Write Explicit or Recursive Functions - Algebra 1

A phone battery is at 80% charge and loses 5 percentage points of charge each hour while in use. Express the battery charge $B(t)$ (in percent) as a function of time $t$ (in hours).
Answer: $B(t)=80-5t$ 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 phone battery is at 80% charge and loses 5 percentage points of charge each hour while in use,' we identify the independent variable as t (hours) and dependent variable as B (battery charge). The relationship is linear because there's a constant rate of loss. The rate is -5 from 'loses 5 percentage points each hour,' and the initial amount is 80 from 'at 80% charge.' Putting this together: B(t) = 80 - 5t. This formula lets us calculate battery charge for any value of hours! Choice A is correct because it matches the linear structure with the initial charge subtracted by the loss rate times time. Choice B gives an exponential function when the context describes linear decay: 'loses 5 percentage points' indicates constant subtraction, which means linear, not exponential. Percent growth means exponential, but fixed points mean linear! 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!
A video game character has 80 health points and loses 6 health points each minute. Express remaining health $H(t)$ as a function of time $t$ (in minutes).
Answer: $H(t)=80-6t$ 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 = $a·b^x$ 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!
A ball is dropped from a height of 64 feet. After each bounce, it reaches a height that is $\frac{3}{4}$ of its previous height. Which function represents the height $h(n)$ of the ball after $n$ bounces?
Answer: $h(n) = 64 \left(\frac{3}{4}\right)^n$ The correct answer is A. Before any bounces (n = 0), the ball is at 64 feet. After 1 bounce (n = 1), the height is 64 × (3/4) = 48 feet. After 2 bounces (n = 2), the height is 64 × (3/4)² = 36 feet. The pattern shows h(n) = 64(3/4)ⁿ. Choice B would give h(1) = 64(3/4)⁰ = 64, meaning the ball doesn't lose height after the first bounce. Choice C represents linear decay rather than exponential decay. Choice D uses the reciprocal ratio 4/3, which would make the ball gain height with each bounce.
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 $t$ seconds is modeled by $h(t)=-16t^2+40t+6$ . Which formula represents this height function?
Answer: $h(t)=-16t^2+40t+6$ 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!
A gym membership costs $50 to join plus $30 per month. After 6 months of membership, the monthly rate decreases to $25. Which function represents the total cost $C(m)$ after $m$ months of membership?
Answer: $C(m) = \begin{cases} 50 + 30m & \text{if } m \leq 6 \\ 230 + 25(m-6) & \text{if } m > 6 \end{cases}$ The correct answer is C. For the first 6 months, the cost is $50 + $30m. After 6 months, the total cost is $50 + $30(6) = $230. For months beyond 6, the additional cost is $25 per month for each month beyond 6, so the total cost is $230 + $25(m-6). Choice A restarts the calculation ignoring previous payments. Choice B incorrectly adds 25m for all months rather than just the additional months beyond 6. Choice D is equivalent to choice C when simplified (50 + 180 + 25(m-6) = 230 + 25(m-6)), but choice C is the more direct representation.
A water tank contains 600 liters and drains at a constant rate of 15 liters per minute. Express the remaining volume $V(t)$ (in liters) as a function of time $t$ (in minutes).
Answer: $V(t)=600-15t$ 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 'tank contains 600 liters and drains at a constant rate of 15 liters per minute,' we identify the independent variable as t (time in minutes) and dependent variable as V(t) (remaining volume). The relationship is linear because we have a starting amount that decreases at a constant rate. The initial volume is 600 from 'contains 600 liters,' and the rate is -15 (negative because draining) from 'drains...15 liters per minute.' Putting this together: V(t) = 600 - 15t. This formula lets us calculate the remaining water after any time! Choice A is correct because it properly identifies 600 as the starting volume and subtracts 15t to represent the volume drained after t minutes, matching the decreasing linear relationship. Choice B has the right numbers but adds instead of subtracts: the context says 'drains,' which means the volume decreases, so we subtract 15t, not add it. When you see 'drains,' 'loses,' or 'decreases,' that's subtraction! 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.).
