ExamView - M3Sem1_review_2016

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M3Sem1Rev

Multiple Choice

Identify the choice that best completes the statement or answers the question.

____ 1. Lena made 32 ounces of a fruit drink mix using pineapple juice and grapefruit juice. The number of ounces of pineapple juice in the fruit drink mix is 5 more than 2 times the number of ounces of grapefruit juice in the fruit drink mix. Which graph shows the number of ounces pineapple juice, x, and the number of ounces of grapefruit juice, y, in the fruit drink mix? What system of equations was used to create the graph?

a. x+ y = 32

x−5 = 2y

c. 5x+ 2y = 32

x−5 = 2y

b. x+ y = 32

x+5 = 2y

d. 5x+ 2y = 32

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____ 2. What is the vertex of f x( )? Is it a maximum or a minimum?

a. Ê_ËÁÁ0, −2ˆ_¯˜˜; minimum

b. Ê_ËÁÁ3, −5ˆ_¯˜˜; minimum

c. Ê_ËÁÁ−2, 0ˆ_¯˜˜; minimum d. Ê_ËÁÁ8, 0ˆ_¯˜˜; maximum

____ 3. Without using graphing technology, sketch the parent graph and translate it to obtain a graph of

y+4= |x−5|.

a. c.

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____ 4. A website allows its users to submit and edit content in an online encyclopedia. The graph shows the number of articles a t( ) in the encyclopedia t months after the website went live. How many articles were in the

encyclopedia when it went live?

a. 0 b. 30 c. 60 d. 180

____ 5. Ted’s comic book collection, which was worth $1300 five years ago, has been increasing in value by 12% per year since then. Which expression gives the current value of the collection?

a. 1300 1.12( )5 c. 1300 1.12( )( )5

b. 1300 .12( )5 d. 1300 1ÈÍÍÍ_Î +( ).12 ( )5 ˘_˚˙˙˙

____ 6. Which of the following best describes the sum of ax2 +_bx+_{c and mx}2+_nx+_{p, where x is a variable and a,}

b, c, m, n, and p are real numbers?

a. The sum is a constant.

b. The sum is an exponential expression. c. The sum is a polynomial.

d. Nothing can be determined about the sum without more information.

____ 7. Multiply. (d + 7)(d – 7)

a. d2_{– 49} _c. _2d2_{– 49d}

b. d2_{+ 14d + 49} _d. _d2_{– 14d – 49}

____ 8. Describe the transformation of the graph f x( ) =2x2 to g x( ) =2x2−4.

a. f(x) is translated 4 units down. c. f(x) is translated 4 units to the left.

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____ 9. Which function is NOT a translation of f x( ) =x2+17?

a. f x( ) = (x−4)2+17 c. f x( ) _{= −}x2₋17

b. f x( ) =x2−4 d. f x( ) = x+ 1

2 Ê Ë

ÁÁÁÁÁÁ ˆ_¯˜˜˜˜˜˜2 ____ 10. Which function’s graph is the widest parabola?

a. y= 1₈x2 c. y=3x2

b. y₌ 1

3x 2

d. y₌8x2

____ 11. Find the domain and range.

a. D: all real numbers R: y≥3

c. D: −10≤x≤10 R: y≥ −5 b. D: x≥ −5

R: all real numbers

d. D: all real numbers R: y≥ −5

____ 12. Which equation below allows you to solve 2x2₋₁₅₌_{x using the zero product property?}

a. (2x+5)(x−3)=0 b. (2x−5)(x−3)=0 c. (2x+5)(x+3)=0 d. (2x−5)(x+3)=0

____ 13. Solve −4x2 =5x+9.

a. x= −1 or x _{= −}1

4 c. x= −

5± 119 8

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____ 14. Find the root(s) of the equation 16x₋16₌4x2.

a. x= −2 or x ₌2 c. x₌4 or x= −4

b. x= −4 only d. x₌2 only

____ 15. When the quadratic formula is applied to 2x2+_3x−₄=_{0, what is the numerator of the simplified answer?}

a. 3± 41 b. −3± 41 c. 3± 38 d. −3± −23

____ 16. Write the quadratic function c x( ) =x2 −16x+84 in vertex form.

a. c x( ) ₌ (x₋16)2₋172 c. c x( ) =(x−8)2+84 b. c x( ) = (x−8)2+20 d. c x( ) =(x−8)2+148

____ 17. Factor to find the zeros of the function y =5x2−6x+1.

a. 1, 5 c. ₋1, ₋5

b. 1

5, 1 d. −1, −

1 5

____ 18. Which of the following data sets is best described by a linear model?

a. {(₋2,4),(₋1,9),(0,16),(1,25)} c. {(₋2,12), (₋1,10), (0,8),(1,6)} b. {(₋2,₋1),(₋1,0),(0,1),(1,0)} d. {(₋2,1),(₋1,0),(0,1),(1,4)}

____ 19. You buy a used truck for $22,000. It depreciates at the rate of 10% per year. What is the value of the truck after 4 years?

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____ 20. What kind of function best describes the following graph?

a. An absolute value function b. A cube root function c. A square root function d. A step function

____ 21. Compare the graph of g(x)=x2+6 with the graph of f(x)₌x2. a. The graph of g(x) is wider.

b. The graph of g(x) is narrower.

c. The graph of g(x) is translated 6 units down from the graph of f(x). d. The graph of g(x) is translated 6 units up from the graph of f(x).

