑢−𝑣
sin 𝑢 − sin 𝑣 = 2 cos (
) sin (
)
2
2
𝑢+𝑣
𝑢−𝑣
cos 𝑢 + cos 𝑣 = 2 cos (
) cos (
)
2
2
𝑢+𝑣
𝑢−𝑣
cos 𝑢 − cos 𝑣 = −2 sin (
) sin (
)
2
2
sin 𝑢 + sin 𝑣 = 2 sin (
1
sin 𝑢 cos 𝑣 = 2 [sin(𝑢 + 𝑣) + sin(𝑢 − 𝑣)]
cos 𝑢 sin 𝑣 =
n!
(n  r )!r !
 n C0a nb0  n C1a n1b1  n C2a n2b 2    n Cr a nrb r    n Cn a 0b n
1
[sin(𝑢 + 𝑣) − sin(𝑢 − 𝑣)]
2
I. CHAPTER 1:
Determine if the function is even/odd/neither:
1.
y  3x x 2  2
Find f 1 ( x)
x2
4. f ( x) 
x 3
6. Find the domain of: 𝑓(𝑥) =
2.
f ( x)  2 x  9  4
5.
f ( x) 
3. g ( x) 
3 x3
1  x2
2x  5
3x  9
√𝑥−2
√𝑥+1
7. Evaluate 𝑓(−5), 𝑓(3), 𝑎𝑛𝑑 𝑓(4) for the piecewise below. Graph.
−|𝑥 + 2| − 3 𝑖𝑓 𝑥 < −2
. 𝑓(𝑥) = {(𝑥)2 − 7 𝑖𝑓 − 2 ≤ 𝑥 < 4
3
5 √3𝑥 − 4 𝑖𝑓 𝑥 ≥ 4
II. CHAPTER 2:
1. Find all asymptotes of the graph of each rational
function:
2𝑥
a. 𝑓(𝑥) = 3𝑥 2 +1
2𝑥 2
b. 𝑓(𝑥) = 𝑥 2 −1
3. Sketch the graph of the rational functions: a. 𝑓(𝑥) =
d. 𝑓(𝑥) =
𝑥 2 +𝑥−2
2. Find all asymptotes and holes: 𝑓(𝑥) = 𝑥 2 −𝑥−6
𝑥 2 −𝑥
2𝑥−1
𝑥
𝑥
b. 𝑓(𝑥) = 𝑥 2 −𝑥−2
𝑥 2 −9
c. 𝑓(𝑥) = 𝑥 2 −2𝑥−3
𝑥+1
4. Write the polynomial 𝑓(𝑥) = 𝑥 4 − 𝑥 2 − 20: (a) as the product of factors that are irreducible over the
rationals (b) as the product of linear factors and quadratic factors that are irreducible over the reals, and
(c) in completely factored form.
5. Find a polynomial function with real coefficients that has the given zeros (there are many correct
answers): 2, 2, 4 − 𝑖
6. Use the given zero to find all zeros of the function: 𝑔(𝑥) = 3𝑥 3 − 4𝑥 2 + 8𝑥 + 8 Given zero: 1 − √3𝑖.
7. State the linear factorization for the function and then state the zeros: 𝑓(𝑥) = 2𝑥 4 + 4𝑥 3 − 18𝑥 2 −
4𝑥 + 16
II. CHAPTER 3: EXPONENTIAL/LOGARITHM FUNCTIONS:
Solve for x:
1
1. 2 x 
64
2. 33 x 1  9 x  4
1
3.  
4
x2
 82 x  9
4. Match the function with its graph for the following:
i.
ii.
iii.
iv.
