Math 2 Quiz: Building Function Models

What this quiz covers

This quiz focuses on Building Function Models, giving you a quick way to practice the rules, question types, and explanations that matter most for Math 2.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. Exponential growth with factor $r = 1.2$ per hour, since $1000 \cdot (1.2)^4 = 2074$ approximately
2. Linear growth with rate 537 bacteria per hour, since the total increase over 4 hours is 1,074 bacteria
3. Quadratic growth with $a = 67.25$, since the second differences in bacteria count are constant
4. Exponential growth with factor $r = 1.44$ per 2-hour period, equivalent to $r \approx 1.2$ per hour (correct answer)

Question 2

1. Exponential growth with rate $r = 1.5$; the limitation is that exponential growth is unrealistic for long-term user adoption due to market saturation (correct answer)
2. Linear growth with slope $m = 312.5$; the limitation is that linear models cannot capture the accelerating growth pattern observed
3. Quadratic growth with $a > 0$; the limitation is that quadratic models eventually decrease after reaching a maximum value
4. Exponential decay with rate $r = 0.67$; the limitation is that decay models predict decreasing users over time

Question 3

1. Linear function with $334.14 added per year, representing simple interest rather than compound interest on the account
2. Exponential function $B(t) = 5000(1.05)^t$ representing 5% annual compound interest, since $(1.05)^2 \approx 1.1025$ (correct answer)
3. Quadratic function representing variable interest rates that increase over time, with acceleration in balance growth observed
4. Exponential function $B(t) = 5000(1.1025)^{t/2}$ representing 10.25% interest compounded every two years instead of annually

Question 4

1. Linear function $P(x) = 5x + 80$; predicting $580 profit, assuming fixed costs and constant marginal profit per drink (correct answer)
2. Quadratic function $P(x) = -0.05x^2 + 8x - 40$; predicting $460 profit, accounting for diminishing returns at high volumes
3. Exponential function $P(x) = 80(1.22)^{x/20}$; predicting $1,247 profit, modeling accelerating profit growth with volume
4. Square root function $P(x) = 50\sqrt{x} - 70$; predicting $430 profit, representing economies of scale with diminishing marginal returns

Question 5

1. Linear model $R(q) = 120q + 280$ with the assumption that market conditions and competitive landscape remain constant indefinitely (correct answer)
2. Exponential model $R(q) = 400(1.3)^{q-1}$ with the assumption that growth rates can sustain exponentially without market saturation effects
3. Quadratic model $R(q) = 20q^2 + 100q + 280$ with the assumption that acceleration in growth will continue without any market limitations
4. Logarithmic model $R(q) = 280 + 160\ln(q)$ with the assumption that revenue growth will naturally slow over time due to market maturity

Question 6

1. Linear model $C(x) = -0.075x + 57.5$ representing constant reduction in cost per unit as volume increases
2. Exponential decay model $C(x) = 85(0.95)^{x/100}$ representing percentage-based cost reductions due to economies of scale
3. Quadratic model $C(x) = 0.0001x^2 - 0.09x + 59$ representing initial cost reductions followed by increasing marginal costs
4. Inverse/rational model $C(x) = 20 + \frac{3000}{x}$ representing fixed costs spread over more units plus variable costs (correct answer)

Question 7

1. Linear growth $P(t) = 50 + 19.95t$ predicting approximately 170 rabbits, assuming constant monthly additions to the population
2. Exponential growth $P(t) = 50(1.3)^t$ predicting approximately 214 rabbits, assuming constant proportional growth rate each month (correct answer)
3. Quadratic growth $P(t) = 2.45t^2 + 12.5t + 50$ predicting approximately 163 rabbits, assuming accelerating but bounded population increases
4. Exponential growth $P(t) = 50(1.69)^{t/2}$ predicting approximately 186 rabbits, assuming growth rate varies with environmental carrying capacity

Question 8

1. Linear drainage $V(t) = 1000 - 90.33t$ representing constant outflow rate regardless of water pressure changes in the tank
2. Exponential decay $V(t) = 1000(0.9)^t$ representing drainage rate proportional to current water volume and pressure (correct answer)
3. Quadratic drainage $V(t) = 1000 - 50t - 20.17t^2$ representing accelerating drainage due to increasing hole size over time
4. Square root model $V(t) = 1000 - 100\sqrt{t}$ representing drainage rate proportional to square root of pressure according to Torricelli's law

Question 9

1. Linear function because the average velocity appears constant when calculated over the entire 8-second interval from the given data
2. Exponential function because the altitude values are increasing at an accelerating rate, suggesting exponential rocket propulsion effects
3. Quadratic function because rocket motion involves constant acceleration, and the second differences in altitude are approximately constant (correct answer)
4. Cubic function because rocket engines provide variable thrust, creating non-uniform acceleration that requires higher-order polynomial modeling