To calculate the total cost of an online order, you take the item price $p$ (in dollars), apply a 20% discount, then add $7 shipping, and then add 8% sales tax to the discounted price plus shipping. Describe the steps to calculate the final total.
Answer: 1) Compute $0.80p$ 2) Add $7$ 3) Multiply the result by $1.08$ 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. Sometimes the best way to describe a relationship is with calculation steps rather than a single formula: if there are multiple operations like 'take 20% off, then add shipping, then add tax,' listing the steps in order (1. multiply original by 0.80, 2. add $7.50, 3. multiply result by 1.08) can be clearer than one complicated formula. The context 'apply a 20% discount, then add $7 shipping, and then add 8% sales tax to the discounted price plus shipping' involves multiple operations that need to happen in a specific order: discount first, shipping second, tax last. To write clear steps: (1) Apply 20% discount means multiply by 0.80 (keeping 80%), (2) Add $7 shipping to the discounted price, (3) Apply 8% tax means multiply the total by 1.08. Following these steps in order with any input value will give the correct output! Choice B is correct because it lists steps in correct order: compute 0.80p (20% off), add 7 (shipping), then multiply by 1.08 (8% tax on the subtotal). Choice C has the calculation steps in the wrong order: it adds shipping before applying the discount, but the context says discount first. If you add $7 to p then multiply by 0.80, you're discounting the shipping too, which gives a different (wrong) answer. Order matters in multi-step calculations! When describing calculation steps, think like you're teaching someone who knows math but doesn't know this specific problem: what do they calculate first, what do they do with that result, what's next? Number your steps (1, 2, 3) and make each one complete. Test your steps with sample numbers to make sure they work! 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!
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 $a_n$ (in dollars) in the jar after $n$ weeks, where $a_1$ is the amount after week 1.
Answer: $a_1=40,\ a_{n+1}=a_n+15$ 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].
A car rental company charges $35 per day plus $0.15 per mile driven. However, if a customer drives more than 200 miles in a day, they pay a flat rate of $75 for that day regardless of mileage. Which function correctly represents the daily cost $C(m)$ for driving $m$ miles?
Answer: $C(m) = \begin{cases} 35 + 0.15m & \text{if } m \leq 200 \\ 75 & \text{if } m > 200 \end{cases}$ The correct answer is B. For 200 miles or fewer, the cost follows the standard formula of $35 + $0.15m. For more than 200 miles, the cost is a flat $75 regardless of mileage. Choice A ignores the flat rate condition entirely. Choice C incorrectly adds the mileage charge to the flat rate for high mileage. Choice D attempts to charge the flat rate plus additional mileage beyond 200 miles, which contradicts the 'regardless of mileage' condition in the problem.
A car rental company charges a $40 flat fee plus $28 per day. What is the formula for the total cost $C(d)$ in dollars based on the number of days $d$ ?
Answer: $C(d)=40+28d$ 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 'charges a $40 flat fee plus $28 per day,' we identify the independent variable as d (number of days) and dependent variable as C (total cost). The relationship is linear because we have a fixed fee plus a rate per day. The flat fee is $40 from '$40 flat fee,' and the daily rate is $28 from '$28 per day.' Putting this together: C(d) = 40 + 28d. This formula lets us calculate total cost for any number of rental days! Choice C is correct because it properly identifies d as days and includes both the flat fee ($40) and the per-day rate ($28d) in the right order: fixed fee + rate × quantity. Choice A has the right idea but swaps the coefficients: it uses 40d + 28 instead of 40 + 28d. When you see 'per' or 'each,' that's multiplication (like $28 per day = 28×days), and this amount should multiply the variable d, not the constant 40! 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!
A phone plan costs $25 each month plus $0.05 per text message sent. What function models the monthly cost $C(t)$ (in dollars) in terms of the number of texts $t$ ?
Answer: $C(t)=25+0.05t$ 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 $25 each month plus $0.05 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 $25 from 'costs $25 each month,' and the per-text rate is $0.05 from '$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) and the per-text rate ($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.).
A sequence is defined by this situation: You start with 12 points in a game, and each new level gives you 5 times as many points as the previous level. Write both an explicit formula and a recursive definition for the sequence $a_n$ , where $a_1=12$ is the points after level 1.