𝑓(𝑥) = 4𝑥
𝑓(𝑥) = 4−𝑥
𝑓(𝑥) = −4𝑥
𝑓(𝑥) = 4𝑥 + 1
5. Graph the function by hand, identify any asymptotes and intercepts and determine whether the function
is increasing or decreasing: a) 𝑔(𝑥) = 6−𝑥
b) 𝑓(𝑥) = 0.3𝑥 − 2
6. Find the domain, vertical asymptote, and x-intercept of the logarithmic function and sketch its graph by
hand: 𝑓(𝑥) = log 2 (𝑥 − 1) + 6
7. Use the properties of logarithms to rewrite and simplify the logarithmic expression: a) ln(5𝑒 −2 )
b) log10 0.002
8. Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple
of logarithms (assume all variables positive): a) log
5√ 𝑦
𝑥2
b) ln
𝑥+3
𝑥𝑦
c) ln
𝑥𝑦 5
√𝑧
1
9. Condense the expression to the logarithm of a single quantity: a) 2 ln(2𝑥 − 1) − 2 ln(𝑥 + 1)
1
b) ln 3 + 3 ln(4 − 𝑥 2 ) − ln 𝑥
c) 3[ln 𝑥 − 2 ln(𝑥 2 + 1)] + 2 ln 5
10. Solve the equation for x without using a calculator:
a) log 2 (𝑥 − 1) = 3
b) ln(2𝑥 + 1) = −4
11. Solve the exponential equation algebraically:
a) 3𝑒 −5𝑥 = 132
b) 14𝑒 3𝑥+2 = 560
c) −4(5𝑥 ) = −68
d) 𝑒 2𝑥 − 7𝑒 𝑥 + 10 = 0
12. Solve the logarithmic equation algebraically:
a) log10 (𝑥 − 1) = log10 (𝑥 − 2) − log10 (𝑥 + 1)
b) log10 (𝑥 + 2) − log10 𝑥 = log10 (𝑥 + 5)
13. Find the exact value of the logarithm without using a calculator:
a) log 2 (−4)
b) log 5 75 − log 5 3
c) ln 𝑒 3 − ln 𝑒 7
d) 2 ln 𝑒 4
e) ln
1
√𝑒
III. CHAPTER 4: TRIG FUNCTIONS Note: Make sure to leave answers exact when appropriate.
1. Given a point on the terminal side of an angle in standard position, determine the exact values of the 6
trigonometric functions: a) (5, −12)
b) (−4, 10)
2. Find the values of the 6 trigonometric functions of 𝜃.
𝜋
a) csc 𝜃 = 4 given cot 𝜃 < 0
b) sin 𝜃 = 0 given 2 ≤ 𝜃 ≤
c) tan 𝜃 is undefined given 𝜋 ≤ 𝜃 ≤ 2𝜋
3𝜋
2
3. Evaluate the trigonometric function for the given quadrantal angle.
3𝜋
a) sec 𝜋
b) cot 2
c) sec 0
d) cot 𝜋
4. Evaluate the sine, cosine, and tangent of the angle (without a calculator).
11𝜋
17𝜋
a) −750°
b) −240°
c) 4
d) − 6
5. Find two solutions of the equation. Give you answer in radians (0 ≤ 𝜃 < 2𝜋)
1
a) sin 𝜃 = − 2
b) csc 𝜃 =
2√3
c) cot 𝜃 = −1
3
d) sec 𝜃 = −
2
𝑥
3
3
6. Find the exact value: a) tan (arccos )
b) cos [arcsin (− )]
3
5
7. Write each of the following as an algebraic expression in terms of x:
1
a) sin(arccos 3𝑥)
0≤𝑥≤3
b) cot(arccos 3𝑥)
c) csc (arctan
2√3
1
0≤𝑥≤3
)
√7
8. Find the exact value of the expression without using a calculator.
1
a) arcsin 2
b) arcsin 0
1
c) arctan
√3
3
e) cos −1 (−
d) arctan(−1)
√2
)
2
f) arccos (− 2)
9. Use the law of sines to solve the triangle. If 2 solutions exist, find both.
a) 𝐴 = 75°, 𝑎 = 2.5, 𝑏 = 16.5
b) 𝐵 = 115°, 𝑎 = 9, 𝑏 = 14.5
c) 𝐴 = 15°, 𝑎 = 5, 𝑏 = 10
10. A tree stands on a hillside of slope 28° from the
horizontal. From a point 75 feet down the hill, the angle
of elevation to the top of the tree is 45°. Find the height
of the tree. (see diagram)
11. Use law of cosines to solve the triangle:
a) 𝑎 = 9, 𝑏 = 12, 𝑐 = 20
b) 𝐵 = 150°, 𝑎 = 10, 𝑐 = 20
12. If 𝑎 = 12 and 𝐵 = 28°, determine the values of b that will produce: a) 2 triangles
b) 1 triangle
c) no triangles
13. Two planes leave Washington, D.C.’s Dulles International Airport at approximately the same time. One
is flying at 425 miles per hour at a bearing of 355°, and the other is flying at 530 miles per hour at a
bearing of 67°. Determine the distance between the planes after they have flown for 2 hours.