____ 22. Eliud has 65 water bottles for the 5K race participants. In addition, he is planning to buy some

boxes of water bottles that have 14 water bottles each. All of the boxes cost the same. Eliud is not sure yet about the number of boxes of water bottles he wants to buy, but he has enough money to buy up to 5 of them. Write a function to describe how many water bottles Eliud can buy. Let x represent the number of

boxes of water bottles Eliud buys. Find a reasonable domain and range for the function. a. f(x)=65x+14; D: {5}; R: {135}

b. f(x)=14x+65; D: {0, 1, 2, 3, 4}; R: {65, 79, 93, 107, 121} c. f(x)=65x+14; D: {1, 2, 3, 4}; R: {79, 93, 107, 121, 135}

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____ 23. The growth of a population of bacteria can be modeled by an exponential function. The graph models the population of the bacteria colony P t( ) as a function of the time t, in weeks, that has passed. The initial

population of the bacteria colony was 500. What is the domain of the function? What does the domain represent in this context?

a. The domain is the real numbers greater than 500. The domain represents the time, in weeks, that has passed.

b. The domain is the real numbers greater than 500. The domain represents the population of the colony after a given number of weeks.

c. The domain is the nonnegative real numbers. The domain represents the time, in weeks, that has passed.

d. The domain is the nonnegative real numbers. The domain represents the population of the colony after a given number of weeks.

____ 24. Which is the average rate of change over the interval [0, 4]?

Equation A

Equation B

f(x)₌2x₋1

a. A: 4, B: 2 c. A: 8, B: 16

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____ 25. Which is the average rate of change over the interval [2,3]?

Equation A

Equation B

f(x)= −x+2

a. A: 7, B: 1 c. A: 3, B:₋1

b. A: 7, B:₋1 d. A: ₋1, B: 7

____ 26. The table shows the height of a sassafras tree at each of two ages. What was the tree’s average rate of growth during this time period?

Age (years) Height (meters)

4 2

10 5

a. 0.4 meter per year b. 0.5 meter per year c. 2 meters per year d. 2.5 meters per year

____ 27. Which is the inverse of y =x−3?

a. y₌ 1

x₋3 c. y= −x+3

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____ 28. A movie theater charges $8.50 for an adult ticket to an evening showing of a popular movie. To help the local animal shelter, the theater management has agreed to reduce the price of each adult ticket by $0.50 for every can of pet food a customer contributes to a collection barrel in the theater lobby. Which of the following shows both an equation in which y represents the cost of an adult ticket in dollars for a customer who

contributes x cans of pet food, and the graph of the cost if a customer brings in 2, 5, 8, or 10 cans of pet food?

a. y= 8.5−0.50x c. y= 8.5+0.50x

b. y ₌ 9x₋0.5 d. y_{= −}9x₋0.5

____ 29. What must be done to the graph of f(x)= | |x to obtain the graph of the function g(x)= 3

7|x+4| −10?

a. The graph of f(x) is shifted left 4 units, horizontally shrunk by a factor of 3

7, and shifted down 10 units.

b. The graph of f(x) is shifted right 4 units, vertically shrunk by a factor of 3

c. The graph of f(x) is shifted left 4 units, vertically shrunk by a factor of 3

d. The graph of f(x) is shifted left 4 units, vertically shrunk by a factor of 3

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____ 30. Solve (13x₋7)2= 110.

a. −7− 110

26 ,

−7+ 110

26 c.

−7− 110

13 ,

−7+ 110

13

b. 7− 110

13 ,

7+ 110

13 d.

7− 110

26 ,

7+ 110 26

____ 31. Find the standard form of the equation for the conic section represented by x2+y2−10x+6y=47.

a. Ê_ËÁÁy−3ˆ_¯˜˜2−(x+5)2 =9 c.

y+3 Ê

ËÁÁ ˆ¯˜˜2

9 +

x−5 ( )2

1 =1

b. Ê_ËÁÁy+3ˆ_¯˜˜2+(x−5)2 =81 d.

y+3 Ê

ËÁÁ ˆ¯˜˜2

9 −

x−5 ( )2

9 =1

____ 32. Write the standard form of the equation of the circle with radius 7 and center at (0, 0).

a. x2+y2 = 7 c. x

2

14 +

x2

14 = 1

b. x2+y2 = 49 d. x2+y2 = 14

____ 33. Write the standard equation of a circle with its center at the origin and radius 3.

a. x2+y2 = 6 c. x2+y2 = 9

b. x 2

6 +

y2

6 = 1 d. x

2+

y2 = 3

____ 34. Find the first 5 terms of the sequence with a₁ ₌6 and a_n =2a_n₋₁−1 for n≥2.

a. 1, 2, 3, 4, 5 c. 6, 12, 24, 48, 96

b. 6, 7, 8, 9, 10 d. 6, 11, 21, 41, 81

____ 35. Write a recursive rule for the sequence.

a_n = −2+6n

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____ 36. Julio is training for a swimming race. The first part of his training schedule is shown. Is this training schedule an arithmetic sequence? Explain. If Julio’s training schedule starts on a Tuesday and he swims every two days, on which day will he swim for 2.95 miles?

Session 1 2 3 4 5 6

Swimming distance (mi) 0.25 0.55 0.85 1.15 1.55 1.85

a. Julio’s training schedule is an arithmetic sequence, because a constant increase of 0.3 occurs between the sessions. Julio will swim 2.95 miles on Thursday.

b. Julio’s training schedule is not an arithmetic sequence, because the increase between session numbers and corresponding distances is not the same. Julio will swim 2.95 miles on Thursday.

c. Julio’s training schedule is an arithmetic sequence, because a constant increase of 0.3 occurs between the sessions. Julio will swim 2.95 miles on Saturday.

d. Julio’s training schedule is an arithmetic sequence, because a constant increase of 0.3 occurs between the sessions. Julio will swim 2.95 miles on Monday.

