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Algebra

Algebra Help: Write Explicit Or Recursive Functions

Review real example questions for Write Explicit Or Recursive Functions in Algebra.

Question 1

A ball is thrown upward from a platform 4 feet high with an initial upward velocity of 24 ft/s. Its height (in feet) after ttt seconds is modeled by a quadratic function with gravity −16t2-16t^2−16t2. What is the formula for the height function h(t)h(t)h(t)?

  1. h(t)=16t2+24t+4h(t)=16t^2+24t+4h(t)=16t2+24t+4
  2. h(t)=−16t2+24t+4h(t)=-16t^2+24t+4h(t)=−16t2+24t+4
  3. h(t)=−16t2+4t+24h(t)=-16t^2+4t+24h(t)=−16t2+4t+24
  4. h(t)=−24t2+16t+4h(t)=-24t^2+16t+4h(t)=−24t2+16t+4
Explanation: This question tests your ability to translate a real-world situation into mathematical language—either as an explicit formula, a recursive process, or a series of calculation steps. The key to writing functions from context is identifying what changes (independent variable, like time or number of items) and what you're calculating (dependent variable, like cost or height), then finding the mathematical relationship between them using clues in the language. This context involves motion with gravity ('ball is thrown upward'), which tells us this is a quadratic function. The standard form for height with gravity is h(t) = -16t² + v₀t + h₀, where v₀ is initial velocity and h₀ is initial height. From the context we extract: initial height = 4 feet ('from a platform 4 feet high'), initial velocity = 24 ft/s ('initial upward velocity of 24 ft/s'), giving us h(t) = -16t² + 24t + 4. Choice A is correct because it properly includes the gravity term (-16t²), the initial velocity term (24t), and the initial height (4), matching the standard physics formula for projectile motion. Choice B has positive 16t² instead of negative, which would mean the ball accelerates upward forever—that violates the laws of physics! Gravity always pulls down, so the t² term must be negative in height functions. Context translation cheat sheet: 'per/each' = multiply by that rate, 'plus/in addition' = add, 'starts at' = initial value, 'doubles/triples' = multiply by 2 or 3 (exponential), 'percent' = divide by 100 for decimal. These phrases are your clues for turning words into math!

Question 2

A phone plan costs \25eachmonthpluseach month pluseachmonthplus$0.05pertextmessagesent.Whatfunctionmodelsthemonthlycostper text message sent. What function models the monthly costpertextmessagesent.WhatfunctionmodelsthemonthlycostC(t)(indollars)intermsofthenumberoftexts(in dollars) in terms of the number of texts(indollars)intermsofthenumberoftextst$?

  1. C(t)=0.05(t+25)C(t)=0.05(t+25)C(t)=0.05(t+25)
  2. C(t)=25+0.05tC(t)=25+0.05tC(t)=25+0.05t
  3. C(t)=0.05+25tC(t)=0.05+25tC(t)=0.05+25t
  4. C(t)=25t+0.05C(t)=25t+0.05C(t)=25t+0.05
Explanation: This question tests your ability to translate a real-world situation into mathematical language—either as an explicit formula, a recursive process, or a series of calculation steps. An explicit expression gives you a direct formula to calculate the output from the input without needing any previous values: if the context says 'costs 45perdayplus45 per day plus 45perdayplus25 fee,' you can write C(d) = 45d + 25, where d is days and C is cost. Just plug in any number of days and calculate immediately! Looking at the context 'costs 25eachmonthplus25 each month plus 25eachmonthplus0.05 per text message,' we identify the independent variable as t (number of texts) and dependent variable as C (monthly cost). The relationship is linear because we have a fixed monthly cost plus a constant rate per text. The fixed monthly cost is 25from′costs25 from 'costs 25from′costs25 each month,' and the per-text rate is 0.05from′0.05 from '0.05from′0.05 per text message.' Putting this together: C(t) = 25 + 0.05t. This formula lets us calculate monthly cost for any number of texts! Choice B is correct because it properly identifies t as the variable for texts and includes both the fixed monthly cost (25)andtheper−textrate(25) and the per-text rate (25)andtheper−textrate(0.05) in the standard form: fixed cost + rate × variable. Choice C swaps the coefficients: 0.05 + 25t would mean 5 cents fixed plus $25 per text—that would be an incredibly expensive texting plan! Always match the numbers to their meanings: 'per' indicates the rate that multiplies the variable. To write explicit functions from context: (1) Underline key phrases like 'per,' 'starts at,' 'plus,' 'each'—these tell you operations and values, (2) Identify what changes (independent variable x, n, t) and what you're calculating (dependent variable C, h, P), (3) Determine function type: constant rate = linear, percent/doubling = exponential, area/motion = quadratic, (4) Extract numbers from context and plug into the right form (y = mx + b for linear, y = a·b^x for exponential, etc.).