14. Find the area of the triangle given: a) 𝑎 = 4, 𝑏 = 5, 𝑐 = 7
b) 𝐴 = 27°, 𝑏 = 5, 𝑐 = 8
15. Use transformations to describe how the graph of the function is related to a basic trigonometric graph.
Graph 2 periods:
3
𝜋
1
b. 𝑓(𝑥) = − 2 csc(𝜋𝑥)
a. 𝑓(𝑥) = − cos (3𝑥 + 2 ) + 4
𝑥
𝜋
c. 𝑓(𝑥) = 2 cot (2 − 8 )
16. Describe how the graph of the function is related to a basic inverse trigonometric graph. State the
domain and range:
1
a. 𝑦 = sin−1(3𝑥 − 1) + 2
b. 𝑦 = cos−1 (2𝑥 + 1) − 3
c. 𝑦 = − arctan ( 𝑥)
2
2𝜋
17. Find the length of the arc intercepted by a central angle of 3 radians in a circle with a radius 2 inches.
18. From the top of a 150-ft building Flora observes a car moving toward her. If the angel of depression of
the car changes from 18° to 42° during the observation, how far does the car travel?
19. A windshield wiper on a Plymouth Acclaim is 20 inches long and has a blade 16 inches long. If the
wiper sweeps through an angle of 110°, how large an area does the wiper blade clean?
IV. CHAPTER 5: VERIFYING TRIG IDS AND SOLVING TRIG EQNS.
1. Find the exact value of
a) cos 285°
b) tan (−
13𝜋
12
)
3
5
2. Find the exact value of sin(𝛼 − 𝛽) if tan 𝛼 = − 4 and cos 𝛽 = 13 .
𝜋
Given 2 < 𝛼 < 𝜋 and
3𝜋
2
< 𝛽 < 2𝜋.
𝑥
8
3. Find the exact value of cos 2 given tan 𝑥 = 15 and 𝜋 < 𝑥 <
2
3𝜋
3
2
4. Find the exact value of sin 2𝜃 given sin 𝜃 = − and
3𝜋
2
< 𝜃 < 2𝜋
√3
2
5. Solve sin 𝜃 + 𝑐𝑜𝑠𝜃 = 1
6. Solve sin 𝑥 cos 2𝑥 − cos 𝑥 sin 2𝑥 =
7. Solve 4 sin2 𝑥 = 2 cos 𝑥 + 1
8. Write the expression as the sine, cosine or
tangent of an angle: a) 2 sin 100° cos 100°
2 tan 40°
b) 1−tan2 40°
9. Simplify the expression to a single term:
a) (1 − 2 sin2 𝜃)2 + 4 sin2 𝜃 cos2 𝜃
b) 1 − 4 sin2 𝑥 cos2 𝑥
Confirm the identity for the following:
10. cos 3𝑥 = 4 cos3 𝑥 − 3 cos 𝑥
(1+tan 𝜃)
(1+cot 𝜃)
12. (1−tan 𝜃) + (1−cot 𝜃) = 0
14. tan (𝑥 +
3𝜋
tan 𝑥−1
) = 1+tan 𝑥
4
16. Verify:
cos2 2𝑥 − sin 2𝑥 = 2 sin 𝑥 cos 𝑥 − cos2 2𝑥
11. 2 sin 𝜃 cos3 𝜃 + 2 sin3 𝜃 cos 𝜃 = sin 2𝜃
cos(−𝑥)
13. sec(−𝑥)+tan(−𝑥) = 1 + sin 𝑥
15. Write sin 3𝑥 + cos 3𝑥 in terms of sin 𝑥 and
cos 𝑥 only.
