This packet belongs to:__________________________________________

Algebra 2 CC Midterm Review January 2018

WHEN/WHERE: Wednesday, January 24th, 12:00-1:30 in room ___________

BRING WITH YOU: Your graphing calculator Two pencils and two pens (black & blue

only)

REVIEW:

Thursday, 1/18 in class

Friday, 1/19 in class

FORMAT: 22 QuestionsWHAT IT COUNTS FOR:

Does not count as part of your 2nd marking period grade

Counts as 4% of your average for the year

10 Multiple-choice questions (2 points each) 5 show all work questions (2 points each) 5 show all work questions (4 points each) 2 show all work questions (6 points each)

Page # Topics to Study:Unit 1 – Algebraic Essentials 3

Multiply binomial by a trinomial

Unit 2 – Functions 3-5

Composition of functions Average rate of change Domain increasing/decreasing

Unit 3 – Linear Functions 5-7

Write equation of inverse of a linear function Solve 3 X 3 system of equations

Unit 4 – Exponential & Logarithmic Functions

7-10

Equations with Common Bases Exponential Growth/Decay word problems (Applications) Operations on exponents and Rules of exponents

(Multiplying, Fractional exponents, Negative exponents) Log equations Properties of Logarithms (Power and Product Rules)

Unit 5 – Sequences & Series 11-12

Recursive definition Sum of a series (arithmetic and geometric)

Unit 6 – Quadratic Functions 13-16

Factor by grouping Quadratic inequalities Graph a parabola, write equation of parabola, and find focus Vertex form of a parabola (CTS) Center, radius of circle with CTS Quadratic linear system algebraically

1

2

Unit 1 – Algebraic EssentialsTopic to study:

Multiply binomial by a trinomial

1. Use any method write the product of the two polynomials in standard form.

(3 x2−3 x+1)(4 x+3)

2. Use any method write the product of the two polynomials in standard form.

(2 x2+3 x−1)(5 x−2)

Unit 2 – FunctionsTopics to study:

Composition of functions Average rate of change

Domain increasing/decreasing

3. If

a) Find f ¿

b)

4. For the functions f ( x )=x+6 and g ( x )=x2+7a) Find g (f (−4 ))

b) f (g ( x ) )3

5. An astronaut drops a rock off the edge of a cliff on the Moon. The distance, , in meters, the rock travels after t seconds can be modeled by the function d (t )=0.8 t2. What is the average speed, in meters per second, of the rock between 5 and 10 seconds after it was dropped?

6. Given the functions and , shown below:

Which function has the greatest average rate of change over the interval [0,3] ? Justify your answer.

AVERAGE RATE OF CHANGE

For a function over the domain interval , the function's average rate of change is calculated by:

g ( x )=x2−2 x

4

7. For the graph of pictured below, over which interval is only decreasing?

For what intervals is only increasing?

Unit 3 – Linear FunctionsTopics to study:

Write equation of inverse of a linear function Solve 3 X 3 system of equations

8. Find g−1(x ), the inverse of g ( x )=−2 x+5.

1) g−1 ( x )=¿ −x2

+ 52

2) g−1 ( x )=−x2

−52

3) g−1 ( x )=2 x−5

4) g−1 ( x )= 1−2x+5

5

9. a) Write the equation of the inverse of f ( x )=2x−1. The use of the accompanying grid is optional.

10. Solve the following system of equations. Carefully show how you arrived at your answers.

6

11. Solve the following system of equations. Carefully show how you arrived at your answers.

Unit 4 – Exponential & Logarithmic FunctionsTopics to study:

Equations with Common Bases Exponential Growth/Decay word problems (Applications)

Operations on exponents and Rules of exponents (Multiplying, Fractional exponents, Negative exponents)

Log equations Properties of Logarithms (Power and Product Rules)

Can you create a common base?

Multiply exponents

Cancel Bases

Solve

7

12. Solve algebraically for x: 27 9 2x x 13.

Solve for x: ( 14 )

x

=81− x

14.

Which expression is equivalent to ?1)

2)

3)

4)

15.

Which expression is equivalent to ?1)

2)

3)

4)

16. Which expression is equivalent to(2−3 x4 y−2 )−1 ?1) 6 y2

x4

2) y2

8 x4

3) 8 y2

x4

4) y2

6 x4

17.The expression is equivalent to1)

2)

3)

4)

8

18. The Franklins inherited $3,500, which they want to invest for their child’s future college expenses. If they invest it at 8.25% with interest compounded monthly, determine the value of the account, in dollars, after 5

years. Use the formula , where of the investment after t years, invested, interest rate, and of times compounded per year.

19. Five thousand dollars is invested at an interest rate of 3.5% compounded continuously. No money is deposited or withdrawn from the account. Determine, to the nearest cent, how much this investment will be worth in 18 years.

9

20. If , then x is equal to:

1) 72) 123) 644) 81

21. If , find, to the nearest tenth, the value of x.

22. Solve algebraically for the exact

value of x: .

23. If , then x is equal to:

1) 72) 123) 244) 32

24. Solve for x:

25. Currently, the population of the metropolitan Waterville area is 62,700 and is increasing at an annual rate of

3.25%. This situation can be modeled by the equation , where represents the total population and t represents the number of years from now. Determine how many years, to the nearest tenth, it will take for the original population to reach 100,000. [Only an algebraic solution can receive full credit.]

10

Unit 5 – Sequences and SeriesTopics to study:

Recursive definition Sum of a series (arithmetic and geometric)

26. Write the first five terms of

an = 3an−1+ 4 and a1 = 2

27. For the recursively defined sequence

and , the value of is

28. Find the first three terms of the recursive

sequence:

t1=−2tn=(t n−1)2+3

29. A bouncy ball rebounds to 90% of the height of the preceding bounce. Craig drops a bouncy ball from a height of 20 feet. Which of the following is a recursive formula that models the height of the ball after it was dropped?