____ 37. What is the 20th term in the following geometric sequence? –2, –6, –18, –54, –162, ...

a. 1,162,261,467 c. 2,324,522,934

b. –6,973,568,802 d. –2,324,522,934

____ 38. Find the first five terms of the sequence recursively defined as the function with a₁ = −10 and a_n =a_n₋₁−3. a. –10, –7, –4, –1, 2

b. –10, –13, –16, –19, –22 c. –13, –16, –19, –22, –25 d. –10, 30, –90, 270, –810

Multiple Response

Identify one or more choices that best complete the statement or answer the question.

____ 1. Determine which functions have a minimum value that is greater than zero.

a. f(x)=x2−_6x+₅ b. f(x)=x2+_4x+₇ c. f(t)=t2+_8t−₁₀ d. f(n)=n2+10n+11 e. f(p)=p2−_2p+₈

____ 2. Identify the quadratic equations below that have non-real solutions.

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____ 3. Which of the following statements correctly describe the graph of f(x)=2x2+_8x−_2?

a. The maximum value of the function is 10. b. The minimum value of the function is ₋10. c. The axis of symmetry is the line x= −2. d. The axis of symmetry is the line x₌2. e. The graph is a parabola that opens up. f. The graph is a parabola that opens down.

____ 4. If f(x)= −1

8x+5, which of the following statements about g(x), the inverse of f(x), are true?

a. g(₋2.125)₌57 b. g(₋0.5)₌44 c. g(₋0.375)₌37 d. g(0.125)₌39 e. g(0.625)=45 f. g(1.125)₌40

____ 5. A person’s body mass index (BMI) is calculated by dividing the person’s mass in kilograms by the person’s height in meters. The table shows the median BMI for U.S. males from age 2 to age 12. For which intervals is the average rate of change in the median BMI positive?

Age (years) Median BMI

2 16.575 4 15.641 6 15.367 8 15.769 10 16.625 12 17.788

a. age 2 to age 4 b. age 4 to age 6 c. age 6 to age 8 d. age 8 to age 10 e. age 10 to age 12

____ 6. Which of the following equations, when rewritten in the form Ê_ËÁÁx−pˆ_¯˜˜2 =q, have a value of q that is a perfect

square?

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____ 7. Find all solutions of the equation x2

= −484.

a. x=22 b. x_{= −}22 c. x₌22i d. x = −22i

____ 8. Which points are on the parabola with focus (0, 4) and directrix y= −4?

a. (₋8, 4) b. (0,₋4) c. (0, 4) d. (4, 1) e. (9, 12) f. (12, 9)

Numeric Response

1. Find the positive root of the equation 4x2+11x=20.

2. What value of d makes the equation −2+3i+12i=9i−(2−di) true?

Matching

Factor each quadratic function and match it with the correct description of its zeros.

a. One positive zero d. Two positive zeros

b. One negative zero e. Two negative zeros

c. One positive zero and one negative zero, where the positive zero has the larger absolute value

f. One positive zero and one negative zero, where the negative zero has the larger absolute value

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Match each interval with the corresponding rate of change for the function f x( ) graphed below.

a. ₋3 e. 1

b. ₋2 f. 2

c. ₋1 g. 3

d. 0 h. 4

____ 2. From x= −3 to x= −2

____ 3. From x_{= −}2 to x₌1

____ 4. From x₌1 to x₌2

____ 5. From x_{= −}1 to x₌0

Match each number with its equivalent form.

a. ₋8 e. ₋2i

b. −2 f. 8i

c. −2 2 g. 2i 2

d. −20 2 h. 20i 2

____ 6. −82

____ 7. 3 −8

____ 8. −8

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Match each equation with the description of the circle it represents.

a. center: (₋7, 2); radius 3 e. center: (₋4,₋5); radius 4 b. center: (₋7,₋2); radius 3 f. center: (4, 5); radius 2 c. center: (₋2, 7); radius 3 g. center: (5,₋4); radius 2 d. center: (2,₋7); radius 3 h. center: (5, 4); radius 2

____ 10. (x−4)2+_(y−₅₎2 =₄

____ 11. (x+7)2+_(y−₂₎2 =₉

____ 12. x2−_10x+_y2−_8y= −₃₉

Short Answer

1. Graph y−3x <5 3x−y≥7 Ï

Ì Ó ÔÔÔÔÔ

ÔÔÔÔÔ .

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2. For a certain model of car, the average gas mileage m, in miles per gallon, is modeled by the function

m(s)= −0.04s2+_5.2s−_{134, where s is the speed of the car in miles per hour. Graph the function. What is the} maximum gas mileage? Explain your answer.

3. Write 12h2−31h+20 as the product of two factors. What are the zeros of the related function?

4. Write the quadratic function c x( ) =x2 −8x+26 in vertex form. What is the equation of the axis of symmetry of the graph?

5. At a convenience store, bottles of water cost $1.20 each. The function f x( ) ₌1.2x gives the cost of buying x bottles. Graph the function and give its reasonable domain and range in this context.

6. Write an equation for the inverse of y = 18x – 7.

7. Find the inverse function.

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8. Find the inverses of f(x)₌3x₋9 and h(x)= 1

3x+3. Show your work. What do you notice?

