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!

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.).

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!

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].

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!

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!

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' suggests explicit. 'Add $10 to yesterday's amount' suggests recursive. The language tells you which form fits!

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 $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 '$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·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!

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!

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' suggests explicit. 'Add $10 to yesterday's amount' suggests recursive. The language tells you which form fits!