Question 3

A student starts a savings challenge with 25onday1andadds25 on day 1 and adds 25onday1andadds4 each new day. Write a recursive definition for the amount saved ana_nan​ (in dollars) on day nnn.

  1. a1=25,  an+1=an+4a_1=25,\; a_{n+1}=a_n+4a1​=25,an+1​=an​+4
  2. a1=25,  an+1=4ana_1=25,\; a_{n+1}=4a_na1​=25,an+1​=4an​
  3. an=25+4na_n=25+4nan​=25+4n
  4. a1=4,  an+1=an+25a_1=4,\; a_{n+1}=a_n+25a1​=4,an+1​=an​+25
Explanation: This question tests your ability to translate a real-world situation into mathematical language—either as an explicit formula, a recursive process, or a series of calculation steps. A recursive definition describes how to get each term from the previous one, which is perfect for sequential processes: if you 'start with 200 and add 50 each week,' that becomes a₁ = 200, aₙ₊₁ = aₙ + 50. You need both the starting value and the rule for what comes next. The context describes a sequential process: 'starts a savings challenge with 25onday1andadds25 on day 1 and adds 25onday1andadds4 each new day.' We need a starting point—that's 25from′starts...with25 from 'starts...with 25from′starts...with25 on day 1'—and a rule for each step. Since we 'add 4eachnewday,′eachnewvalueisthepreviousvalueplus4.Writingthisasarecursivedefinition:a1=25,an+1=an+4.Tofindthe5thday′samount,we′dstartat25andapplytherulefourtimes!ChoiceAiscorrectbecauseitincludesboththeinitialvalue(a1=25)andtherecurrencerelation(an+1=an+4)thatmatchesthecontextofadding4 each new day,' each new value is the previous value plus 4. Writing this as a recursive definition: a₁ = 25, aₙ₊₁ = aₙ + 4. To find the 5th day's amount, we'd start at 25 and apply the rule four times! Choice A is correct because it includes both the initial value (a₁ = 25) and the recurrence relation (aₙ₊₁ = aₙ + 4) that matches the context of adding 4eachnewday,′eachnewvalueisthepreviousvalueplus4.Writingthisasarecursivedefinition:a1​=25,an+1​=an​+4.Tofindthe5thday′samount,we′dstartat25andapplytherulefourtimes!ChoiceAiscorrectbecauseitincludesboththeinitialvalue(a1​=25)andtherecurrencerelation(an+1​=an​+4)thatmatchesthecontextofadding4 each day. Choice B has the recursive rule backwards: it starts with 4andadds4 and adds 4andadds25 each time, but the context says we start with 25andadd25 and add 25andadd4. When writing recursive definitions, the starting value comes from 'starts with' or 'begins at,' not from the amount that changes! For recursive definitions, remember the two-part recipe: you MUST have both (1) the starting value(s)—look for 'starts at,' 'begins with,' 'initially'—and (2) the rule relating each term to the previous—look for 'add,' 'multiply by,' 'increases by' followed by a description. Write it as: a₁ = [starting value], aₙ₊₁ = [rule using aₙ]. If you're stuck choosing between explicit and recursive, ask: does each value only depend on which step you're at (explicit), or does each value depend on the previous value (recursive)? 'Day 5 costs 45′suggestsexplicit.′Add45' suggests explicit. 'Add 45′suggestsexplicit.′Add10 to yesterday's amount' suggests recursive. The language tells you which form fits!