9. Solve each equation over the set of complex numbers. Write the solutions using the imaginary unit i as

necessary.

a. x2−₁₆=₀ b. x2+₃₆=₀ c. x2−₁₇=₀ d. x2+₅=₀

10. A parabola has focus (0,₋2) and directrix y=2. a. Write the equation of the parabola.

b. Graph the parabola, the focus, and the directrix below.

c. The vertex of a parabola is the midpoint of the perpendicular segment from the focus to the directrix. Use the graph from part b to find the vertex of the parabola. Explain your answer.

11. Write a recursive rule for the sequence. −6, 1, 8, 15, . . .

12. For a science experiment, Abdullah measures a bean stalk as it grows. In the first 4 weeks, he measures the stalk as 3, 6, 9, and 12 centimeters. Write an explicit rule for the height of the bean stalk in centimeters after

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Problem

1. Martha incorrectly graphed the function g x( ) = −1

4(x−4)

2+

3 using transformations of the graph of

f x( ) ₌x2

. First she stated what transformations to perform, and then she drew the graph. Identify and correct any errors.

(1) A vertical shrink by a factor of 1 4 (2) A shift left 4 units

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2. Consider the circle in the diagram below.

a. What are the coordinates of point A?

b. Write expressions for CA and PA.

c. Use the Pythagorean Theorem and your answers from part b to write a relationship between the side lengths of CAP that uses the variables x, y, h, k , and r.

Essay

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2. Use this data about engine size in liters x and highway fuel economy in miles per gallon y for several 2005 automobiles.

Discuss with your partner what a good trend line for these data would look like. Using a ruler, add a trend line to the scatter plot. Extend the line to intersect the y-axis.

Estimate the slope of your line. Explain how you did this.

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M3Sem1Rev

Answer Section

MULTIPLE CHOICE

1. ANS: A PTS: 1 DIF: DOK 2

2. ANS: B

The vertex is Ê_ËÁÁ3, −5ˆ_¯˜˜. It is a minimum.

Feedback

A This is the point where the function intersects the y-axis. B That’s correct!

C This is one of the points where the function intersects the x-axis. D This is one of the points where the function intersects the x-axis.

PTS: 1 DIF: DOK 1 NAT: F-IF.C.7b* STA: F-IF.7b* KEY: absolute value function | graph of a function | function | vertex

3. ANS: B PTS: 1 DIF: Level B REF: MAL20308

TOP: Lesson 2.7 Use Absolute Value Functions and Transformations

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4. ANS: B

The website went live when t=0, so the a t( )-intercept will give the number of articles. The a t( )-intercept is 30. There were 30 articles in the encyclopedia when it went live.

Feedback

A The encyclopedia did not have 0 articles when it went live. B That’s correct!

C The encyclopedia did not have 60 articles when it went live. D The encyclopedia did not have 180 articles when it went live.

PTS: 1 DIF: DOK 1 NAT: F-IF.C.7e* | MP.4

STA: F-IF.7e* | MP.4

KEY: intercepts | exponential functions | graph of a function | function | modeling

5. ANS: A PTS: 1 DIF: DOK 2 NAT: F-LE.A.2 | A-CED.A.1

STA: F-LE.2 | A-CED.1 6. ANS: C

Polynomials are closed under addition, so the sum of two quadratic polynomials will be a polynomial.

Feedback

A Whether this is true or not depends on the values of a, b, c, m, n, and p.

B Neither addend contains an exponent of x, so the sum will not contain an exponent of x. C That’s correct!

D It is possible to give a general description of this sum with the given information.

PTS: 1 DIF: DOK 1 NAT: A-APR.A.1 STA: A-APR.1

KEY: polynomial operations | quadratic polynomials

7. ANS: A PTS: 1 DIF: DOK 1 OBJ: Special Products of Binomials

NAT: A-APR.A.1 STA: A-APR.1 LOC: MTH.C.10.05.08.03.01.003 TOP: Multiplying Binomials

KEY: polynomial | binomial | multiplication | special product | FOIL

8. ANS: A PTS: 1 DIF: DOK 1 NAT: F-BF.B.3

STA: F-BF.3

9. ANS: C PTS: 1 DIF: DOK 1 NAT: F-BF.B.3

STA: F-BF.3

10. ANS: A PTS: 1 DIF: DOK 1 NAT: F-BF.B.3

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11. ANS: D

The graph opens upward. The vertex is (3,₋5), so the minimum is –5. The range begins at the minimum value.

Feedback

A The range begins at the minimum or maximum value. B The domain is all the x-values. The range is all the y-values.

C You may not be able to see the entire graph, but that does not mean the graph stops. D Correct!

PTS: 1 DIF: Average REF: 120c65f2-4683-11df-9c7d-001185f0d2ea OBJ: 8-1.5 Finding Domain and Range LOC: MTH.C.10.07.06.003 | MTH.C.10.07.06.004 TOP: 8-1 Identifying Quadratic Functions KEY: quadratic

MSC: DOK 2 12. ANS: A

2x2−₁₅=_x

2x2−_x−₁₅=₀ 2x+5

( )(x−3)=0

Feedback A That’s correct!

B This is the factored form of the expression 2x2−_11x+_{15, not 2x}2−_x−_15. C This is the factored form of the expression 2x2+_11x+_{15, not 2x}2−_x−_15. D This is the factored form of the expression 2x2+_x−_{15, not 2x}2−_x−_15.

