math1 final exam practice - Berkeley City College · Objective: (4.4) Solve Rational Inequalities Use the Rational Zeros Theorem to ﬁnd all the real zeros of the polynomial function

Berkeley City College Math 1 Precalculus - Final Exam PreparationPractice Problems

Name__________________________________________Please print your name as it appears on the class roster.

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

List the intercepts for the graph of the equation.1) 4x2 + y2 = 4 1)Objective: (1.2) Find Intercepts from an Equation

2) y = x2 + 13x + 40 2)Objective: (1.2) Find Intercepts from an Equation

Write the standard form of the equation of the circle.3)

x

y

(3, 5) (7, 5)

x

y

(3, 5) (7, 5)

3)

Objective: (1.4) Write the Standard Form of the Equation of a Circle

Write the standard form of the equation of the circle with radius r and center (h, k).4) r = 6; (h, k) = (2, -3) 4)Objective: (1.4) Write the Standard Form of the Equation of a Circle

Find the center (h, k) and radius r of the circle with the given equation.5) 5(x + 6)2 + 5(y + 2)2 = 30 5)Objective: (1.4) Write the Standard Form of the Equation of a Circle

Find the value for the function.6) Find f(x - 1) when f(x) = 4x2 - 3x + 3. 6)Objective: (2.1) Find the Value of a Function

Instructor: K Pernell 1

7) Find f(x + 1) when f(x) = x2 - 8x - 2

. 7)

Objective: (2.1) Find the Value of a Function

Find the domain of the function.

8) g(x) = 3xx2 - 25

8)

Objective: (2.1) Find the Domain of a Function Defined by an Equation

9) h(x) = x - 1x3 - 81x

9)

Objective: (2.1) Find the Domain of a Function Defined by an Equation

10) f(x) = 13 - x 10)Objective: (2.1) Find the Domain of a Function Defined by an Equation

For the given functions f and g, find the requested function and state its domain.11) f(x) = 7x - 4; g(x) = 9x - 9Find f ∙ g.

11)

Objective: (2.1) Form the Sum, Difference, Product, and Quotient of Two Functions

Solve the problem.12) Find (fg)(4) when f(x) = x - 3 and g(x) = -5x2 + 12x - 4. 12)


13) Find fg(-2) when f(x) = 2x - 5 and g(x) = 3x2 + 14x + 4. 13)


Find and simplify the difference quotient of f, f(x + h) - f(x)h

, h≠ 0, for the function.

14) f(x) = 3x2 14)Objective: (2.1) Form the Sum, Difference, Product, and Quotient of Two Functions

2

The graph of a function f is given. Use the graph to answer the question.15) For what numbers x is f(x) > 0?

100

-100 100

-100

15)

Objective: (2.2) Obtain Information from or about the Graph of a Function

16) How often does the line y = 1 intersect the graph?

5

-5 5

-5

16)


Answer the question about the given function.17) Given the function f(x) = 5x2 + 10x + 8, is the point (-1, 3) on the graph of f? 17)


18) Given the function f(x) = 4x2 + 8x - 6, is the point (-2, 2) on the graph of f? 18)Objective: (2.2) Obtain Information from or about the Graph of a Function

Find the average rate of change for the function between the given values.19) f(x) = x2 + 7x; from 1 to 5 19)

Objective: (2.3) Find the Average Rate of Change of a Function

Write the equation of a sine function that has the given characteristics.20) The graph of y = x2, shifted 6 units upward 20)

Objective: (2.5) Graph Functions Using Vertical and Horizontal Shifts

21) The graph of y = x, shifted 7 units to the right 21)Objective: (2.5) Graph Functions Using Vertical and Horizontal Shifts

3

Graph the function by starting with the graph of the basic function and then using the techniques of shifting,compressing, stretching, and/or reflecting.

22) f(x) = (x - 3)2 - 4

x-10 -5 5 10

y10

5

-5

-10

x-10 -5 5 10

y10

5

-5

-10

22)


23) f(x) = (x + 6)3 + 7

x-10 -5 5 10

y10

5

-5

-10

x-10 -5 5 10

y10

5

-5

-10

23)


Use the accompanying graph of y = f(x) to sketch the graph of the indicated equation.24) y = - 2f(x + 5) + 4

x-10 -5 5 10

y10

5

-5

-10

y = f(x)

x-10 -5 5 10

y10

5

-5

-10

y = f(x)

x-10 -5 5 10

y10

5

-5

-10

x-10 -5 5 10

y10

5

-5

-10

24)

Objective: (2.5) Graph Functions Using Compressions and Stretches

4

Determine the slope and y-intercept of the function.25) h(x) = -5x - 3 25)

Objective: (3.1) Graph Linear Functions

Use the slope and y-intercept to graph the linear function.26) g(x) = -2x + 1

x-5 5

y

5

-5

x-5 5

y

5

-5

26)

Objective: (3.1) Graph Linear Functions

Find the vertex and axis of symmetry of the graph of the function.27) f(x) = -x2 - 6x + 5 27)

Objective: (3.3) Identify the Vertex and Axis of Symmetry of a Quadratic Function

Determine, without graphing, whether the given quadratic function has a maximum value or a minimum value andthen find that value.

