MRS21 – Algebra 2/Trigonometry Exam 6 Review Sheet

Topics: • Relations and Functions • Domain and Range of a Function • Function Notation • Composition of Functions • Finding the Inverse of a Relation Algebraically and Graphically • Inverse Variation • Equation of a Circle (𝑥 − ℎ)! + (𝑦 − 𝑘)! = 𝑟! • Transformations of Functions −𝑓 𝑥 ,    𝑓 −𝑥 , 𝑓 𝑥± ℎ , 𝑎𝑓 𝑥 ,      𝑓 𝑎𝑥  • Laws of Exponents • Fractional and Negative Exponents • Solving Equations with Fractional or Negative Exponents

Be sure to thoroughly prepare for the exam by reviewing (and re-doing) problems in your class notes and homework assignments. Note that exams are cumulative, meaning that some questions on this exam will cover topics that were tested on previous exams and quizzes.

Practice Problems:

1. If f (x) = 2x and g(x) = x − 4 , what is the value of g f (3)( ) ?

2. What is the inverse of the equation y = 3x + 2 ? (1) y = 1

3 x − 2 (2) x = 13 y + 2

3 (3) 3y = x + 2 (4) x = 3y + 2

3. Solve algebraically for z: 272z−8 = 19

⎛⎝⎜

⎞⎠⎟3z

4. On the accompanying diagram, draw a mapping of a relation from Set A to Set B that not a function. Explain why the relationship you drew is not a function. 5. Given f (x) = x2 + 3 and g(x) = 2x − 5 , find (a) g f (−2)( ) (b)

6. Which graph represents a one-to-one function? (1) (2) (3) (4)

7. Write an equation of the circle shown in the diagram below.

8. Write the equation x2 + y2 −10x +16y + 8 = 0 in center-radius form and state the coordinates of the

center and the length of the radius.

9. If R is inversely proportional to A, and R = 4 when A = 100 , what is the value of R when A = 250 ?

10. To balance a seesaw, the distance, in feet, a person is from the fulcrum is inversely proportional to the person’s weight, in pounds. Bill, who weighs 150 pounds, is sitting 4 feet away from the fulcrum. If Dan weighs 120 pounds, how far from the fulcrum should he sit to balance the seesaw?

11. If R varies inversely as S, when S is doubled, R is multiplied by (1) 12 (2) 2 (3) 14 (4) 4

12. The relation defined by the set of ordered pairs is not a function. Which of the ordered pairs listed below, if omitted from this relation, will make the resulting set a function

(1) (–2, 2) (2) (1, 4) (3) (4, 1) (4) (0, –1) 13. What is the inverse of the function 52)( −= xxf ?

Math.–Course III–Aug. ’03 [6]

38 In the accompanying diagram of circle O, istangent to the circle at A; is a secant; diameter intersects chord at E;chords , , and are drawn; m = 46;and m is 32 more than m .

Find:

a m [2]b m∠BAC [2]c m∠P [2]d m∠DEC [2]e m∠PDA [2]

39 a Given the equation y = 2x.(1) On graph paper, sketch and label the graph

of the equation y = 2x in the interval –3 ≤ x ≤ 3. [2]

(2) On the same set of axes, reflect the graphdrawn in part a(1) in the line y = x andlabel it c. [2]

(3) What is the equation of the graph drawn inpart a(2)? [2]

b Using logarithms, solve for x to the nearesthundredth: 5 x = 1,325 [4]

40 Find all values of θ in the interval 0° ≤ θ < 360°that satisfy the equation 3 cos 2θ = 7 cos θ. Expressyour answer to the nearest tenth of a degree ornearest ten minutes. [10]

41 a Solve for x and express the roots in simplest a + bi form: [6]

b Given: f(x) = and g(x) = 6x – 3Find:(1) g(f(10)) [2](2) (f ° g)(x) [2]

42 a A spinner is divided into six equal sections andlabeled as shown in the accompanying diagram.

(1) Determine the probability of getting a letter in one spin. [1]

(2) Determine the probability of getting noletters in three spins. [2]

(3) Determine the probability of getting atleast two letters in three spins. [3]

b The table below shows the scores that a class ofstudents received on their latest review quiz.

Find the standard deviation of these scores tothe nearest tenth. [4]

A B

1 D

2 C

2 5x +

9 6x x+ = _2

AB

A P

D

C

BO

E

ABBCDADABCAB

BDAEOCPDC

PA

Score Frequency95 690 785 880 4

Regents Exam Questions A2.A.43: Defining Functions Name: ________________________ www.jmap.org

1

A2.A.43: Defining Functions: Determine if a function is one-to-one, onto, or both

1 Which function is QRW one-to-one?1) {(0,1),(1,2),(2,3),(3,4)}2) {(0,0),(1,1),(2,2),(3,3)}3) {(0,1),(1,0),(2,3),(3,2)}4) {(0,1),(1,0),(2,0),(3,2)}

2 Which diagram represents a one-to-one function?

1)

2)

3)

4)

3 Which graph represents a one-to-one function?

1)

2)

3)

4)

4 Which function is one-to-one?1) f([) [_ _2) f([) 2[

3) f([) [ 2

4) f([) sin[

5 Which function is one-to-one?1) k([) [ 2 � 22) g([) [ 3 � 23) f([) [_ _ � 24) j([) [ 4 � 2

Regents Exam Questions A2.A.37: Defining Functions Name: ________________________ www.jmap.org

1

A2.A.37: Defining Functions: Define a relation and function

1 Which diagram represents a relation in which each member of the domain corresponds to only one member of its range?

1) 3)

2) 4)

2 On the accompanying diagram, draw a mapping of a relation from set A to set B that is not a function. Explain why the relationship you drew is not a function.