PTS: 1 DIF: DOK 1 NAT: A-REI.B.4b STA: A-REI.4b

KEY: solving quadratic equations | factoring

13. ANS: D PTS: 1 DIF: DOK 2 NAT: A-REI.B.4b

STA: A-REI.4b

14. ANS: D PTS: 1 DIF: DOK 2

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15. ANS: B 2x2+_3x−₄=₀

x= −3± 3

2−_{4 2}( )( )−₄ 2 2( )

= −3± 9₄− −( 32)

= −3±₄ 41

Feedback

A Do not forget that the b-term in the numerator of the quadratic formula is negative.

B That’s correct!

C It seems you multiplied b by 2 instead of squaring it.

D It seems you forgot the negative sign in front of the c-value, ₋4, when multiplying underneath the square root.

KEY: quadratic equations | quadratic formula

16. ANS: B PTS: 1 DIF: DOK 1 NAT: F-IF.C.8a

STA: F-IF.8a

17. ANS: B PTS: 1 DIF: DOK 1 NAT: F-IF.C.8a

STA: F-IF.8a

18. ANS: C PTS: 1 DIF: DOK 1 NAT: F-LE.A.1

STA: F-LE.1

19. ANS: B PTS: 1 DIF: DOK 2 NAT: F-LE.A.2 | F-LE.A.1c

STA: F-LE.2 | F-LE.1.c LOC: NCTM.PSSM.00.MTH.9-12.ALG.1.c KEY: exponential decay

20. ANS: B

The domain and range of the function are the real numbers. Since the domain and range are not bounded, the function is neither an absolute value function nor a square root function. The function is not a step function because it does not contain any breaks and does not have any constant pieces. Thus, the function is a cube root function.

Feedback

A An absolute value function is a piecewise defined function with two linear pieces that meet at a vertex. Its range is always bounded on one side.

B That’s correct!

C The domain of a square root function is always bounded on one side.

D A step function is a piecewise defined function with a finite number of linear constant pieces that contains breaks.

PTS: 1 DIF: DOK 1 NAT: F-IF.C.7b* STA: F-IF.7b* KEY: cube root function | function | graph of a function

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OBJ: Finding Reasonable Domain and Range of a Function NAT: F-IF.B.5 STA: F-IF.5 LOC: MTH.C.10.07.01.005 | MTH.C.10.07.01.007

TOP: Writing Functions KEY: function | domain | range 23. ANS: C

The domain is the set of inputs of the function. Thus, the domain is the set of t-values. In the graph, it is

shown that t can take on any nonnegative real value. Thus, the domain is the nonnegative real numbers. The

domain represents the amount of time that has passed since the population was initially measured.

Feedback

A The domain is the set of inputs of the function. B The domain is the set of inputs of the function. C That’s correct!

D The domain is the set of inputs of the function. Notice that the population is being modeled as a function of time.

PTS: 1 DIF: DOK 2 NAT: F-IF.B.5* | MP.4

STA: F-IF.5* | MP.4 KEY: function | domain | graph of a function | modeling

24. ANS: A PTS: 1 DIF: DOK 2 NAT: F-IF.B.6

STA: F-IF.6

25. ANS: B PTS: 1 DIF: DOK 2 NAT: F-IF.B.6

STA: F-IF.6 26. ANS: B

average rate of growth= change in height_{change in age} = ₁₀5−₋2₄ = 3₆ =0.5 meter per year

Feedback

A You need to find the rate of change in height to change in age. B That’s correct!

C You found the rate of change in age to change in height. You need to find the rate of change in height to change in age.

D You need to find the rate of change in height to change in age.

STA: F-IF.6* | MP.4 KEY: average rate of change from a table | modeling NOT: Data taken from height-versus-age graph of sassafras tree number 2 at

http://www.yale.edu/fes519b/totoket/allom/allom.htm

27. ANS: D PTS: 1 DIF: DOK 1 NAT: F-BF.B.4

STA: F-BF.4

28. ANS: A PTS: 1 DIF: DOK 2 NAT: A-CED.A.2 | F-BF.A.1a | F-LE.A.2

STA: A-CED.2 | F-BF.1a | F-LE.2

LOC: NCTM.PSSM.00.MTH.9-12.ALG.3.a | NCTM.PSSM.00.MTH.9-12.DAP.1.d | NCTM.PSSM.00.MTH.9-12.DAP.2.b | NCTM.PSSM.00.MTH.9-12.DAP.2.e

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29. ANS: C

Follow the order of operations to apply the transformations. First, notice that 4 is being added to x inside the

absolute value bars. So, the graph of f(x) is shifted left 4 units. Now, notice that the absolute value

expression is being multiplied by 3

7. So, the graph of f(x) is being vertically shrunk by a factor of 3

7. Finally, 10 is being subtracted from the first term of f(x). So, the graph of f(x) is being shifted down 10 units.

Feedback

A Recall that a horizontal shrink occurs when x is multiplied by a constant k , where

0<k<1, before any horizontal shifts occur.

B In horizontal shifts of the form f(x+k), where k is a constant, the graph is moved in

the opposite direction of the sign of k .

C That’s correct!

D In vertical shifts of the form f(x)+k , where k is a constant, the graph is moved in the

same direction of the sign of k .