Question 4

A student starts a savings jar with \40.Eachweek,thestudentadds. Each week, the student adds .Eachweek,thestudentadds$15morethanthepreviousweek’stotalamountinthejar.Writearecursivedefinitionfortheamountofmoneymore than the previous week’s total amount in the jar. Write a recursive definition for the amount of moneymorethanthepreviousweek’stotalamountinthejar.Writearecursivedefinitionfortheamountofmoneya_n(indollars)inthejarafter(in dollars) in the jar after(indollars)inthejarafternweeks,whereweeks, whereweeks,wherea_1$ is the amount after week 1.

  1. a1=40, an+1=an+15a_1=40,\ a_{n+1}=a_n+15a1​=40, an+1​=an​+15
  2. a1=15, an+1=an+40a_1=15,\ a_{n+1}=a_n+40a1​=15, an+1​=an​+40
  3. an=40+15na_n=40+15nan​=40+15n
  4. an+1=15n+40a_{n+1}=15n+40an+1​=15n+40
Explanation: This question tests your ability to translate a real-world situation into mathematical language—either as an explicit formula, a recursive process, or a series of calculation steps. A recursive definition describes how to get each term from the previous one, which is perfect for sequential processes: if you 'start with 200 and add 50 each week,' that becomes a₁ = 200, aₙ₊₁ = aₙ + 50. You need both the starting value and the rule for what comes next. The context describes a sequential process: 'starts with 40′and′adds40' and 'adds 40′and′adds15 more than the previous week's total.' We need a starting point—that's 40from′startswith40 from 'starts with 40from′startswith40'—and a rule for each step. Since we 'add 15′eachtime,eachnewvalueisthepreviousvalueplus15.Writingthisasarecursivedefinition:a1=40,an+1=an+15.Tofindthe5thweek′samount,we′dstartat40andadd15fourtimes!ChoiceAiscorrectbecauseitincludesboththeinitialvalue(a1=40)andtherecurrencerelation(an+1=an+15)thatmatchesadding15' each time, each new value is the previous value plus 15. Writing this as a recursive definition: a₁ = 40, aₙ₊₁ = aₙ + 15. To find the 5th week's amount, we'd start at 40 and add 15 four times! Choice A is correct because it includes both the initial value (a₁ = 40) and the recurrence relation (aₙ₊₁ = aₙ + 15) that matches adding 15′eachtime,eachnewvalueisthepreviousvalueplus15.Writingthisasarecursivedefinition:a1​=40,an+1​=an​+15.Tofindthe5thweek′samount,we′dstartat40andadd15fourtimes!ChoiceAiscorrectbecauseitincludesboththeinitialvalue(a1​=40)andtherecurrencerelation(an+1​=an​+15)thatmatchesadding15 each week to the previous total. Choice B reverses the values: it starts with 15andadds15 and adds 15andadds40 each time, which doesn't match the problem description at all. When you see 'starts with X and adds Y each time,' X is your initial value and Y is what you add in the recursive rule! For recursive definitions, remember the two-part recipe: you MUST have both (1) the starting value(s)—look for 'starts at,' 'begins with,' 'initially'—and (2) the rule relating each term to the previous—look for 'add,' 'multiply by,' 'increases by' followed by a description. Write it as: a₁ = [starting value], aₙ₊₁ = [rule using aₙ].

Question 5

A video game score starts at 80 points and increases by 25 points each level completed. Write an explicit formula for the score S(n)S(n)S(n) after completing nnn levels, where n=0n=0n=0 means no levels completed yet.