28) f(x) = x2 + 2x - 2 28)Objective: (3.3) Find the Maximum or Minimum Value of a Quadratic Function

29) f(x) = -x2 - 2x - 6 29)Objective: (3.3) Find the Maximum or Minimum Value of a Quadratic Function

Solve the problem.30) You have 220 feet of fencing to enclose a rectangular region. Find the dimensions of the

rectangle that maximize the enclosed area.30)

Objective: (3.3) Find the Maximum or Minimum Value of a Quadratic Function

31) You have 324 feet of fencing to enclose a rectangular region. What is the maximum area? 31)Objective: (3.3) Find the Maximum or Minimum Value of a Quadratic Function

32) The quadratic function f(x) = 0.0038x2 - 0.45x + 36.90 models the median, or average, age,y, at which U.S. men were first married x years after 1900. In which year was thisaverage age at a minimum? (Round to the nearest year.) What was the average age atfirst marriage for that year? (Round to the nearest tenth.)

32)

Objective: (3.3) Find the Maximum or Minimum Value of a Quadratic Function

5

Use the figure to solve the inequality.33) f(x) < 0

x-16 -12 -8 -4 4 8 12 16

y16

12

8

4

-4

-8

-12

-16

(-4, 0) (3, 0)x-16 -12 -8 -4 4 8 12 16

y16

12

8

4

-4

-8

-12

-16

(-4, 0) (3, 0)

33)

Objective: (3.5) Solve Inequalities Involving a Quadratic Function

Solve the inequality.34) x2 - 8x ≥ 0 34)

Objective: (3.5) Solve Inequalities Involving a Quadratic Function

35) x2 - 64 ≤ 0 35)Objective: (3.5) Solve Inequalities Involving a Quadratic Function

Form a polynomial whose zeros and degree are given.36) Zeros: -3, -2, 2; degree 3 36)

Objective: (4.1) Identify the Real Zeros of a Polynomial Function and Their Multiplicity

37) Zeros: -4, -2, -1, 1; degree 4 37)Objective: (4.1) Identify the Real Zeros of a Polynomial Function and Their Multiplicity

For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axisat each x -intercept.

38) f(x) = 2(x - 7)(x + 5)3 38)Objective: (4.1) Identify the Real Zeros of a Polynomial Function and Their Multiplicity

39) f(x) = 2(x2 + 3)(x + 2)2 39)Objective: (4.1) Identify the Real Zeros of a Polynomial Function and Their Multiplicity

40) f(x) = 13x(x2 - 5) 40)

Objective: (4.1) Identify the Real Zeros of a Polynomial Function and Their Multiplicity

Find the x- and y-intercepts of f.41) f(x) = (x + 1)(x - 6)(x - 1)2 41)

Objective: (4.1) Analyze the Graph of a Polynomial Function

6

42) f(x) = -x2(x + 6)(x2 + 1) 42)Objective: (4.1) Analyze the Graph of a Polynomial Function

Find the power function that the graph of f resembles for large values of |x|.43) f(x) = 7x - x3 43)

Objective: (4.1) Analyze the Graph of a Polynomial Function

Find the vertical asymptotes of the rational function.

44) f(x) = 3x(x - 4)(x - 8)

44)

Objective: (4.2) Find the Vertical Asymptotes of a Rational Function

45) f(x) = x - 416x - x3

45)

Objective: (4.2) Find the Vertical Asymptotes of a Rational Function

Give the equation of the horizontal asymptote, if any, of the function.

46) h(x) = 8x2 - 5x - 2

5x2 - 4x + 846)

Objective: (4.2) Find the Horizontal or Oblique Asymptotes of a Rational Function

47) h(x) = 3x3 - 4x - 72x + 2

47)


48) g(x) = x + 8x2 - 49

48)


Find the indicated intercept(s) of the graph of the function.