PTS: 1 DIF: DOK 1 NAT: F-BF.B.3 STA: F-BF.3

KEY: absolute value function | vertical stretch | horizontal shifts | vertical shifts | transformations

30. ANS: B PTS: 1 DIF: DOK 2 NAT: A-REI.B.4a

STA: A-REI.4a TOP: Complete the Square KEY: square | solve | complex | quadratic

31. ANS: B PTS: 1 DIF: DOK 2 NAT: G-GPE.A.1

STA: G-GPE.1

32. ANS: B PTS: 1 DIF: DOK 2 NAT: G-GPE.A.1

STA: G-GPE.1 TOP: Graph and Write Equations of Circles KEY: equation | circle | radius

33. ANS: C PTS: 1 DIF: DOK 1 NAT: G-GPE.A.1

STA: G-GPE.1 TOP: Write and Graph Equations of Circles KEY: equation | circle | radius

OBJ: Finding Terms of a Sequence by Using a Recursive Formula

NAT: F-BF.A.2 STA: F-BF.2 LOC: MTH.C.13.06.01.01.003 TOP: Introduction to Sequences

35. ANS: C PTS: 1 DIF: DOK 2 NAT: F-BF.A.1a | F-BF.A.2

STA: F-BF.1a | F-BF.2

TOP: Translate Between Recursive and Explicit Rules for Sequences KEY: sequence | recursive rule

36. ANS: C PTS: 1 DIF: DOK 2 OBJ: Application

NAT: F-BF.A.2 STA: F-BF.2

LOC: MTH.P.06.006 | MTH.C.13.06.01.01.01.003 | MTH.C.13.06.01.01.004 | MTH.C.13.06.01.01.007 TOP: Terms of Arithmetic Sequences KEY: arithmetic sequence

37. ANS: D PTS: 1 DIF: DOK 1 NAT: F-BF.A.2

STA: F-BF.2 TOP: Relate Geometric Sequences to Exponential Functions KEY: geometric sequence

38. ANS: B PTS: 1 DIF: DOK 1 NAT: F-IF.A.3

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MULTIPLE RESPONSE 1. ANS: B, E

A: x2−_6x+₅=₀

x2−_6x+₉−₄=₀

(x−3)2−₄=₀

The minimum of this function is ₋4.

B: x2+4x+7=0

x2+_4x+₄+₃=₀

(x+2)2+₃=₀

The minimum of this function is 3.

C: t2+_8t−₁₀=₀

t2+_8t+₁₆−₂₆=₀

(t+4)2−₂₆=₀

D: n2+10n+11=0

n2+_10n+₂₅−₁₄=₀

(n+5)2−₁₄=₀

E: p2−_2p+₈=₀

p2−_2p+₁+₇=₀

(p−1)2+₇=₀

The minimum of this function is 7.

Feedback

Correct That’s correct!

Incorrect Make sure you have completed the square properly.

PTS: 2 DIF: DOK 1 NAT: A-SSE.B.3b*

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2. ANS: C, D

After putting the equation in standard form, use the discriminant to determine whether each equation has real solutions or non-real solutions.

A: x2+_3x−₂₅= −₇

x2+_3x−₁₈=₀

b2−_4ac=₃2−_{4 1}( )(−₁₈) =9+72

=81

Since the discriminant is not negative, the equation x2 +_3x−₂₅= −_{7 has real solutions.}

B: −x2+_7x+₁=₁₃ −x2+_7x−₁₂=₀

b2 −_4ac=₇2−₄( )−₁ (−₁₂) =49+48

=97

Since the discriminant is not negative, the equation −x2 +_7x+₁=_{13 has real solutions.}

C: x2+2x= −5

x2+_2x+₅=₀

b2−_4ac=₂2−_{4 1}( )( )₅ =4−20 = −16

Since the discriminant is negative, the equation x2+_2x= −_{5 has non-real solutions.}

D: 2x2+_x+₁₃=₀

b2 −_4ac=₁2−_{4 2}( )( )₁₃ =1−104 = −103

Since the discriminant is negative, the equation 2x2+_x+₁₃=_{0 has non-real solutions.}

E: −2x2+_4x+₉=₁₁ −2x2+_4x−₂=₀

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Since the discriminant is not negative, the equation −2x2+_4x+₉=_{11 has real solutions.}

Feedback Correct That’s correct!

Incorrect Use the discriminant to determine if each equation has real solutions or non-real solutions.

KEY: solving quadratic equations | using the discriminant of the quadratic formula 3. ANS: B, C, E

A, B: Complete the square:

f(x)=2x2+_8x−₂ =2 xÊ 2+_4x

Ë

ÁÁ ˆ_¯˜˜ −2 =2 xÊ 2+_4x+₄−₄

Ë

ÁÁ ˆ_¯˜˜ −2 =2 x( +2)2−10

The vertex of the parabola is Ê_ËÁÁ−2,−10ˆ_¯˜˜. Since the coefficient of x2_{is positive, the vertex is a} minimum value, and the function has no maximum value.

C, D: The axis of symmetry is a vertical line passing through the vertex of the parabola. The vertex of the parabola is Ê_ËÁÁ−2,−10ˆ_¯˜˜, so the axis of symmetry is x_{= −}2.

E, F: Since the coefficient of x2_{is positive, the parabola opens up.}

Feedback

Incorrect Complete the square to rewrite the equation and reveal the properties of the graph.

PTS: 2 DIF: DOK 2 NAT: F-IF.C.8a STA: F-IF.8a

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4. ANS: A, B, D

The inverse is g(x)= −8x+40.

A: g(−2.125)= −8(−2.125)+40 =17+40

=57

B: g(−0.5)= −8(−0.5)+40 =4+40 =44

C: g(−0.375)= −8(−0.375)+40 =3+40

=43 43≠37

D: g(0.125)= −8(0.125)+40 = −1+40 =39

E: g(0.625)= −8(0.625)+40 = −5+40 =35 35≠45

F: g(1.125)= −8(1.125)+40 = −9+40 =31 31≠40

Feedback

Incorrect Find g(x), the inverse of f(x). Then substitute each given x-value into g(x) and

check if the corresponding given g(x)-value is correct.