  1. S(n)=80(25)nS(n)=80(25)^nS(n)=80(25)n
  2. S(n)=25+80nS(n)=25+80nS(n)=25+80n
  3. S(n)=80n+25S(n)=80n+25S(n)=80n+25
  4. S(n)=80+25nS(n)=80+25nS(n)=80+25n
Explanation: This question tests your ability to translate a real-world situation into mathematical language—either as an explicit formula, a recursive process, or a series of calculation steps. An explicit expression gives you a direct formula to calculate the output from the input without needing any previous values: if the context says 'costs 45perdayplus45 per day plus 45perdayplus25 fee,' you can write C(d) = 45d + 25, where d is days and C is cost. Just plug in any number of days and calculate immediately! Looking at the context 'score starts at 80 points and increases by 25 points each level completed,' we identify the independent variable as n (number of levels completed) and dependent variable as S(n) (score). The relationship is linear because we have a starting score plus a constant increase per level. The initial score is 80 from 'starts at 80 points,' and the rate is 25 from 'increases by 25 points each level.' Putting this together: S(n) = 80 + 25n. This formula lets us calculate the score after any number of levels! Choice B is correct because it properly identifies 80 as the starting score (when n = 0, no levels completed) and 25n as the points gained from completing n levels, matching the context perfectly. Choice C has the same mathematical result but writes it as 25 + 80n, which reverses the roles: this would mean starting at 25 points and gaining 80 per level, but the context clearly states we start at 80 and gain 25 per level. The order matters for understanding! Context translation cheat sheet: 'per/each' = multiply by that rate, 'plus/in addition' = add, 'starts at' = initial value, 'doubles/triples' = multiply by 2 or 3 (exponential), 'percent' = divide by 100 for decimal. These phrases are your clues for turning words into math!

Question 6

A ball is thrown upward from a platform 6 feet high with an initial upward velocity of 40 ft/s. Its height in feet after ttt seconds is modeled by h(t)=−16t2+40t+6h(t)=-16t^2+40t+6h(t)=−16t2+40t+6. Which formula represents this height function?

  1. h(t)=−16t2−40t+6h(t)=-16t^2-40t+6h(t)=−16t2−40t+6
  2. h(t)=16t2+40t+6h(t)=16t^2+40t+6h(t)=16t2+40t+6
  3. h(t)=−16t2+40t+6h(t)=-16t^2+40t+6h(t)=−16t2+40t+6
  4. h(t)=−40t2+16t+6h(t)=-40t^2+16t+6h(t)=−40t2+16t+6
Explanation: This question tests your ability to translate a real-world situation into mathematical language—either as an explicit formula, a recursive process, or a series of calculation steps. The key to writing functions from context is identifying what changes (independent variable, like time or number of items) and what you're calculating (dependent variable, like cost or height), then finding the mathematical relationship between them using clues in the language. This context involves motion with gravity, which tells us this is a quadratic function. Quadratic functions model situations where acceleration is constant, like objects under gravity. The form is h(t) = -16t² + v₀t + h₀, and from the context we extract: gravity coefficient = -16 (standard for feet), initial velocity v₀ = 40 from 'initial upward velocity of 40 ft/s,' and initial height h₀ = 6 from 'platform 6 feet high' = h(t) = -16t² + 40t + 6. Choice B is correct because it includes the negative coefficient -16t² (gravity pulls down), the positive initial velocity term +40t (thrown upward), and the initial height +6, matching the standard form for projectile motion. Choice A has +16t² instead of -16t²: gravity always pulls objects down, which requires a negative coefficient on t². The negative sign is crucial for modeling falling objects! Context translation cheat sheet: 'per/each' = multiply by that rate, 'plus/in addition' = add, 'starts at' = initial value, 'doubles/triples' = multiply by 2 or 3 (exponential), 'percent' = divide by 100 for decimal. These phrases are your clues for turning words into math!

Question 7

A sequence is described as follows: you start with 60 points, and each new level gives you 15 more points than the previous level. Write a recursive definition for the sequence ana_nan​.

