Algebra 3 Quiz: Selecting A Function Model

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Selecting A Function ModelQuestion 1 of 20

A company buys a machine for $20,000. The machine loses 12% of its current value each year.

Which function family best models the machine's value $V$ as a function of time $t$ in years?

Linear:

V(t)=20000-2400t

, because 12% of $20,000 is $2,400 each year.

Exponential:

V(t)=20000(0.88)^t

, because the value is multiplied by 0.88 each year.

Power:

V(t)=20000t^{-0.12}

, because a negative exponent represents decay.

Quadratic:

V(t)=20000(1-0.12t)^2

, because the depreciation is applied twice.

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Algebra 3 Quiz

Algebra 3 Quiz: Selecting A Function Model

Practice Selecting A Function Model in Algebra 3 with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Selecting A Function Model, giving you a quick way to practice the rules, question types, and explanations that matter most for Algebra 3.

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.

All questions

Question 1

A company buys a machine for $20,000. The machine loses 12% of its current value each year.

Which function family best models the machine's value $V$ as a function of time $t$ in years?

Linear: $V(t)=20000-2400t$ , because 12% of $20,000 is $2,400 each year.
Exponential: $V(t)=20000(0.88)^t$ , because the value is multiplied by 0.88 each year. (correct answer)
Power: $V(t)=20000t^{-0.12}$ , because a negative exponent represents decay.
Quadratic: $V(t)=20000(1-0.12t)^2$ , because the depreciation is applied twice.

Explanation: Whenever a quantity changes by a fixed percent of its current value each period, you are in exponential decay territory. The key signal here is "loses 12% of its current value each year"—the 12% is taken from the value that remains, not from the original $20,000. The correct model is exponential: after one year, the machine retains

100\%-12\%=88\%

of its value, so its value is multiplied by

0.88

. After

t

years, $

V(t)=20000(0.88)^t.

That matches the choice: “Exponential:

V(t)=20000(0.88)^t$, because the value is multiplied by 0.88 each year.” The linear choice,

V(t)=20000-2400t

, assumes the machine loses exactly

12\%

of the original $20,000, or $2,400, every year. That would be a fixed dollar amount, but the problem says 12% of the current value, which shrinks each year. The power choice,

20000t^{-0.12}

, misuses a negative exponent; percent decay is not a power function, and it would not even be defined at

t=0

. The quadratic choice,

20000(1-0.12t)^2

, incorrectly treats depreciation as something applied twice rather than as repeated multiplication by 0.88. Study tip: when you see "percent of its current value" or "grows/decays by a percent each year," write the multiplier as

1 \pm r

and use an exponential function. Fixed dollar change? Then linear. That distinction is the whole question.

Question 2

In a sound system, each 10-fold increase in actual sound intensity corresponds to an increase of 10 decibels in perceived loudness.

Which function family best models perceived loudness $L$ as a function of sound intensity $I$ , and why?

Linear, because equal increases in intensity produce equal increases in loudness.
Exponential, because loudness increases by a factor of 10 when intensity increases by 10.
Logarithmic, because equal multiplicative changes in intensity correspond to equal additive changes in loudness. (correct answer)
Power, because loudness is proportional to the 10th power of intensity.

Explanation: When you see a statement like "each 10-fold increase in intensity corresponds to an increase of 10 decibels," focus on how the input changes versus how the output changes. Here, the input

I

changes multiplicatively (multiply by 10), while the output

L

changes additively (add 10). That pairing—multiplicative input, additive output—is the signature of a logarithmic relationship. So

L = k \log_{10}(I) + c

describes this exactly: each time

I

is multiplied by 10,

\log_{10}(I)

increases by 1, and with a constant

k=10

L

increases by 10. The linear choice is a trap because it assumes equal additive increases in intensity cause equal increases in loudness, but the passage describes equal multiplicative increases. The exponential choice reverses the pattern: exponential growth means a constant multiplier in the output for a constant addition in the input, which is the opposite of this situation. The power choice misunderstands the decibel scale entirely; if

L

were proportional to

I^{10}

, multiplying

I

by 10 would multiply

L

10^{10}

, not add 10. As a study tip, whenever a real-world scale says "a 10-fold increase in the original quantity adds a fixed amount to the measured value," think logarithmic. The key is recognizing which variable is multiplied and which is added.