V. CHAPTER 6: SYSTEMS OF EQUATIONS AND MATRICES
1. Use inverse matrices to solve the system of linear equations:
𝑥 + 3𝑦 − 𝑧 = 13
𝑥 − 𝑦 + 𝑧 = −6
a) { 2𝑥 − 5𝑧 = 23
b) { 2𝑥 − 3𝑦 = −7
4𝑥 − 𝑦 − 2𝑧 = 14
−𝑥 + 3𝑦 − 3𝑧 = 11
2. Use your calculator to find the RREF of the system and solve:
2𝑥 + 3𝑦 + 3𝑧 = 3
3𝑥 + 21𝑦 − 29𝑧 = −1
a) {6𝑥 + 6𝑦 + 12𝑧 = 13
b) {
2𝑥 + 15𝑦 − 21𝑧 = 0
12𝑥 + 9𝑦 − 𝑧 = 2
1 2
7 5 2
3. Find AB if possible: a) 𝐴 = [5 −4] , 𝐵 = [
]
0 1 0
6 0
−6 5
4. Find the inverse of the matrix if it exists: a) [
]
−5 4
−2 4
8 5
5. Find the determinant of the matrix: a) [
] b) [−6 0
2 −4
5 3
6. Use Cramer’s Rule to solve the system:
−2𝑥 + 3𝑦 − 5𝑧 = −11
𝑥 + 2𝑦 = 5
a) {
b) { 4𝑥 − 𝑦 + 𝑧 = −3
−𝑥 + 𝑦 = 1
−𝑥 − 4𝑦 + 6𝑧 = 15
7
3 −2 0
b) 𝐴 = [
],𝐵 = [ 5
1 4 9
−1
−1 −2 −2
−1
b) [ 3
7
9 ] c) [ 3
10
1
4
7
1
1 0 −2
c) [ 0 1 0 ]
2]
4
−2 0 1
0
3]
3
20
−6]
CHAPTER 7
For each conic re-write into standard form, sketch the graph and then provide the important information.
Circle: center and radius
Parabola: vertex, focus, directrix, axis of symmetry
Ellipse: center, vertices, co-vertices, foci, and eccentricity
Hyperbola: center, vertices, foci, and equations of asymptotes
1.  y 2  x  12 y  28  0
2. 4𝑥 2 + 4𝑦 2 + 24𝑥 − 12𝑦 − 19 = 0
3. 9𝑥 2 + 16𝑦 2 − 36𝑥 − 80𝑦 − 8 = 0
4. 𝑥 2 − 40 = −𝑦 2
5. −25𝑥 2 + 16𝑦 2 − 400 = 0
6. 25𝑥 2 + 9𝑦 2 − 150𝑥 + 36𝑦 + 36 = 0
7. 3𝑥 2 − 18𝑥 + 𝑦 + 32 = 0
8. 𝑥 2 − 16𝑦 2 − 2𝑥 − 128𝑦 − 271 = 0
Use the information provided to write the standard form equation of each circle.
9. The endpoints of the diameter are (13, 5) and (-3,-5).
10. The center is at (9, 5) and passes through the point (16,-2).
11. The center lies on the y-axis and is tangent to the x-axis and the line y  10 .
Use the information provided to write the standard form equation of each parabola.
23 

12. The vertex is (-7,-3) and the focus is  7,  
8 

19
 13 
13. The focus is at   , 0  and the directrix is x  
4
 4 
Use the information provided to write the standard form equation of each ellipse.
15.
14.
Use the information provided to write the standard form equation of each hyperbola.
17.
16.
18. Match the graph of the conic to the appropriate equation:
a.
b.
i) 𝑥 2 − 𝑥𝑦 + 𝑦 2 = 2
ii)145𝑠 2 + 120𝑥𝑦 + 180𝑦 2 − 900 = 0
iii) 2𝑥 2 − 72𝑥𝑦 + 23𝑦 2 + 100𝑥 − 50𝑦 = 0
iv) 16𝑥 2 − 24𝑥𝑦 + 9𝑦 2 − 5𝑥 − 90𝑦 + 25 = 0
d.
d.
CHAPTER 10: SEQUENCES AND SERIES - use chapter 10 review and test!