49) y-intercept of f(x) = (5x - 15)(x - 3)x2 + 9x- 19

49)

Objective: (4.3) Analyze the Graph of a Rational Function

50) x-intercepts of f(x) = (x - 2)(2x + 9)x2 + 5x - 5

50)


7

Graph the function.

51) f(x) = 2x(x - 3)(x - 5)

x-8 -4 4 8

y40

20

-20

-40

x-8 -4 4 8

y40

20

-20

-40

51)


52) f(x) = x2 + x - 30x2 - x - 20

x-10 -8 -6 -4 -2 2 4 6 8 1 0

y

1 0

8

6

4

2

-2

-4

-6

-8

-10

x-10 -8 -6 -4 -2 2 4 6 8 1 0

y

1 0

8

6

4

2

-2

-4

-6

-8

-10

52)


Solve the inequality.53) 2x2 + 3x < 20 53)

Objective: (4.4) Solve Polynomial Inequalities

54) x - 7x + 9

< 0 54)

Objective: (4.4) Solve Rational Inequalities

55) x + 18x

< 9 55)


8

56) 8x7 - x

≥ 4x 56)


Use the Rational Zeros Theorem to find all the real zeros of the polynomial function. Use the zeros to factor f over thereal numbers.

57) f(x) = x4 - 24x2 - 25 57)Objective: (4.5) Find the Real Zeros of a Polynomial Function

58) f(x) = x3 + 3x2 - 4x - 12 58)Objective: (4.5) Find the Real Zeros of a Polynomial Function

59) f(x) = 4x3 - 3x2 + 16x - 12 59)Objective: (4.5) Find the Real Zeros of a Polynomial Function

Find the intercepts of the function f(x).60) f(x) = x3 + 2x2 - 5x - 6 60)

Objective: (4.5) Find the Real Zeros of a Polynomial Function

61) f(x) = -x2(x + 6)(x2 + 1) 61)Objective: (4.5) Find the Real Zeros of a Polynomial Function

Information is given about a polynomial f(x) whose coefficients are real numbers. Find the remaining zeros of f.62) Degree 4; zeros: 4 - 5i, 8i 62)

Objective: (4.6) Use the Conjugate Pairs Theorem

Form a polynomial f(x) with real coefficients having the given degree and zeros.63) Degree: 3; zeros: -2 and 3 + i. 63)

Objective: (4.6) Find a Polynomial Function with Specified Zeros

64) Degree: 4; zeros: -1, 2, and 1 - 2i. 64)Objective: (4.6) Find a Polynomial Function with Specified Zeros

For the given functions f and g, find the requested composite function value.65) f(x) = x + 5, g(x) = 2x; Find (f ∘ g)(0). 65)

Objective: (5.1) Form a Composite Function

66) f(x) = 2x + 4, g(x) = 2x2 + 1; Find (g ∘ g)(1). 66)Objective: (5.1) Form a Composite Function

67) f(x) = 2x + 7, g(x) = -2/x; Find (g ∘ f)(3). 67)Objective: (5.1) Form a Composite Function

For the given functions f and g, find the requested composite function.68) f(x) = 7x + 6, g(x) = 5x - 1; Find (f ∘ g)(x). 68)


9

69) f(x) = 3x - 1

, g(x) = 83x; Find (f ∘ g)(x). 69)


70) f(x) = x + 4, g(x) = 8x - 8; Find (f ∘ g)(x). 70)Objective: (5.1) Form a Composite Function

Indicate whether the function is one-to-one.71) {(5, -3), (6, -3), (7, -7), (8, 9)} 71)

Objective: (5.2) Determine Whether a Function Is One-to-One

72) {(4, 5), (-5, -4), (8, -3), (-8, 3)} 72)Objective: (5.2) Determine Whether a Function Is One-to-One

Decide whether or not the functions are inverses of each other.

73) f(x) = 8x - 5, g(x) = x + 85

73)

Objective: (5.2) Find the Inverse of a Function Defined by an Equation

74) f(x) = 2x - 2, g(x) = 12x + 1 74)


75) f(x) = 2x2 + 1, g(x) = x - 12

75)


The function f is one-to-one. Find its inverse.76) f(x) = x2 + 4, x ≥ 0 76)


77) f(x) = 5x - 73

77)


78) f(x) =3x + 7 78)


Solve the equation.

79) 27 - 3x = 14

79)

Objective: (5.3) Solve Exponential Equations

10

80) 2x2 - 3= 64 80)Objective: (5.3) Solve Exponential Equations

Change the exponential expression to an equivalent expression involving a logarithm.81) 73 = 343 81)

Objective: (5.4) Change Exponential Statements to Logarithmic Statements & Logarithmic Statementsto Exponential Statements

82) 52 = x 82)Objective: (5.4) Change Exponential Statements to Logarithmic Statements & Logarithmic Statements

to Exponential Statements

Change the logarithmic expression to an equivalent expression involving an exponent.