Regents Exam Questions A2.A.49: Equations of Circles Name: ________________________ www.jmap.org

1

A2.A.49: Equations of Circles: Write the equation of a circle from its graph

1 Which equation represents the circle shown in the graph below that passes through the point (0,�1)?

1) ([ � 3)2 � (\ � 4)2 162) ([ � 3)2 � (\ � 4)2 183) ([ � 3)2 � (\ � 4)2 164) ([ � 3)2 � (\ � 4)2 18

2 Write an equation of the circle shown in the diagram below.

3 A circle shown in the diagram below has a center of (�5,3) and passes through point (�1,7).

Write an equation that represents the circle.

4 Write an equation of the circle shown in the graph below.

Regents Exam Questions A2.A.43: Defining Functions Name: ________________________ www.jmap.org

1

A2.A.43: Defining Functions: Determine if a function is one-to-one, onto, or both

1 Which function is QRW one-to-one?1) {(0,1),(1,2),(2,3),(3,4)}2) {(0,0),(1,1),(2,2),(3,3)}3) {(0,1),(1,0),(2,3),(3,2)}4) {(0,1),(1,0),(2,0),(3,2)}

2 Which diagram represents a one-to-one function?

1)

2)

3)

4)

3 Which graph represents a one-to-one function?

1)

2)

3)

4)

4 Which function is one-to-one?1) f([) [_ _2) f([) 2[

3) f([) [ 2

4) f([) sin[

5 Which function is one-to-one?1) k([) [ 2 � 22) g([) [ 3 � 23) f([) [_ _ � 24) j([) [ 4 � 2

Regents Exam Questions A2.A.43: Defining Functions Name: ________________________ www.jmap.org

1

A2.A.43: Defining Functions: Determine if a function is one-to-one, onto, or both

1 Which function is QRW one-to-one?1) {(0,1),(1,2),(2,3),(3,4)}2) {(0,0),(1,1),(2,2),(3,3)}3) {(0,1),(1,0),(2,3),(3,2)}4) {(0,1),(1,0),(2,0),(3,2)}

2 Which diagram represents a one-to-one function?

1)

2)

3)

4)

3 Which graph represents a one-to-one function?

1)

2)

3)

4)

4 Which function is one-to-one?1) f([) [_ _2) f([) 2[

3) f([) [ 2

4) f([) sin[

5 Which function is one-to-one?1) k([) [ 2 � 22) g([) [ 3 � 23) f([) [_ _ � 24) j([) [ 4 � 2

Regents Exam Questions A2.A.43: Defining Functions Name: ________________________ www.jmap.org

1

A2.A.43: Defining Functions: Determine if a function is one-to-one, onto, or both

1 Which function is QRW one-to-one?1) {(0,1),(1,2),(2,3),(3,4)}2) {(0,0),(1,1),(2,2),(3,3)}3) {(0,1),(1,0),(2,3),(3,2)}4) {(0,1),(1,0),(2,0),(3,2)}

2 Which diagram represents a one-to-one function?

1)

2)

3)

4)

3 Which graph represents a one-to-one function?

1)

2)

3)

4)

4 Which function is one-to-one?1) f([) [_ _2) f([) 2[

3) f([) [ 2

4) f([) sin[

5 Which function is one-to-one?1) k([) [ 2 � 22) g([) [ 3 � 23) f([) [_ _ � 24) j([) [ 4 � 2

14. Given the graph of y = f (x) ,

draw the graph of each of the following:

(a) y = f (x)+ 4

(b) y = f (−x)

(c) y = f (x + 6)− 4

(d) y = f (x − 5)

(e) y = − f (x)

(f) y = 2 f (x − 7)

15. Which diagram represents a relation in which each member of the domain corresponds to only one member of its range?

(1) (2) (3) (4)

16. Find the domain for each function:

(a) 6)( −= xxf (b) x

xxg−

+=752)( (c)

91)( 2 −

+=xxxh

17. Which graph represents a function? (1) (2) (3) (4)

18. If xxxf 3)( 2 += and 3)( += xxg , find each of the following: (a) )2)(( −gf ! (b) )2)(( −fg ! (c) ))(( xgf ! (d) ))(( xfg !

19. If , then is equivalent to (1) (2) (3) (4) 20. On the accompanying set of axes, 21. The graph below represents the function graph the function and its . State the domain and range of this function.

inverse, .

22. Simplify and express with positive exponents only: (a) a2b−3

a−4b2 (b)

x3y−2( )2

x−3y3

23. Evaluate:

(a) 6−2 (b) 2(4)2 −3(4)0 (c) 23

!"#

$%&

−3

(d) −27( )−43 (e)

1125!"#

$%&

−13

24. Solve and check:

(a) x13 = 4 (b) x

25 = 9 (c) b

32 = 8

(d) 9a−34 =13

(e) 5x12 + 7 = 22 (f) 14− 4b

13 = 2

y

x

1

2

3

4

5

6

7

8

9

10

11

12

–1–2–3–4–5–6–7–8–9–10–11–12 1 2 3 4 5 6 7 8 9 10–1

–2

–3

–4

–5

–6

–7

–8

–9

–10

–11

–12

11 12

Math. B – June ’05 [7] [OVER]

Part II

Answer all questions in this part. Each correct answer will receive 2 credits. Clearly indicate thenecessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For allquestions in this part, a correct numerical answer with no work shown will receive only 1 credit. [12]

21 The graph of the function g(x) is shown on the accompanying set ofaxes. On the same set of axes, sketch the image of g(x) under the trans-formation D2.