Question 10

1. Linear decay $C(t) = 100 - 13.07t$ with complete elimination in approximately 7.65 hours, assuming constant elimination rate
2. Exponential decay $C(t) = 100(0.6)^{t/2}$ with half-life of approximately 1.44 hours, assuming first-order elimination kinetics (correct answer)
3. Quadratic decay $C(t) = 100 - 6t - 1.53t^2$ with elimination rate accelerating over time due to metabolic saturation
4. Exponential decay $C(t) = 100(0.775)^t$ with half-life of approximately 2.75 hours, assuming continuous elimination processes

Question 11

1. Exponential model assuming unlimited market reach and sustained viral growth, with doubling occurring every 10 days consistently
2. Linear model $U(d) = 240d$ assuming steady user acquisition rate of 240 users per day without market saturation (correct answer)
3. Quadratic model assuming accelerating user growth due to network effects, with growth rate increasing over time
4. Logarithmic model assuming initial rapid growth followed by natural slowdown as market approaches saturation point

Question 12

1. Exponential growth $E(t) = 50(1.9)^{t-1}$ for hours 1-4, then constant function $E(t) = 275$ after Hour 6
2. Linear growth $E(t) = 45t + 5$ for hours 1-4, then logistic growth function to model plateau transition
3. Quadratic growth $E(t) = -2.5t^2 + 52.5t$ for hours 1-4, then parabolic extension to plateau
4. Linear growth $E(t) = 45t + 5$ for hours 1-4, then piecewise function with constant $E(t) = K$ after Hour 6 (correct answer)

Question 13

1. Linear model $h(t) = -8t + 92$ because the average rate of change from $t = 1$ to $t = 3$ is zero
2. Exponential model $h(t) = 68(1.25)^t$ because the height increases then decreases, showing exponential behavior with decay
3. Quadratic model $h(t) = -16t^2 + 84t + 16$ because projectile motion follows parabolic paths due to constant acceleration (correct answer)
4. Piecewise linear model because the ball travels upward linearly, then downward linearly after reaching maximum height

Question 14

1. Linear function $A(w) = 200w - 2w$, with domain $0 < w < 100$, because area increases linearly with width
2. Quadratic function $A(w) = w(200-2w)$, with domain $0 < w < 100$, because area depends on two varying dimensions (correct answer)
3. Quadratic function $A(w) = w(200-w)$, with domain $0 < w < 200$, because three sides total 200 feet
4. Linear function $A(w) = w(200-2w)$, with domain $0 < w < 100$, because the relationship simplifies to linear form

Question 15

1. Piecewise function combining linear, exponential, and quadratic portions to match each distinct growth phase (correct answer)
2. Single exponential function, because business revenue typically follows exponential patterns over long time periods
3. Single quadratic function, because it can model both increasing and decreasing rates of growth within one equation
4. Single linear function with a large slope, because the overall trend shows consistent revenue increase

Question 16

1. Option A linear, Option B exponential; Option A: $V(t) = 10000 + 300t$, Option B: $V(t) = 10000(1.028)^t$; Option B is better (correct answer)
2. Option A linear, Option B exponential; Option A: $V(t) = 10000 + 300t$, Option B: $V(t) = 10000(1.028)^t$; Option A is better
3. Both linear functions; Option A: $V(t) = 10000(1 + 0.03t)$, Option B: $V(t) = 10000(1 + 0.028t)$; Option A is better
4. Both exponential functions; Option A: $V(t) = 10000(1.03)^t$, Option B: $V(t) = 10000(1.028)^t$; Option A is better

Question 17

1. Linear function $A(t) = 100 - 6.25t$, because the substance decreases by 12.5 grams every 8 years
2. Quadratic function $A(t) = 100 - 6.25t^2$, because the decay rate accelerates over time due to compound effects
3. Exponential function $A(t) = 100(0.5)^{8t}$, because radioactive decay always follows exponential patterns with base 0.5
4. Exponential function $A(t) = 100(0.5)^{t/8}$, because the amount decreases by a constant percentage over equal intervals (correct answer)

Question 18

1. Linear function, because the app gains users at an increasing rate each month
2. Quadratic function, because the monthly user gains form the sequence 1000, 2000, 4000 which increases quadratically
3. Exponential function, because social media growth always follows exponential patterns due to network effects
4. Exponential function, because the monthly user gains double each month, indicating exponential total growth (correct answer)

Question 19

1. Linear function $P(x) = mx + b$ where $m > 0$, because profit increases with price
2. Exponential function $P(x) = ab^x$ where $b > 1$, because profit grows rapidly with optimal pricing
3. Quadratic function $P(x) = a(x-2)(x-8)$ where $a < 0$, because profit has two zeros and one maximum (correct answer)
4. Exponential function $P(x) = ab^x$ where $0 < b < 1$, because profit decreases due to customer loss