Question 3

A new video is posted online. The number of views grows slowly at first, then very rapidly, then more slowly as it approaches about 50,000 views.

Which function family best models the number of views as a function of time since posting?

Exponential growth, because the rapid middle portion is the most important feature.
Logarithmic, because the growth slows after the initial increase.
Quadratic, because the growth speeds up and then slows down.
Logistic, because the growth is S-shaped and has an upper limit. (correct answer)

Explanation: Whenever you see a question about modeling growth, start by identifying the overall shape and whether there is a limit. The passage describes views that increase slowly, then very rapidly, then slow down as they approach about 50,000 views. That pattern—an S-shaped curve with a horizontal upper bound—is the signature of logistic growth. A logistic model begins with a slow rise, enters a phase of rapid increase, then levels off near a carrying capacity. Here, the 50,000-view ceiling is exactly that capacity, so logistic is the best match. Exponential growth is tempting because of the rapid middle, but exponential functions grow without limit and never slow down, so it cannot account for the leveling off. Logarithmic growth does slow after an initial increase, but it is concave downward from the start and never has the explosive middle stage described. Quadratic growth changes its rate at a constant pace—it can speed up or slow down, but not both—and it has no fixed upper limit. Only logistic combines an initially slow rise, a rapid surge, and a final plateau at a maximum. On exam day, whenever you see phrases like "approaches a maximum" or "levels off," think logistic. If there is no limit, then consider exponential or logarithmic depending on whether the growth rate is increasing or decreasing.

Question 4

A taxi company charges a $3.50 pickup fee plus $2.10 per mile driven.

Which function family best models the total fare F as a function of miles m, and why?

Quadratic, because the fare includes both a per-mile charge and a fixed pickup fee.
Linear, but not proportional, because the fare has a constant per-mile rate and a nonzero pickup fee. (correct answer)
Exponential, because the fare increases by a constant percentage with each additional mile.
Step, because the fare increases by $2.10 only after each full mile is completed.

Explanation: When you see a real-world situation described as a fixed fee plus a constant rate, you should immediately think about the shape of the graph: a constant rate of change is the signature of a linear function. Here the total fare is

F(m)=3.50+2.10m

, so each mile adds the same $2.10, and the $3.50 pickup fee shifts the graph away from the origin. That is exactly linear, but not proportional, because proportional relationships require

F(0)=0

. The quadratic family would require a variable rate or an

m^2

term; no squaring or acceleration appears here. Exponential growth would mean the fare multiplies by a constant percentage per mile, but here it increases by a constant dollar amount, not a percent. The step-function model is tempting if you think meters are rounded up, but the problem says the company charges per mile driven, implying a continuous, smooth line; if there were rounding to whole miles, a step function would apply, but that is not stated. As a study tip, when deciding function families, ask: constant rate of change? Linear. Constant percentage? Exponential. Fixed jumps at intervals? Step. Extra nonzero starting value? Linear but not proportional. This quick check will keep you from confusing the key categories on the exam.

Question 5

A Ferris wheel rotates at a constant speed. A rider's height above the ground increases, then decreases, then repeats the same pattern over and over.

Which function family best models the rider's height $h$ as a function of time $t$ ?

Sinusoidal, because the height is periodic and repeats indefinitely. (correct answer)
Logarithmic, because the increase slows and then reverses.
Quadratic, because height goes up and then down.
Logistic, because the height approaches a maximum and then a minimum.