83) log 218

= -3 83)


84) ln x = 4 84)Objective: (5.4) Change Exponential Statements to Logarithmic Statements & Logarithmic Statements

to Exponential Statements

85) ln 1e5

= -5 85)


Find the exact value of the logarithmic expression.

86) log4 164

86)

Objective: (5.4) Evaluate Logarithmic Expressions

87) log 5 5 87)

Objective: (5.4) Evaluate Logarithmic Expressions

Solve the equation.88) log3 (x2 - 2x) = 1 88)

Objective: (5.4) Solve Logarithmic Equations

89) 7 + 9 ln x = 4 89)Objective: (5.4) Solve Logarithmic Equations

90) ln x + 5 = 3 90)Objective: (5.4) Solve Logarithmic Equations

11

91) e x + 7 = 5 91)


Write as the sum and/or difference of logarithms. Express powers as factors.

92) log 4x3

y892)

Objective: (5.5) Write a Logarithmic Expression as a Sum or Difference of Logarithms

93) log 3

716

q2p93)


94) log 4mn19

94)


Express as a single logarithm.95) 3 loga (2x + 1) - 2 loga (2x - 1) + 2 95)

Objective: (5.5) Write a Logarithmic Expression as a Single Logarithm

Use the Change-of-Base Formula and a calculator to evaluate the logarithm. Round your answer to two decimalplaces.

96) log8.7 7.6 96)

Objective: (5.5) Evaluate Logarithms Whose Base Is Neither 10 Nor e

Solve the equation.97) log (3 + x) - log (x - 5) = log 3 97)


98) log3 x + log3(x - 24) = 4 98)Objective: (5.6) Solve Logarithmic Equations

Solve the equation. Express irrational answers in exact form and as a decimal rounded to 3 decimal places.99) 3 x = 41 - x 99)

Objective: (5.6) Solve Exponential Equations

Convert the angle in degrees to radians. Express the answer as multiple of !.100) 144° 100)

Objective: (6.1) Convert from Degrees to Radians and from Radians to Degrees

101) 87° 101)Objective: (6.1) Convert from Degrees to Radians and from Radians to Degrees

12

Convert the angle in radians to degrees.

102) - 11!6

102)


103) 349! 103)


Find the exact value. Do not use a calculator.104) cos 2! 104)

Objective: (6.2) Find the Exact Values of the Trigonometric Functions of Quadrantal Angles

105) tan (19!) 105)Objective: (6.2) Find the Exact Values of the Trigonometric Functions of Quadrantal Angles

106) cos 16!3

106)

Objective: (6.2) Find the Exact Values for Integer Multiples of !/6 = 30°, !/4 = 45°, and !/3 = 60°

107) sec 19!4

107)


Find the exact value of the expression. Do not use a calculator.

108) tan 7!4

+ tan 5!4

108)


109) sin 135° - sin 270° 109)Objective: (6.2) Find the Exact Values for Integer Multiples of !/6 = 30°, !/4 = 45°, and !/3 = 60°

110) tan 150° cos 210° 110)Objective: (6.2) Find the Exact Values for Integer Multiples of !/6 = 30°, !/4 = 45°, and !/3 = 60°

Name the quadrant in which the angle θ lies.111) cot θ < 0, cos θ > 0 111)

Objective: (6.3) Determine the Signs of the Trigonometric Functions in a Given Quadrant

In the problem, sin θ and cos θ are given. Find the exact value of the indicated trigonometric function.

112) sin θ = 14, cos θ =

154

Find cot θ. 112)

Objective: (6.3) Find the Values of the Trigonometric Functions Using Fundamental Identities

13

Find the exact value of the indicated trigonometric function of θ.

113) tan θ = - 85, θ in quadrant II Find cos θ. 113)

Objective: (6.3) Find Exact Values of the Trig Functions of an Angle Given One of the Functions andthe Quadrant of the Angle

Without graphing the function, determine its amplitude or period as requested.

