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Algebra Help: Write Explicit Or Recursive Functions

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

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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 tt seconds is modeled by a quadratic function with gravity −16t2-16t^2. What is the formula for the height function h(t)h(t)?

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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 tt seconds is modeled by a quadratic function with gravity −16t2-16t^2. What is the formula for the height function h(t)h(t)?

  1. h(t)=16t2+24t+4h(t)=16t^2+24t+4
  2. h(t)=−16t2+24t+4h(t)=-16t^2+24t+4 (correct answer)
  3. h(t)=−16t2+4t+24h(t)=-16t^2+4t+24
  4. h(t)=−24t2+16t+4h(t)=-24t^2+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 student starts a savings jar with $40. Each week, the student adds $15 more than the previous week's total amount in the jar. Write a recursive definition for the amount of money ana_n (in dollars) in the jar after nn weeks, where a1a_1 is the amount after week 1.

  1. a1=40, an+1=an+15a_1=40,\ a_{n+1}=a_n+15 (correct answer)
  2. a1=15, an+1=an+40a_1=15,\ a_{n+1}=a_n+40
  3. an=40+15na_n=40+15n
  4. an+1=15n+40a_{n+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 'adds $15 more than the previous week's total.' We need a starting point—that's $40 from 'starts with $40'—and a rule for each step. Since we 'add $15' 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 each week to the previous total. Choice B reverses the values: it starts with $15 and adds $40 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 3

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) after completing nn levels, where n=0n=0 means no levels completed yet.

  1. S(n)=80(25)nS(n)=80(25)^n
  2. S(n)=25+80nS(n)=25+80n
  3. S(n)=80n+25S(n)=80n+25
  4. S(n)=80+25nS(n)=80+25n (correct answer)

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 $45 per day plus $25 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 4

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 tt seconds is modeled by h(t)=−16t2+40t+6h(t)=-16t^2+40t+6. Which formula represents this height function?

  1. h(t)=−16t2−40t+6h(t)=-16t^2-40t+6
  2. h(t)=16t2+40t+6h(t)=16t^2+40t+6
  3. h(t)=−16t2+40t+6h(t)=-16t^2+40t+6 (correct answer)
  4. h(t)=−40t2+16t+6h(t)=-40t^2+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 5

A student saves money each week. In week 1, the student saves $40. Each following week, the student saves $15 more than the previous week. Write a recursive definition for the sequence $a_n,where, where a_nistheamountsavedinweekis the amount saved in weekn$ (in dollars).

  1. a1=40, an+1=an+15a_1=40,\ a_{n+1}=a_n+15 (correct answer)
  2. a1=15, an+1=an+40a_1=15,\ a_{n+1}=a_n+40
  3. a1=40, an+1=15ana_1=40,\ a_{n+1}=15a_n
  4. an+1=an+15a_{n+1}=a_n+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. Each following week, the student saves $15 more than the previous week.' We need a starting point—that's 40 from 'In week 1, the student saves $40'—and a rule for each step. Since we 'saves $15 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' suggests explicit. 'Add $10 to yesterday's amount' suggests recursive. The language tells you which form fits!

Question 6

A gym membership costs $35 to sign up plus $18 per month. What is the formula for the total cost $C(m)(indollars)after(in dollars) afterm$ months?

  1. C(m)=35m+18C(m)=35m+18
  2. C(m)=18+35mC(m)=18+35m
  3. C(m)=35+18mC(m)=35+18m (correct answer)
  4. C(m)=35⋅18mC(m)=35\cdot 18^m

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 $45 per day plus $25 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 $35 to sign up plus $18 per month,' we identify the independent variable as m (number of months) and dependent variable as C(m) (total cost). The relationship is linear because we have a one-time fee plus a constant monthly rate. The sign-up fee is $35 from 'costs $35 to sign up,' and the monthly rate is 18from′18 from '18 per month.' Putting this together: C(m) = 35 + 18m. This formula lets us calculate the total cost after any number of months! Choice C is correct because it properly identifies $35 as the initial sign-up fee and $18m as the total monthly charges for m months, matching the structure 'initial fee + rate × time.' Choice A reverses the coefficients: it would mean $35 per month plus an 18sign−upfee,butthecontextclearlystates′18 sign-up fee, but the context clearly states '35 to sign up' (one-time) and '$18 per month' (recurring). Always match the numbers to their correct roles! 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⋅bxa·b^x for exponential, etc.).