Explanation: Whenever you see a word problem describing a repeating pattern over time, your first thought should be periodicity. A Ferris wheel's height goes up, down, and then repeats the exact same cycle endlessly, so you are looking for a function family that naturally models cycles. That family is sinusoidal. Sine and cosine functions are built to repeat at regular intervals: as time increases, the height rises, falls, and returns to the same values in a smooth, periodic wave. This matches the Ferris wheel's constant rotation. Now look at the traps. "Logarithmic, because the increase slows and then reverses" misunderstands logarithmic growth: a logarithm increases forever, never reverses or repeats. "Quadratic, because height goes up and then down" describes only one rise-and-fall arc, like a ball thrown once, but a quadratic does not repeat that pattern indefinitely. "Logistic, because the height approaches a maximum and then a minimum" confuses logistic growth with a cycle: a logistic function approaches a single maximum level and levels off, then it stays there — it doesn't swing back down, and it certainly doesn't oscillate. Your study tip: whenever you see words like "repeats," "cycle," "periodic," or "oscillates," circle sinusoidal. If a graph or description only shows one turn, think quadratic; if it shows endless repetition, think sine or cosine.

Question 6

A savings account earns 4% interest per year, compounded annually. No additional deposits or withdrawals are made.

Which function family best models the account balance as a function of years since the initial deposit, and why?

Linear, because the balance increases by the same dollar amount each year.
Exponential, because each year the balance is multiplied by the constant factor $1.04$ . (correct answer)
Quadratic, because the interest is calculated on an ever-increasing balance.
Logarithmic, because the dollar amount of growth slows as the balance grows.

Explanation: Whenever you see a question about how an account balance grows, ask yourself: is the change additive or multiplicative each year? Here, 4% interest compounded annually means the balance is multiplied by

1.04

every year. After

t

years, the balance is

B(t)=B_0(1.04)^t,

where

B_0

is the initial deposit. That repeated multiplication by a constant factor is the signature of an exponential function, so the correct model is the exponential choice. The linear choice is tempting because interest is paid yearly, but linear growth would require the same dollar amount to be added each year. In reality, 4% of a larger balance is a larger dollar amount, so the yearly increase grows. The quadratic choice also misses the structure: "calculated on an ever-increasing balance" describes compounding, but the pattern is multiplicative, not polynomial. The logarithmic choice gets the direction wrong — logarithmic growth slows over time, but this balance grows faster in dollar terms each year, not slower. For exam day, remember: if a situation says "multiplied by a constant factor" or "increases by a fixed percent," model it exponentially. If it says "increases by a fixed dollar amount," model it linearly. The key is always deciding whether the change is by a constant amount or a constant factor.

Question 7

For small angles, the period of a pendulum is proportional to the square root of its length. A 1 m pendulum has a period of about 2.0 s, and a 4 m pendulum has a period of about 4.0 s.

Which function family best models period $P$ as a function of length $L$ ?

Linear, because doubling the length from 1 m to 2 m would double the period.
Exponential, because the period increases by a constant factor when length increases by a constant amount.
Quadratic, because $P=kL^2$ , so the period is proportional to the square of length.
Power, because $P=kL^{1/2}$ , so the period is a constant multiple of $\sqrt{L}$ . (correct answer)

Explanation: When a question describes a relationship in words like "proportional to the square root of," you should translate that directly into a function equation. Proportional means "constant multiple," so

P

is proportional to

L^{1/2}

means

P = kL^{1/2}

. This is a power function, because the independent variable

L

is raised to a constant exponent. Check the given data: if

P = k\sqrt{L}

, then using

P=2.0

when

L=1

gives

k=2.0

. For

L=4

P=2.0\sqrt{4}=4.0

, exactly matching the passage. So the correct model is the Power choice. Why the others fail: Linear would require

P = mL

, so doubling length doubles period; here doubling length from 1 m to 2 m should give about

2.0\sqrt{2}\approx 2.83

s, not 4.0 s. Exponential would require

P = ab^L

, where equal length increases multiply the period by a constant factor; the data do not show that, and the square-root relationship is not exponential. Quadratic would be

P = kL^2

, so with

k=2.0

from the first pair,

L=4

would give

P=32

s, not 4.0 s; it confuses squaring with square rooting. Your study takeaway: when you see "proportional to" plus a root or power, write

y = kx^n

and verify with the given values. That instantly identifies the function family.

Question 8

A delivery service charges a $5 pickup fee plus $2 per mile for the first 10 miles. Beyond 10 miles, the per-mile rate drops to $1.50, and the pickup fee still applies.

Which function family best models the total cost C as a function of distance d in miles?