114) y = -2 sin 13x Find the amplitude. 114)

Objective: (6.4) Determine the Amplitude and Period of Sinusoidal Functions

115) y = -3 cos 14x Find the period. 115)

Objective: (6.4) Determine the Amplitude and Period of Sinusoidal Functions

Match the given function to its graph.116) 1) y = sin 2x 2) y = 2 cos x

3) y = 2 sin x 4) y = cos 2xA B

x-2π -π π 2π

y3

2

1

-1

-2

-3

x-2π -π π 2π

y3

2

1

-1

-2

-3

x-2π -π π 2π

y3

2

1

-1

-2

-3

x-2π -π π 2π

y3

2

1

-1

-2

-3

C D

x-2π -π π 2π

y3

2

1

-1

-2

-3

x-2π -π π 2π

y3

2

1

-1

-2

-3

x-2π -π π 2π

y3

2

1

-1

-2

-3

x-2π -π π 2π

y3

2

1

-1

-2

-3

116)

Objective: (6.4) Graph Sinusoidal Functions Using Key Points

14

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Answer the question.117) Which one of the equations below matches the graph?

A) y = 4 cos 2x B) y = 2 cos 14x C) y = 4 sin 1

2x D) y = 4 cos 1

2x

117)


118) Which one of the equations below matches the graph?

A) y = 2 cos 3x B) y = 2 sin 13x

C) y = -2 sin 13x D) y = 2 cos 1

3x

118)



Solve the problem.119) What is the y-intercept of y = csc x? 119)

Objective: (6.5) Graph Functions of the Form y = A tan(ωx) + B and y = A cot(ωx) + B

Find the exact value of the expression.

120) cos-1 32

120)

Objective: (7.1) Find the Exact Value of an Inverse Sine, Cosine, or Tangent Function

15

121) cos-1 - 32

121)

Objective: (7.1) Find the Exact Value of an Inverse Sine, Cosine, or Tangent Function

Find the inverse function f-1 of the function f.122) f(x) = 3 cos x + 6 122)

Objective: (7.1) Find the Inverse Function of a Trigonometric Function

123) f(x) = 2 tan(10x - 3) 123)Objective: (7.1) Find the Inverse Function of a Trigonometric Function

Find the exact solution of the equation.124) 4 cos-1 x = ! 124)

Objective: (7.1) Solve Equations Involving Inverse Trigonometric Functions

Find the exact value of the expression.

125) cos sin-1 12

125)

Objective: (7.2) Find the Exact Value of Expressions Involving the Inverse Sine, Cosine, and TangentFunctions

126) sec sin-1 - 49

126)

Objective: (7.2) Find the Exact Value of Expressions Involving the Inverse Sine, Cosine, and TangentFunctions

Solve the equation on the interval 0 ≤ θ < 2π.127) 2 cos θ + 3 = 2 127)

Objective: (7.3) Solve Equations Involving a Single Trigonometric Function

Solve the equation. Give a general formula for all the solutions.128) sin θ = 1 128)

Objective: (7.3) Solve Equations Involving a Single Trigonometric Function

Solve the equation on the interval 0 ≤ θ < 2π.129) cos2 θ + 2 cos θ + 1 = 0 129)

Objective: (7.3) Solve Trigonometric Equations Quadratic in Form

130) 2 sin2 θ = sin θ 130)Objective: (7.3) Solve Trigonometric Equations Quadratic in Form

Simplify the trigonometric expression by following the indicated direction.131) Rewrite in terms of sine and cosine: tan x ∙ cot x 131)

Objective: (7.4) Use Algebra to Simplify Trigonometric Expressions

16

132) Multiply sin θ1 - cos θ

by 1 + cos θ1 + cos θ

132)

Objective: (7.4) Use Algebra to Simplify Trigonometric Expressions

The polar coordinates of a point are given. Find the rectangular coordinates of the point.

133) 7, 2!3

133)

Objective: (9.1) Convert from Polar Coordinates to Rectangular Coordinates

134) -5, 3!4

134)

Objective: (9.1) Convert from Polar Coordinates to Rectangular Coordinates

The rectangular coordinates of a point are given. Find polar coordinates for the point.135) (0, -8) 135)

Objective: (9.1) Convert from Rectangular Coordinates to Polar Coordinates

136) (- 3, -1) 136)Objective: (9.1) Convert from Rectangular Coordinates to Polar Coordinates

The letters x and y represent rectangular coordinates. Write the equation using polar coordinates (r, θ).137) x2 + 4y2 = 4 137)

Objective: (9.1) Transform Equations between Polar and Rectangular Forms

138) y2 = 16x 138)Objective: (9.1) Transform Equations between Polar and Rectangular Forms

Transform the polar equation to an equation in rectangular coordinates. Then identify and graph the equation.139) r = 5

r-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

654321

-1-2-3-4-5-6

r-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

654321

-1-2-3-4-5-6

139)