Question 7

A bacteria culture starts with 300 bacteria and triples every hour. What function models the population P(t)P(t) after tt hours?

  1. P(t)=3⋅300tP(t)=3\cdot 300^t
  2. P(t)=300⋅3tP(t)=300\cdot 3^t (correct answer)
  3. P(t)=300⋅t3P(t)=300\cdot t^3
  4. P(t)=300+3tP(t)=300+3t

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 $45 per day plus $25 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! This context involves 'triples every hour,' which tells us this is an exponential function. Exponential: constant multiplication; the form is P(t) = a · b^t, and from the context we extract: initial 300 from 'starts with 300 bacteria' and growth factor 3 from 'triples,' so P(t) = 300 · 3^t. Choice B is correct because it matches the exponential structure with the initial population multiplied by the growth factor raised to time. Choice A gives a linear function when the context describes exponential growth: 'triples' indicates constant multiplication, which means exponential, not linear. Percent growth means exponential! 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⋅bxa·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 8

A school club has 300 members and membership increases by 5% each year. What function models the number of members M(t)M(t) after tt years?

  1. M(t)=300+0.05tM(t)=300+0.05t
  2. M(t)=300(1.05)tM(t)=300(1.05)^t (correct answer)
  3. M(t)=300(0.95)tM(t)=300(0.95)^t
  4. M(t)=300⋅5tM(t)=300\cdot 5^t

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 $45 per day plus $25 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! This context involves 'increases by 5% each year,' which tells us this is an exponential function. Exponential functions involve constant multiplication (like percent growth), not constant addition. The form is M(t) = a·b^t, and from the context we extract: initial value a = 300 from 'has 300 members' and growth factor b = 1.05 from '5% increase' (100% + 5% = 105% = 1.05) = M(t) = 300(1.05)^t. Choice B is correct because it matches the exponential structure with the right initial value (300) and growth factor (1.05 for 5% increase). Choice A gives a linear function M(t) = 300 + 0.05t when the context describes exponential growth: 'increases by 5%' indicates percent growth, which means exponential, not linear. Percent growth means exponential! 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⋅bxa·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 gym membership costs $25 to join and then $15 each month. Which expression represents the total cost $C(m)afterafterm$ months?

  1. C(m)=15m+25C(m)=15m+25 (correct answer)
  2. C(m)=25+15C(m)=25+15
  3. C(m)=25m+15C(m)=25m+15
  4. C(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 $45 per day plus $25 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 $25 to join and then 15eachmonth,′weidentifytheindependentvariableasm(months)anddependentvariableasC(totalcost).Therelationshipislinearbecausethere′saconstantratepermonthplusafixedfee.Therateis15from′15 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 '15 each month,' and the initial fee is 25 from '25tojoin.′Puttingthistogether:C(m)=25+15m.Thisformulaletsuscalculatetotalcostforanyvalueofmonths!ChoiceBiscorrectbecauseitmatchesthelinearstructurewiththejoiningfeeaddedtothemonthlyratetimesmonths.ChoiceAhastherightideabutswitchesthenumbers:thecontextsays′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 '25 to join' (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⋅bxa·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 10

A video game character has 80 health points and loses 6 health points each minute. Express remaining health H(t)H(t) as a function of time tt (in minutes).

  1. H(t)=80+6tH(t)=80+6t
  2. H(t)=80−6tH(t)=80-6t (correct answer)
  3. H(t)=6−80tH(t)=6-80t
  4. H(t)=80⋅6tH(t)=80\cdot 6^t

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 $45 per day plus $25 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 'has 80 health points and loses 6 health points each minute,' we identify the independent variable as t (time in minutes) and dependent variable as H (remaining health). The relationship is linear because we have a starting amount that decreases at a constant rate. The initial health is 80 from 'has 80 health points,' and the rate is -6 per minute from 'loses 6 health points each minute.' Putting this together: H(t) = 80 - 6t. This formula lets us calculate remaining health at any time! Choice C is correct because it properly identifies the starting health (80) and subtracts the health lost over time (6t), matching the context of losing 6 points per minute. Choice A has the right numbers but uses addition instead of subtraction: the context says 'loses,' which means subtract, not add. When you see 'loses,' 'decreases,' or 'drains,' that's subtraction—the opposite of 'gains' or 'adds'! 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⋅bxa·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!