Piecewise-linear, because the slope changes at 10 miles but the cost remains continuous. (correct answer)
Exponential, because the cost grows quickly for the first 10 miles and then slows.
Linear, because the cost is always a flat fee plus a constant rate per mile.
Quadratic, because the cost increases by a smaller amount with each mile after 10 miles.

Explanation: This question tests how you match a word problem to a function family. When a rule changes at a boundary, ask: What is the rate of change before and after? Does the cost jump at the boundary? For

0\le d\le 10

C(d)=5+2d

, so the slope is 2. At 10 miles, the cost is

5+20=25

. For

d>10

, the cost is

25+1.5(d-10)

, which simplifies to

1.5d+10

, so the slope is 1.5. Both pieces are straight lines, and both give 25 at

d=10

, so the graph is continuous while the slope changes: that is piecewise-linear. The exponential choice confuses "grows quickly then slows" with exponential growth; exponential growth means the rate itself changes multiplicatively, but here each interval has a fixed per-mile rate. The linear choice is only true within each interval, not over the whole domain, because the per-mile rate changes at 10 miles. The quadratic choice assumes the slope keeps changing every mile, but after 10 miles the slope is constant at 1.5, so that piece is linear. The piecewise-linear family captures both constant rates and the single break point. Study tip: Whenever a problem has a threshold like "first 10 miles" or "after 10 miles," write the formula for each side and test the boundary. If the slope changes but the boundary value matches, choose piecewise-linear.

Question 9

A manufacturer makes spherical beads. The material cost is proportional to the volume of a bead. In testing, doubling the radius makes the material cost 8 times as large.

Which function family best models material cost $C$ as a function of radius $r$ ?

Cubic, because doubling the radius multiplies the volume, and therefore the cost, by 8. (correct answer)
Quadratic, because doubling the radius quadruples the cost.
Linear, because doubling the radius doubles the cost.
Exponential, because cost grows by a factor of 8 when the radius doubles.

Explanation: When you see a statement like "doubling

r

makes the cost 8 times as large," your job is to connect that behavior to a function family. This is really about recognizing power functions and proportional reasoning. For a sphere, volume is

V=\frac{4}{3}\pi r^3

, so if you double

r

, the volume becomes

\frac{4}{3}\pi (2r)^3 = 8\left(\frac{4}{3}\pi r^3\right)

. The volume, and therefore the material cost, is multiplied by

8

. Since cost is proportional to volume,

C = k r^3

, which is a cubic model. The quadratic choice is a common trap: if cost grew like

r^2

, doubling

r

would quadruple the cost, not multiply it by

8

. The linear choice would mean doubling

r

doubles the cost, but the test says the factor is

8

, not

2

. The exponential choice is also incorrect: exponential functions grow by a constant factor for each equal step in

r

, but here the factor

8

is specifically tied to doubling

r

, and the underlying geometry is a power function, not an exponential one. A useful strategy: whenever a quantity is proportional to a geometric measure like volume, ask what power of the radius is involved. Doubling the radius and observing whether the output multiplies by

2

4

, or

8

tells you whether the function is linear, quadratic, or cubic. Here, the factor

8

points straight to

r^3

Question 10

A ball is dropped from rest. The total distance fallen is about 5 m after 1 second, 20 m after 2 seconds, 45 m after 3 seconds, and 80 m after 4 seconds.

Which function family best models distance $d$ as a function of time $t$ , and why?

Linear, because the ball moves at a constant speed throughout the fall.
Exponential, because the distance increases by a larger amount each second.
Quadratic, because the distance is proportional to $t^2$ under constant acceleration. (correct answer)
Cubic, because the first differences between consecutive distances are increasing.