Objective: (9.2) Identify and Graph Polar Equations by Converting to Rectangular Equations

17

140) r = 2 sin θ

r-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

654321

-1-2-3-4-5-6

r-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

654321

-1-2-3-4-5-6

140)


141) r sin θ = 5

r-5 -4 -3 -2 -1 1 2 3 4 5

654321

-1-2-3-4-5-6

r-5 -4 -3 -2 -1 1 2 3 4 5

654321

-1-2-3-4-5-6

141)


Write the complex number in polar form. Express the argument in degrees, rounded to the nearest tenth, if necessary.142) 3 + i 142)

Objective: (9.3) Convert a Complex Number between Rectangular Form and Polar Form

143) 2 + 2i 143)Objective: (9.3) Convert a Complex Number between Rectangular Form and Polar Form

Find zw or zw

as specified. Leave your answer in polar form.

144) z = 8 cos !6

+ i sin !6

w = 3 cos !2

+ i sin !2

Find zw.

144)

Objective: (9.3) Find Products and Quotients of Complex Numbers in Polar Form

18

145) z = 10(cos 45° + i sin 45°)w = 5(cos 15° + i sin 15°)

Find zw.

145)

Objective: (9.3) Find Products and Quotients of Complex Numbers in Polar Form

Write the expression in the standard form a + bi.146) 2(cos 15° + i sin 15°) 3 146)

Objective: (9.3) Use De Moivre's Theorem

147) (- 3 + i)6 147)Objective: (9.3) Use De Moivre's Theorem

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Match the equation to the graph.148) (y + 2)2 = 8(x + 1)

A)

x-10 -5 5 10

y10

5

-5

-10

x-10 -5 5 10

y10

5

-5

-10

B)

x-10 -5 5 10

y10

5

-5

-10

x-10 -5 5 10

y10

5

-5

-10

C)

x-10 -5 5 10

y10

5

-5

-10

x-10 -5 5 10

y10

5

-5

-10

D)

x-10 -5 5 10

y10

5

-5

-10

x-10 -5 5 10

y10

5

-5

-10

148)

Objective: (10.2) Analyze Parabolas with Vertex at (h, k)


Find an equation for the parabola described.149) Vertex at (8, 7); focus at (8, 3) 149)

Objective: (10.2) Analyze Parabolas with Vertex at (h, k)

19

Write an equation for the graph.150)

x-5 5

y

5

-5

(2, -1)x-5 5

y

5

-5

(2, -1)

150)

Objective: (10.3) Analyze Ellipses with Center at (h, k)

Find the center, foci, and vertices of the ellipse.151) 3x2 + 4y2 - 36x + 32y + 160 = 0 151)


Graph the equation.

152) (x + 1)29

+ (y - 2)24

= 1

x-10 -5 5 10

y

5

-5

x-10 -5 5 10

y

5

-5

152)


Find an equation for the hyperbola described.153) center at (2, 4); focus at (0, 4); vertex at (1, 4) 153)

Objective: (10.4) Analyze Hyperbolas with Center at (h, k)

Find the center, transverse axis, vertices, foci, and asymptotes of the hyperbola.154) x2 - 25y2 + 6x + 50y - 41 = 0 154)


20

Graph the hyperbola.

155) (y + 2)24

- (x - 2)29

= 1

x-5 5

y

5

-5

x-5 5

y

5

-5

155)


Solve the system of equations by substitution.156)

5x - 2y = -1 x + 4y = 35

156)

Objective: (11.1) Solve Systems of Equations by Substitution

Solve the system of equations.157)

x - y + 3z = -164x + z = -5x + 2y + z = -3

157)

Objective: (11.1) Solve Systems of Three Equations Containing Three Variables

Perform the row operation(s) on the given augmented matrix.158) R2 = -4r1 + r2

1 3 104 2 -8

158)

Objective: (11.2) Perform Row Operations on a Matrix

159) (a) R2 = -4r1 + r2(b) R3 = -2r1 + r3(c) R3 = 6r2 + r31 -3 -5 -24 -5 -4 52 5 4 6

159)

Objective: (11.2) Perform Row Operations on a Matrix

Write the partial fraction decomposition of the rational expression.