(1) an=20 ( .90 )n (2) an=20 ( .10 )n (3) a1=20 (4)a1=20

an=.10 (an−1 ) an=.90 (an−1 )

Sum of a Finite Geometric Series

Look on formula sheet for formula!

30. Find the sum of the first 6 terms of the geometric series

(1)

(2)

(3)

Recursive Formula:Example: a1=5

an=2an−1+3

*In the example, the first term is 5 and every term after is 3 more than twice the previous term

A recursive formula always has two parts:1. the starting value for a12. the recursion equation for an as a function of an−1 (the term before it).

11

(4)

31. Which of the following represents the sum of if the arithmetic series has 14 terms?

(1) 1,358(2) 658(3) 679(4) 1,276

32. Which of the following represents the sum ?

(1) (3)

(2) (4)

33. Express each sum using sigma notation. Use i as your index variable. First, consider any patterns you notice amongst the terms involved in the sum. Then, work to put these patterns into a formula and sum.

(a) (b)

Sum of a Finite Arithmetic Series

Sn=n2(a1+an)

n = # of terms, a1=first term, an=last term

*Memorize* NOT on formula sheet!

12

Unit 6 – Quadratic FunctionsTopics to study:

Factor by grouping Quadratic inequalities

Graph a parabola, write equation of parabola, and find focus Vertex form of a parabola (CTS) Center, radius of circle with CTS

Quadratic linear system algebraically

34. When factored completely, equals

1)2)3)

4)

35. The completely factored form of is

1)

2)

3)

4)

36. The solution set of the inequality x2−3 x−4>6 is

1)2)3)4)

37. What is the solution set for the inequality

?1)

2)

3)

4)

13

38. The directrix of the parabola 12 ( y−2 )= (x−5 )2 has the equation ¿−1 .

a) Graph the parabola on the accompanying set of axes.

b) Find the coordinates of the focus of the parabola.

THE LOCUS DEFINITION OF A PARABOLA:

A parabola is the collection of all points equidistant from a fixed point (known as its focus)

and a fixed line (known as its directrix).

THE EQUATION OF A PARABOLAWhen writing the equation of a parabola

when the vertex (h,k) is known and

the distance from the vertex to the focus and directrix, p, is known, we use the

formula

( x−h )2=4 p ( y−k )

14

39. Which equation represents the set of points equidistant from line and point V shown on the graph below?

(1)

(2)

(3)

(4)

Equation of a Circle

40. Determine the center and radius of each circle.

a)

b)

l

15

41. Solve the following systems of equations algebraically:

42. Solve the following systems of equations algebraically:

y=x2−x−6

y+6=3 x

16

Name____________________________________ Date______________Day 1 HW: MIDTERM REVIEW HW #1 Algebra 2 CC Units 1-6

1. Find the product of ( x+3 ) (4 x2−6 x+9 ) and write your solution in standard form.

2. Given (use of the grid is optional)Find the equation of f−1(x).

3. Given the function shown below, over which of the following intervals is the function always increasing?

(1)

(2)

(3)

(4)

4. Which of the following is the solution set to the inequality ?

(1) (3)

(2) (4)

5. Given and

17

a) Find

b)6. Given the functions , and shown below, which has the greatest average rate of change over the interval

?

7. Factor the expression completely.

8. Determine the center and radius of the circle whose equation is

9. Solve the following system of equations algebraically for all values of x, y and z:

18

Name____________________________________ Date______________Day 2 HW: MIDTERM REVIEW HW #2 Algebra 2 CC Units 1-6

1. Which value of k satisfies the equation ?

1) 3)2) 4)

2. If and is a positive integer, then which of the following is equivalent to ?

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

3. Expressed in simplest form, is equivalent to

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

4. Which expression is equivalent to(4−2a3b−4 )−1 ?

(1) 16a3

b4 (2)

b4

16a3 (3)

16b4

a3 (4)

a3

16b4

5. Solve the following equation algebraically.

6. Emily’s parents gave her $2000 to invest for her 20th birthday. She is considering two investment options. Option A will pay her 5.2% interest compounded annually. Option B will pay her 5% compounded continuously.

a. Emily plans to use the money after her 30th birthday in 10 years. Determine how much each option would give her after 10 years.

b. Algebraically determine, to the nearest tenth of a year, how long it would take for option B to double Emily’s initial investment.

19

7. Solve for all values for x:

8. The population of Jamesburg for the years 2010-2013, respectively, was reported as follows:250,000 250,937 251,878 252,822

How can this sequence be recursively approximately modeled?1)

2)

3)

4)

9. For the recursively defined sequence and , find the value of .

10. The formula below can be used to model which scenario?

1) An account that has 5000$ to start will increase in value by 94% each year.2) An account that has 5000$ will decrease in value by 94% each year.3) The initial value of a car is $5000, and its value each of the following years is 6% more.4) The initial value of a car is $5000, and its value each of the following years is 6% less.

11. Given the arithmetic sequence 23, 29, 35, 41, ... , 77.a. Express the sum using sigma notation. Use i as your index variable.

12. Find the sum of the first 8 terms of the series

20

13. A parabola has an equation (x – 6)2 = 12 (y-5) and a directrix of .

a) Graph the parabola on the accompanying set of axes.

b) Find the coordinates of the focus of the parabola.

14. A parabola has a directrix of y = -2 and a focus of (0,4). What are the coordinates of the vertex of the parabola?

15. log 4 log6 1x 16. 2log log 25 2x

21