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5. ANS: C, D, E

As the ages given in the table increase by 2, the median BMIs decrease up to age 6 and then increase after age 6. Only when the median BMIs increase will the average rate of change in the median BMI be positive. So, the average rate of change is positive from age 6 to age 8, from age 8 to age 10, and from age 10 to age 12.

Incorrect Look for intervals where the median BMI is increasing.

STA: F-IF.6* | MP.4 KEY: average rate of change from a table | modeling NOT: Data taken from CDC's “Body Mass Index for Age Tables” at

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6. ANS: A, C, D A: x2−_2x+₅=₁₃

x2−_2x=₈

x2−_2x+₁=₉

x−1 ( )2 =9

B: x2+_8x+₉=₅

x2 +_8x= −₄

x2+_8x+₁₆=₁₂

x+4 ( )2 =12

C: 2x2+12x−29=81

2x2+_12x=₁₁₀

x2+_6x=₅₅

x2 +_6x+₉=₆₄

x+3 ( )2 =64

D: 5x2−_20x+₁₄= −₆

5x2−_20x= −₂₀

x2−4x= −4

x2 −_4x+₄=₀

x−2 ( )2 =0

E: 3x2+36x+88=4

3x2+_36x= −₈₄

x2+_12x= −₂₈

x2+_12x+₃₆=₈

x+6 ( )2 =8

Since 9, 64, and 0 are all perfect squares, A, C, and D meet the criteria. Since 12 and 8 are not perfect squares, B and E do not meet the criteria.

Incorrect Complete the square to rewrite each equation in the desired form.

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7. ANS: C, D

x2 = −₄₈₄

x= ± −484

x= ±22i

Feedback

Incorrect Solve the equation x2 ₌a using the definition of square root, which allows you

to rewrite the equation as x = ± a .

PTS: 2 DIF: DOK 1 NAT: N-CN.A.1 STA: N-CN.1

KEY: imaginary numbers 8. ANS: A, D, F

The formula for the parabola with focus (0, 4) and directrix y= −4 is y= ₁₆1 x2. Check each point.

A: 1

16(−8) 2

= ₁₆1 (64)₌4

B: 1

16(0)

2 = 1

16(0)=0≠ −4

C: 1

16(0)

2 = 1

16(0)=0≠ 4

D: 1

16(4) 2

= ₁₆1 (16)₌1

E: 1

16(9)

2 = 1

16(81)=5 1 16 ≠12

F: 1

16(12) 2

= ₁₆1 (144)₌9

So the points (₋8, 4), (4, 1), and (12, 9) lie on the parabola with focus (0, 4) and directrix y_{= −}4.

Incorrect Write the formula for the parabola with the given focus and directrix. Check each point by substituting.

PTS: 2 DIF: DOK 1 NAT: G-GPE.A.2 STA: G-GPE.2

KEY: parabola | focus | directrix

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2. ANS: 6

PTS: 1 DIF: DOK 2 NAT: N-CN.A.2 STA: N-CN.2

TOP: Complex Numbers and Roots

MATCHING

1. ANS: C PTS: 1 DIF: DOK 1 NAT: A-SSE.B.3a*

STA: A-SSE.3a* KEY: factoring | quadratic functions

2. ANS: F PTS: 1 DIF: DOK 1 NAT: F-IF.B.6*

STA: F-IF.6* KEY: average rates of change from a graph | polynomial functions

3. ANS: B PTS: 1 DIF: DOK 1 NAT: F-IF.B.6*

4. ANS: H PTS: 1 DIF: DOK 1 NAT: F-IF.B.6*

5. ANS: A PTS: 1 DIF: DOK 1 NAT: F-IF.B.6*

6. ANS: F PTS: 1 DIF: DOK 1 NAT: N-CN.A.1

STA: N-CN.1 KEY: imaginary numbers

7. ANS: B PTS: 1 DIF: DOK 1 NAT: N-CN.A.1

8. ANS: G PTS: 1 DIF: DOK 1 NAT: N-CN.A.1

9. ANS: H PTS: 1 DIF: DOK 1 NAT: N-CN.A.1

10. ANS: F PTS: 1 DIF: DOK 2 NAT: G-GPE.A.1

STA: G-GPE.1 KEY: equation of a circle

11. ANS: A PTS: 1 DIF: DOK 2 NAT: G-GPE.A.1

STA: G-GPE.1 KEY: equation of a circle

12. ANS: H PTS: 1 DIF: DOK 2 NAT: G-GPE.A.1

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SHORT ANSWER 1. ANS:

Sample solutions: (5, 0), (7,₋5) Sample non-solutions: (0, 0), (₋5,1)

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2. ANS:

The maximum gas mileage is 35 miles per gallon. The vertex of the parabola is (65, 35). Since the parabola opens down, the vertex is a maximum. The vertex indicates that the car should drive 65 miles per hour to maximize the gas mileage at 35 miles per gallon.