160) x(x - 4)(x - 5)

160)

Objective: (11.5) Decompose P/Q, Where Q Has Only Nonrepeated Linear Factors

21

161) x + 3x3 - 2x2 + x

161)

Objective: (11.5) Decompose P/Q, Where Q Has Repeated Linear Factors

162) 10x + 2(x - 1)(x2 + x + 1)

162)

Objective: (11.5) Decompose P/Q, Where Q Has a Nonrepeated Irreducible Quadratic Factor

Graph the equations of the system. Then solve the system to find the points of intersection.163)

y = x2 - 8x + 16y = -x + 6

x-10 -5 5 10

y10

5

-5

-10

x-10 -5 5 10

y10

5

-5

-10

163)

Objective: (11.6) Solve a System of Nonlinear Equations Using Substitution

Solve the system of equations using substitution.164)

x2 + y2 = 25x + y = -7

164)


165)ln x = 3ln y3x = 27y

165)


Solve using elimination.166)

x2 + y2 = 145x2 - y2 = 17

166)

Objective: (11.6) Solve a System of Nonlinear Equations Using Elimination

22

167) 2x2 + y2 = 17

3x2 - 2y2 = -6

167)

Objective: (11.6) Solve a System of Nonlinear Equations Using Elimination

23

Answer KeyTestname: MATH1_FINAL_EXAM_PRACTICE

1) (-1, 0), (0, -2), (0, 2), (1, 0)2) (-5, 0), (-8, 0), (0, 40)3) (x - 5)2 + (y - 5)2 = 44) (x - 2)2 + (y + 3)2 = 365) (h, k) = (-6, -2); r = 66) 4x2 - 11x + 10

7) x2 + 2x - 7x - 1

8) {x|x ≠ -5, 5}9) {x|x ≠ -9, 0, 9}10) {x|x ≤ 13}11) (f ∙ g)(x) = 63x2 - 99x + 36; all real numbers12) -36

13) 34

14) 3(2x+h)15) [-100, -60), (70, 100)16) three times17) Yes18) No19) 1320) y = x2 + 621) y = x - 722)

x-10 -5 5 10

y10

5

-5

-10

x-10 -5 5 10

y10

5

-5

-10

23)

x-10 -5 5 10

y10

5

-5

-10

x-10 -5 5 10

y10

5

-5

-10

24)

x-10 -5 5 10

y10

5

-5

-10

x-10 -5 5 10

y10

5

-5

-10

25) m = -5; b = - 326)

x-5 5

y

5

-5

x-5 5

y

5

-5

27) (-3, 14) ; x = -328) minimum; - 329) maximum; - 530) 55 ft by 55 ft31) 6561 square feet32) 1959, 23.6 years old33) {x|-4 < x < 3}; (-4, 3)34) (-∞, 0] or [8, ∞)35) [-8, 8]36) f(x) = x3 + 3x2 - 4x - 12 for a = 1

24


37) x4 + 6x3 + 7x2 - 6x - 838) 7, multiplicity 1, crosses x-axis; -5, multiplicity 3,

crosses x-axis39) -2, multiplicity 2, touches x-axis40) 0, multiplicity 1, crosses x-axis;

5, multiplicity 1, crosses x-axis;- 5, multiplicity 1, crosses x-axis

41) x-intercepts: -1, 1, 6; y-intercept: -642) x-intercepts: -6, 0; y-intercept: 043) y = -x344) x = 4, x = 845) x = 0, x = -4

46) y = 85

47) no horizontal asymptotes48) y = 0

49) 0, - 4519

50) (2, 0), - 92, 0

51)

x-8 -4 4 8

y40

20

-20

-40

x-8 -4 4 8

y40

20

-20

-40

52)

x-10 -8 -6 -4 -2 2 4 6 8 1 0

y1 2

1 0

8

6

4

2

-2

-4

-6

-8

-10

-12

x-10 -8 -6 -4 -2 2 4 6 8 1 0

y1 2

1 0

8

6

4

2

-2

-4

-6

-8

-10

-12

53) -4 , 52

54) (-9, 7)55) (-∞, 0) or (3, 6)56) (-∞, 0] or [5, 7)57) -5, 5; f(x) = (x - 5)(x + 5)(x2 + 1)58) -3, -2, 2; f(x) = (x + 3)(x + 2)(x - 2)

59) 34; f(x) = (4x - 3)(x2 + 4)

60) x-intercepts: -3, -1, 2; y-intercept: -661) x-intercepts: -6, 0; y-intercept: 062) 4 + 5i, -8i63) f(x) = x3 - 4x2 - 2x + 2064) f(x) = x4 - 3x3 + 5x2 - x - 1065) 566) 19

67) - 213

68) 35x - 1

69) 9x8 - 3x

70) 2 2x - 171) No72) Yes73) No74) Yes75) Yes; Exclude the interval (-∞, 1)76) f-1(x) = x - 4, x ≥ 4