  1. a1=60,  an+1=15ana_1=60,\; a_{n+1}=15a_na1​=60,an+1​=15an​
  2. a1=15,  an+1=an+60a_1=15,\; a_{n+1}=a_n+60a1​=15,an+1​=an​+60
  3. a1=60,  an+1=an+15a_1=60,\; a_{n+1}=a_n+15a1​=60,an+1​=an​+15
  4. an=60+15na_n=60+15nan​=60+15n
Explanation: This question tests your ability to translate a real-world situation into mathematical language—either as an explicit formula, a recursive process, or a series of calculation steps. A recursive definition describes how to get each term from the previous one, which is perfect for sequential processes: if you 'start with 200 and add 50 each week,' that becomes a₁ = 200, aₙ₊₁ = aₙ + 50. You need both the starting value and the rule for what comes next. The context describes a sequential process: 'start with 60 points, and each new level gives you 15 more points than the previous level.' We need a starting point—that's 60 from 'start with 60 points'—and a rule for each step. Since we get '15 more points than the previous level,' each new value is the previous value plus 15. Writing this as a recursive definition: a₁ = 60, aₙ₊₁ = aₙ + 15. To find the 5th level's points, we'd start at 60 and apply the rule four times! Choice C is correct because it includes both the initial value (a₁ = 60) and the recurrence relation (aₙ₊₁ = aₙ + 15) that matches getting 15 more points each level. Choice A has the recursive rule using multiplication: aₙ₊₁ = 15aₙ means multiply by 15, but the context says '15 more points,' which means addition, not multiplication. When you see 'more than,' that's addition—'times as many' would be multiplication! For recursive definitions, remember the two-part recipe: you MUST have both (1) the starting value(s)—look for 'starts at,' 'begins with,' 'initially'—and (2) the rule relating each term to the previous—look for 'add,' 'multiply by,' 'increases by' followed by a description. Write it as: a₁ = [starting value], aₙ₊₁ = [rule using aₙ]. If you're stuck choosing between explicit and recursive, ask: does each value only depend on which step you're at (explicit), or does each value depend on the previous value (recursive)? 'Day 5 costs 45′suggestsexplicit.′Add45' suggests explicit. 'Add 45′suggestsexplicit.′Add10 to yesterday's amount' suggests recursive. The language tells you which form fits!

Question 8

A gym membership costs 25tojoinandthen25 to join and then 25tojoinandthen15 each month. Which expression represents the total cost C(m)C(m)C(m) after mmm months?

  1. C(m)=15m+25C(m)=15m+25C(m)=15m+25
  2. C(m)=25+15C(m)=25+15C(m)=25+15
  3. C(m)=25m+15C(m)=25m+15C(m)=25m+15
  4. C(m)=40mC(m)=40mC(m)=40m
Explanation: This question tests your ability to translate a real-world situation into mathematical language—either as an explicit formula, a recursive process, or a series of calculation steps. An explicit expression gives you a direct formula to calculate the output from the input without needing any previous values: if the context says 'costs 45perdayplus45 per day plus 45perdayplus25 fee,' you can write C(d) = 45d + 25, where d is days and C is cost. Just plug in any number of days and calculate immediately! Looking at the context 'A gym membership costs 25tojoinandthen25 to join and then 25tojoinandthen15 each month,' we identify the independent variable as m (months) and dependent variable as C (total cost). The relationship is linear because there's a constant rate per month plus a fixed fee. The rate is 15 from '15eachmonth,′andtheinitialfeeis25from′15 each month,' and the initial fee is 25 from '15eachmonth,′andtheinitialfeeis25from′25 to join.' Putting this together: C(m) = 25 + 15m. This formula lets us calculate total cost for any value of months! Choice B is correct because it matches the linear structure with the joining fee added to the monthly rate times months. Choice A has the right idea but switches the numbers: the context says '25tojoin′(fixed)and′25 to join' (fixed) and '25tojoin′(fixed)and′15 each month' (rate), but this choice uses 25 as the rate and 15 as the fixed—'each' means multiply by the variable! To write explicit functions from context: (1) Underline key phrases like 'per,' 'starts at,' 'plus,' 'each'—these tell you operations and values, (2) Identify what changes (independent variable x, n, t) and what you're calculating (dependent variable C, h, P), (3) Determine function type: constant rate = linear, percent/doubling = exponential, area/motion = quadratic, (4) Extract numbers from context and plug into the right form (y = mx + b for linear, y = a·b^x for exponential, etc.). Context translation cheat sheet: 'per/each' = multiply by that rate, 'plus/in addition' = add, 'starts at' = initial value, 'doubles/triples' = multiply by 2 or 3 (exponential), 'percent' = divide by 100 for decimal. These phrases are your clues for turning words into math!

Question 9

A student saves money each week. In week 1, the student saves 40.Eachfollowingweek,thestudentsaves40. Each following week, the student saves 40.Eachfollowingweek,thestudentsaves15 more than the previous week. Write a recursive definition for the sequence ana_nan​, where ana_nan​ is the amount saved in week nnn (in dollars).