Rubric

1 point for the graph; 1 point for the answer; 2 points for the explanation

PTS: 4 DIF: DOK 2 NAT: F-IF.C.7a* | F-IF.B.4* | MP.4 STA: F-IF.7a* | F-IF.4* | MP.4

KEY: graph of a function | function | quadratic function | vertex | maximum 3. ANS:

4h−5

( )(3h−4);5 4 and

4 3

PTS: 1 DIF: DOK 2 NAT: A-SSE.B.3a STA: A-SSE.3a

TOP: Solve ax^2 + bx + c = 0 by Factoring KEY: factor | trinomial 4. ANS:

c x( ) =(x−4)2+10; x₌4

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5. ANS:

domain: {0, 1, 2, 3, 4, 5, …}; range: {0, 1.2, 2.4, 3.6, 4.8, 6.0,…}

PTS: 1 DIF: DOK 1 NAT: F-IF.B.5 STA: F-IF.5

6. ANS:

y= x+7

18

TOP: Use Inverse Functions KEY: inverse relation 7. ANS:

f−1( )x ₌x₋9

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8. ANS:

f(x)=3x−9

y=3x−9

y+9=3x

y+9 3 =x 1

3y+3=x 1

3x+3=y

g(x)= 1

3x+3

h(x)= 1

3x+3

y= 1

3x+3

y−3= 1 3x 3(y−3)=x

3y−9=x

3x−9=y j(x)=3x−9

The inverse of f(x) is h(x), and the inverse of h(x) is f(x). (Stating that f(x) and h(x) are inverses is also

acceptable.)

Rubric

1 point for finding inverse of f(x); 0.5 point for work;

1 point for finding inverse of h(x); 0.5 point for work;

1 point for stating that the inverse of f(x) is h(x) and the inverse of h(x) is f(x)

KEY: linear functions | inverse functions 9. ANS:

a. x= ±4 b. x= ±6i c. x= ± 17 d. x= ±i 5

Rubric

0.5 point for each correct answer

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10. ANS:

a. y= −1₈x2

b.

c. The segment from the focus (0,₋2) to the point (0, 2) has length 4. The midpoint of the segment is the origin, (0, 0). Therefore, the vertex is the origin.

Rubric a. 1 point

b. 1 point for parabola; 0.5 point for focus; 0.5 point for directrix c. 1 point for answer; 1 point for explanation

KEY: parabola | focus | directrix | vertex 11. ANS:

t₁ = −6; t_n =t_n₋₁+7

PTS: 1 DIF: DOK 2 NAT: F-BF.A.1a | F-BF.A.2 STA: F-BF.1a | F-BF.2

LOC: NCTM.PSSM.00.MTH.9-12.ALG.1.a | NCTM.PSSM.00.MTH.9-12.ALG.2.d |

NCTM.PSSM.00.MTH.9-12.ALG.3.b TOP: Use Recursive Rules with Sequences and Functions KEY: sequence | recursive | recursion

12. ANS:

a_n ₌3n

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PROBLEM 1. ANS:

Martha’s interpretation of the horizontal shift is incorrect. Since 4 is being subtracted from x, the parent

function will shift right 4 units.

Martha did not notice that the coefficient of (x−4)2_{is negative. This reflects the parent function about the}

x-axis.

Rubric

1 point recognizing Martha forgot the reflection;

1 point for recognizing the error in Martha’s horizontal shift statement; 1 point for correcting it;

2 points for the correct graph

PTS: 5 DIF: DOK 2 NAT: F-BF.B.3 | MP.3

STA: F-BF.3 | MP.3

2. ANS:

a. The coordinates of point A are (x, k). b. |x−h|; ||y−k||

c. CA2+_PA2 =_CP2

(x−h)2+_(y−_k)2 =_r2

Rubric a. 1 point

b. 1 point for each expression

c. 3 points for writing the equation and arriving at the equation of a circle

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ESSAY

1. ANS:

Sample answer: For the roots to be nonreal, the discriminant, b2 ₋4ac, must be negative, or b2−4ac<0. So,

b2 <4ac, or ac> b 2

4 . This means that when the product ac is greater than one fourth the square of b, the solutions will be nonreal, complex numbers.

KEY: discriminant 2. ANS:

−5; Pick two grid points that the line passes through and use the slope formula.

PTS: 1 DIF: DOK 3 NAT: S-ID.B.6 | F-IF.B.6 STA: S-ID.6 | F-IF.6

3. ANS:

x=1± 6 or x≈ −1.45 and x≈3.45; The solutions of the equation are the x-intercepts of the parabola shown in the graph below. From the graph, you can see that the x-intercepts appear to be consistent with the solutions found using the quadratic formula.

PTS: 1 DIF: DOK 2 NAT: A-REI.B.4b | F-IF.C.7a

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_______ 1.A

_______ 2.B

_______ 3.B

_______ 4.B

_______ 5.A

_______ 6.C

_______ 7.A

_______ 8.A

_______ 9.C

_______ 10.A

_______ 11.D

_______ 12.A

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_______ 14.D

_______ 15.B

_______ 16.B

_______ 17.B

_______ 18.C

_______ 19.B

_______ 20.B

_______ 21.D

_______ 22.D

_______ 23.C

_______ 24.A

_______ 25.B

_______ 26.B

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_______ 28.A

_______ 29.C

_______ 30.B

_______ 31.B

_______ 32.B

_______ 33.C

_______ 34.D

_______ 35.C

_______ 36.C

_______ 37.D

_______ 38.B

_______ 1.B, E

_______ 2.C, D

_______ 3.B, C, E

_______ 4.A, B, D

_______ 5.C, D, E

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_______ 7.C, D

_______ 8.A, D, F

_______ 1.C

_______ 2.F

_______ 3.B

_______ 4.H

_______ 5.A

_______ 6.F

_______ 7.B

_______ 8.G

_______ 9.H

_______ 10.F

_______ 11.A