Answer: Explicit: $a_n=12\cdot 5^{n-1}$ ; Recursive: $a_1=12,\ a_{n+1}=5a_n$ 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 '5 times as many points as the previous level,' which tells us this is an exponential function with growth factor 5. For the explicit form: starting with 12 and multiplying by 5 each level gives us aₙ = 12 · $5^{n-1}$ (we use n-1 because a₁ = 12, not 12·5). For the recursive form: we start with a₁ = 12 and each new term is 5 times the previous, so aₙ₊₁ = 5aₙ. Choice B is correct because it includes both forms correctly: the explicit formula 12 · $5^{n-1}$ gives the right values (a₁ = 12·5⁰ = 12, a₂ = 12·5¹ = 60, etc.) and the recursive definition a₁ = 12, aₙ₊₁ = 5aₙ matches multiplying by 5 each level. Choice A has the recursive rule wrong: aₙ₊₁ = aₙ + 5 means adding 5 each time, but the problem says '5 times as many,' which means multiplication! When you see 'times as many' or 'times more,' that's multiplication in your recursive rule, not addition. 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!
A rideshare app charges a base fee of $3.50 plus $1.20 per mile. What is the formula for the total cost $C(m)$ (in dollars) based on miles traveled $m$ ?
Answer: $C(m)=3.50+1.20m$ 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 rideshare app charges a base fee of $3.50 plus $1.20 per mile,' we identify the independent variable as m (miles) and dependent variable as C (total cost). The relationship is linear because there's a constant rate per mile plus a fixed fee. The rate is 1.20 from '$1.20 per mile,' and the initial fee is 3.50 from 'base fee of $3.50.' Putting this together: C(m) = 3.50 + 1.20m. This formula lets us calculate total cost for any value of miles! Choice B is correct because it properly identifies the variables and matches the linear structure with the base fee added to the per-mile rate times miles. Choice C has the right idea but switches the numbers: the context says '$3.50' for the base and '$1.20 per mile,' but this choice uses 3.50 as the rate and 1.20 as the fee—remember, 'per' 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!
A water tank starts with 500 gallons and is drained at a constant rate of 25 gallons per minute. Express the remaining volume $V(t)$ (in gallons) as a function of time $t$ (in minutes).
Answer: $V(t)=500-25t$ 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 water tank starts with 500 gallons and is drained at a constant rate of 25 gallons per minute,' we identify the independent variable as t (time in minutes) and dependent variable as V (remaining volume). The relationship is linear because there's a constant rate of change. The rate is -25 from 'drained at a constant rate of 25 gallons per minute,' and the initial amount is 500 from 'starts with 500 gallons.' Putting this together: V(t) = 500 - 25t. This formula lets us calculate remaining volume for any value of time! Choice A is correct because it matches the linear structure with the initial volume subtracted by the drainage rate times time. Choice B has the right idea but uses the wrong operation: the context says 'drained,' which means subtraction, but this choice uses addition—when you see words like 'loses' or 'drained,' that's a negative rate! 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.). 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!
A gym membership costs $18 to join plus $12 per month. Which expression represents the total cost $C(m)$ after $m$ months?
Answer: $C(m)=18+12m$ 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 $18 to join plus $12 per month,' we identify the independent variable as m (number of months) and dependent variable as C (total cost). The relationship is linear because we have a one-time joining fee plus a constant monthly rate. The joining fee is $18 from 'costs $18 to join,' and the monthly rate is $12 from '$12 per month.' Putting this together: C(m) = 18 + 12m. This formula lets us calculate total cost for any number of months! Choice C is correct because it properly identifies m as the variable for months and includes both the initial joining fee ($18) and the per-month rate ($12) in the standard form: fixed cost + rate × variable. Choice A reverses the coefficients: 18m + 12 would mean $18 per month plus a $12 joining fee—that's backwards from what the problem states! Always match the 'per' phrase to the variable: '$12 per month' means 12 times m. 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.).
A bacteria culture starts with 300 bacteria and triples every hour. Express the population $P(t)$ as a function of time $t$ (in hours) using an explicit formula.