  1. a1=40, an+1=an+15a_1=40,\ a_{n+1}=a_n+15a1​=40, an+1​=an​+15
  2. a1=15, an+1=an+40a_1=15,\ a_{n+1}=a_n+40a1​=15, an+1​=an​+40
  3. a1=40, an+1=15ana_1=40,\ a_{n+1}=15a_na1​=40, an+1​=15an​
  4. an+1=an+15a_{n+1}=a_n+15an+1​=an​+15
Explanation: This question tests your ability to translate a real-world situation into mathematical language—either as an explicit formula, a recursive process, or a series of calculation steps. A recursive definition describes how to get each term from the previous one, which is perfect for sequential processes: if you 'start with 200 and add 50 each week,' that becomes a₁ = 200, aₙ₊₁ = aₙ + 50. You need both the starting value and the rule for what comes next. The context describes a sequential process: 'A student saves money each week. In week 1, the student saves 40.Eachfollowingweek,thestudentsaves40. Each following week, the student saves 40.Eachfollowingweek,thestudentsaves15 more than the previous week.' We need a starting point—that's 40 from 'In week 1, the student saves 40′—andaruleforeachstep.Sincewe′saves40'—and a rule for each step. Since we 'saves 40′—andaruleforeachstep.Sincewe′saves15 more than the previous week,' each new value is previous plus 15. Writing this as a recursive definition: a₁ = 40, aₙ₊₁ = aₙ + 15. To find the 5th term, we'd start at 40 and apply the rule four times! Choice A is correct because it includes both initial value and recurrence with the starting amount and the addition of 15 each week. Choice D has the recursive rule but is missing the initial value a₁ = 40. Without knowing where to start, we can't use the rule! Recursive definitions ALWAYS need both the starting value(s) and the recurrence relation. For recursive definitions, remember the two-part recipe: you MUST have both (1) the starting value(s)—look for 'starts at,' 'begins with,' 'initially'—and (2) the rule relating each term to the previous—look for 'add,' 'multiply by,' 'increases by' followed by a description. Write it as: a₁ = [starting value], aₙ₊₁ = [rule using aₙ]. If you're stuck choosing between explicit and recursive, ask: does each value only depend on which step you're at (explicit), or does each value depend on the previous value (recursive)? 'Day 5 costs 45′suggestsexplicit.′Add45' suggests explicit. 'Add 45′suggestsexplicit.′Add10 to yesterday's amount' suggests recursive. The language tells you which form fits!

Question 10

A bacteria culture starts with 200 bacteria and triples every hour. What function models the population P(t)P(t)P(t) after ttt hours?

  1. P(t)=200+3tP(t)=200+3tP(t)=200+3t
  2. P(t)=3(200)tP(t)=3(200)^tP(t)=3(200)t
  3. P(t)=200⋅3tP(t)=200\cdot 3^tP(t)=200⋅3t
  4. P(t)=200⋅t3P(t)=200\cdot t^3P(t)=200⋅t3
Explanation: This question tests your ability to translate a real-world situation into mathematical language—either as an explicit formula, a recursive process, or a series of calculation steps. The key to writing functions from context is identifying what changes (independent variable, like time or number of items) and what you're calculating (dependent variable, like cost or height), then finding the mathematical relationship between them using clues in the language. This context involves 'triples every hour,' which tells us this is an exponential function. Exponential functions have constant multiplication—the population is multiplied by the same factor each time period. The form is P(t) = a·b^t, and from the context we extract: initial population a = 200 from 'starts with 200 bacteria' and growth factor b = 3 from 'triples' = P(t) = 200·3^t. Choice C is correct because it properly identifies 200 as the initial population (when t = 0) and 3^t as the growth factor that triples the population each hour, matching the exponential structure for repeated multiplication. Choice A gives a linear function when the context describes exponential growth: 'triples every hour' indicates multiplication by 3 each hour, which means exponential growth, not adding 3 each hour. Percent growth and multiplication patterns mean exponential! If you're stuck choosing between explicit and recursive, ask: does each value only depend on which step you're at (explicit), or does each value depend on the previous value (recursive)? 'Day 5 costs 45′suggestsexplicit.′Add45' suggests explicit. 'Add 45′suggestsexplicit.′Add10 to yesterday's amount' suggests recursive. The language tells you which form fits!