77) f-1(x) = 3x + 75

78) f-1(x) = x3 - 779) {3}80) {3, -3}81) log 7 343 = 382) log 5 x = 2

83) 2-3 = 18

84) e4 = x

85) e-5 = 1e5

86) -3

87) 12

88) {3, -1}25


89) {e -1/3}90) {e6 - 5}91) {ln 5 - 7}92) 3 log 4 x - 8 log 4 y

93) 17

log 3 16 - 2 log 3 q - log 3 p

94) 12log 4 m + 1

2log 4 n - 1

2log 4 19

95) loga a2(2x + 1)3

(2x - 1)2

96) 0.9497) {9}98) {27}

99) ln 4ln 3 + ln 4

≈ 0.558

100) 4π5

101) 29π60

102) -330°103) 680°104) 1105) 0

106) - 12

107) - 2108) 0

109) 2 + 22

110) - 5 36

111) IV112) 15

113) - 5 8989

114) 2115) 8π116) 1B, 2D, 3C, 4A117) A118) B119) none

120) π6

121) 5π6

122) f-1(x) = cos-1 x - 63

123) f-1(x) = 110

tan-1 x2

+ 3

124) 22

125) 32

126) 9 6565

127) 2π3, 4π3

128) θ|θ = π2

+ 2kπ

129) {π}

130) 0, π, π6, 5π6

131) 1

132) 1 + cos θsin θ

133) - 72, 7 32

134) 5 22, -5 2

2

135) 8, - π2

136) 2, - 5π6

137) r2(cos2 θ + 4 sin2 θ) = 4138) r sin2 θ = 16 cos θ

26


139)

r-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

654321

-1-2-3-4-5-6

r-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

654321

-1-2-3-4-5-6

x2 + y2 = 25; circle, radius 5, center at pole

140)

r-5 -4 -3 -2 -1 1 2 3 4 5

654321

-1-2-3-4-5-6

r-5 -4 -3 -2 -1 1 2 3 4 5

654321

-1-2-3-4-5-6

x2 + (y - 1)2 = 1; circle, radius 1,center at (0, 1) in rectangular coordinates

141)

r-5 -4 -3 -2 -1 1 2 3 4 5

54321

-1-2-3-4-5

r-5 -4 -3 -2 -1 1 2 3 4 5

54321

-1-2-3-4-5

y = 5; horizontal line 5 units above the pole

142) 2(cos 30° + i sin 30°)143) 2 2(cos 45° + i sin 45°)

144) 24 cos 2π3

+ i sin 2π3

145) 2(cos 30° + i sin 30°)146) 4 2 + 4 2i147) -64148) D149) (x - 8)2 = -16(y - 7)

150) (x - 2)216

+ (y + 1)29

= 1

151) (x - 6)24

+ (y + 4)23

= 1

center: (6, -4); foci: (7, -4), (5, -4); vertices: (8, -4), (4,-4)

152)

x-10 -5 5 10

y

5

-5

x-10 -5 5 10

y

5

-5

153) (x - 2)2 - (y - 4)23

= 1

154) center at (-3, 1)transverse axis is parallel to x-axisvertices at (-8, 1) and (2, 1)foci at (-3 - 26, 1) and (-3 + 26, 1)

asymptotes of y - 1 = - 15(x + 3) and y - 1 =

15(x + 3)

155)

x-5 5

y

5

-5

x-5 5

y

5

-5

156) x = 3, y = 8; (3, 8)157) x = 0, y = 1, z = -5; (0, 1, -5)158) 1 3 10

0 -10 -48

27


159)1 -3 -5 -20 7 16 130 53 110 88

160) -4x - 4

+ 5x - 5

161) 3x

+ -3x - 1

+ 4(x - 1)2

162) 4x - 1

+ -4x + 2x2 + x + 1

163)

x-10 -5 5 10

y10

5

-5

-10

x-10 -5 5 10

y10

5

-5

-10

(2, 4), (5, 1)164) x = -3, y = -4; x = -4, y = -3

or (-3, -4), (-4, -3)165) x = 3 3, y = 3 or (3 3, 3)166) x = 9, y = 8; x = -9, y = 8; x = 9, y = -8; x = -9, y = -8

or (9, 8), (-9, 8), (9, -8), (-9, -8)167) x = 2, y = 3; x = 2, y = -3; x = -2, y = 3; x = -2, y = -3

or (2, 3), (2, -3), (-2, 3), (-2, -3)

28

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math1 final exam practice - Berkeley City College · Objective: (4.4) Solve Rational Inequalities Use the Rational Zeros Theorem to ﬁnd all the real zeros of the polynomial function