Answer: $P(t)=300\cdot 3^t$ 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 'starts with 300 bacteria and triples every hour,' we identify the independent variable as t (time in hours) and dependent variable as P (population). The relationship is exponential because we multiply by a constant factor each time period. The initial population is 300 from 'starts with 300,' and the growth factor is 3 from 'triples.' Putting this together: P(t) = 300 · $3^t$ . This formula lets us calculate population after any number of hours! Choice C is correct because it properly identifies the exponential structure with initial value 300 and growth factor 3, giving us 300 · $3^t$ where t is the number of hours. Choice B has the base and exponent reversed: 3 · $300^t$ would mean starting with 3 bacteria and multiplying by 300 each hour—that's backwards! In exponential functions, the growth factor (3 for tripling) is the base that gets raised to the time power. 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!
A water tank contains 500 gallons and drains at a constant rate of 30 gallons per minute. Express the remaining volume $V(t)$ (in gallons) as a function of time $t$ (in minutes).
Answer: $V(t)=500-30t$ 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 'contains 500 gallons and drains at a constant rate of 30 gallons per minute,' we identify the independent variable as t (time in minutes) and dependent variable as V (remaining volume). The relationship is linear because we have a constant rate of change. The initial volume is 500 from 'contains 500 gallons,' and the rate is -30 (negative because draining) from 'drains at 30 gallons per minute.' Putting this together: V(t) = 500 - 30t. This formula lets us calculate remaining water for any time! Choice C is correct because it properly identifies that draining means subtraction (negative rate) and includes both the initial volume (500) and the drainage rate (30 gallons per minute) as 500 - 30t. Choice B writes it as addition (500 + 30t), which would mean the tank is filling, not draining! When something decreases at a constant rate, you subtract: initial amount - rate × time. 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.).
A store is running a sale. To find the final price, take 25% off the original price $p$ (in dollars), then add a flat shipping fee of $4$ . Describe the steps to calculate the final price from $p$ .
Answer: 1) Multiply $p$ by $0.75$ . 2) Add $4$ . 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. Sometimes the best way to describe a relationship is with calculation steps rather than a single formula: if there are multiple operations like 'take $20\%$ off, then add shipping, then add tax,' listing the steps in order (1. multiply original by $0.80$ , 2. add $7.50$ , 3. multiply result by $1.08$ ) can be clearer than one complicated formula. The context 'A store is running a sale. To find the final price, take 25% off the original price p (in dollars), then add a flat shipping fee of $4$ ' involves multiple operations that need to happen in a specific order: discount first, then add shipping. To write clear steps: (1) Multiply p by $0.75$ (since 25% off means multiply by $1 - 0.25 = 0.75$ ), (2) Add 4 to the result. Following these steps in order with any input value will give the correct output! Choice C is correct because it lists steps in correct order with the multiplication by $0.75$ for the discount followed by adding the shipping fee. Choice A has the calculation steps in the wrong order: we need to apply the discount before adding shipping to get the right result. If you add shipping first and then discount, you get a different (wrong) answer. Order matters in multi-step calculations! When describing calculation steps, think like you're teaching someone who knows math but doesn't know this specific problem: what do they calculate first, what do they do with that result, what's next? Number your steps (1, 2, 3) and make each one complete. Test your steps with sample numbers to make sure they work! 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!
A student starts a savings challenge with $25 on day 1 and adds $4 each new day. Write a recursive definition for the amount saved $a_n$ (in dollars) on day $n$ .
Answer: $a_1=25,\; a_{n+1}=a_n+4$ 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 $25 on day 1 and adds $4 each new day.' We need a starting point—that's $25 from 'starts...with $25 on day 1'—and a rule for each step. Since we 'add $4 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 $4 each day. Choice B has the recursive rule backwards: it starts with $4 and adds $25 each time, but the context says we start with $25 and add $4. 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' suggests explicit. 'Add $10 to yesterday's amount' suggests recursive. The language tells you which form fits!
A bacteria culture starts with 300 bacteria and triples every hour. What function models the population $P(t)$ after $t$ hours?
Answer: $P(t)=300\cdot 3^t$ 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 = $a·b^x$ 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!

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