Explanation: When you see a pattern of distances over equal time intervals, ask yourself what is changing between the rows. Here, the distances are 5, 20, 45, and 80. Notice that these are exactly

5 \cdot 1^2

5 \cdot 2^2

5 \cdot 3^2

, and

5 \cdot 4^2

, so

d = 5t^2

. That is the signature of a quadratic model. Physically, this makes sense: an object dropped from rest under constant acceleration has

d = \frac12 g t^2

, so distance is proportional to

t^2

. The linear choice is wrong because constant speed would mean equal distances in each 1-second interval, but the ball falls 15 m, then 25 m, then 35 m. The exponential choice is wrong because exponential growth would show a constant multiplier between consecutive distances; here the ratios are

4

2.25

, and

1.78

, not constant. The cubic choice is tempting because first differences

15, 25, 35

are increasing, but that only indicates the function is not linear. To confirm quadratic, check that the second differences are constant:

25-15=10

and

35-25=10

. A cubic would have increasing second differences, not constant ones. Study tip: when given data at equal intervals, always compute differences. Constant first differences mean linear, constant second differences mean quadratic, and constant third differences mean cubic. This test is faster and more reliable than just noticing that values are increasing.

Question 11

A theater currently charges $50 per ticket and averages 2000 attendees. For each $1 increase in ticket price, the theater expects to lose 40 attendees. Assume the relationship between price and attendance is linear.

Which function family best models total revenue $R$ as a function of ticket price $p$ ?

Linear, because revenue is price times a fixed number of tickets sold.
Quadratic, because revenue is price times attendance, and attendance is linear in price. (correct answer)
Exponential, because the loss of attendees compounds with each dollar increase.
Rational, because revenue is price divided by attendance, and attendance decreases linearly.

Explanation: Whenever you see a question about revenue, remember the basic structure: revenue is price times quantity sold. The key is to determine what happens to the quantity when price changes. Here, the attendance is described as linear in price: for each $1 increase, 40 fewer people attend. If

p

is the ticket price, attendance can be written as

A(p)=2000-40(p-50)

, which simplifies to a linear expression like

A(p)=4000-40p

. Total revenue is then

R(p)=p\cdot A(p)=p(4000-40p)=-40p^2+4000p

. Because revenue is the product of price and attendance, and attendance is linear in price,

R

is quadratic. The choice saying revenue is linear because it is price times a fixed number of tickets misses the fact that attendance is not fixed; it changes with price. The exponential choice misreads the steady loss of 40 attendees as compounding, but the problem gives a constant linear decrease, not a percentage. The rational choice reverses the relationship: revenue is price times attendance, not price divided by attendance, so a rational function does not model this situation. A strong study strategy is to ask yourself: "What changes when the input changes?" If a quantity like attendance is linear in price, and you multiply it by price to get revenue, the result will be quadratic. Watch for that pattern on exam day.

Question 12

Twelve volunteers can set up for an event in 3 hours. For any number of volunteers, the product of volunteers and time is constant, assuming everyone works at the same rate.

Which function family best models time $t$ as a function of the number of volunteers $n$ ?

Linear: $t=36-n$ , because each extra volunteer subtracts the same amount of time.
Exponential: $t=36(0.5)^n$ , because doubling volunteers halves the time.
Rational: $t=\frac{36}{n}$ , because time and number of volunteers are inversely proportional. (correct answer)
Logarithmic: $t=36\log(1/n)$ , because time decreases quickly at first and then slowly.

Explanation: Whenever you see a phrase like "the product of volunteers and time is constant," your first instinct should be inverse variation. That wording is a direct signal for a rational model of the form

t=\frac{k}{n}

. Here, the constant is

12 \times 3 = 36

, so the time as a function of volunteers is exactly

t=\frac{36}{n}.

n

increases,

t

decreases proportionally: doubling volunteers halves the time, which is the signature of an inverse relationship. The Linear choice

t=36-n

fails because it claims each extra volunteer subtracts the same number of hours. Actually, the time saved by adding one volunteer depends on how many volunteers you already have: going from 1 to 2 volunteers saves 18 hours, while going from 12 to 13 saves only about 0.23 hours. The Exponential choice

t=36(0.5)^n

also misses the point: at

n=12

, it gives

36(0.5)^{12}

, not 3, so it doesn't even fit the given pair. Exponential functions change by a constant multiplier per added volunteer, not by inverse proportion. The Logarithmic choice

t=36\log(1/n)

would give negative times for

n>1

, which is impossible, and logarithms describe slow growth, not inverse variation. On test day, when you see "product is constant" or "inversely proportional," write

xy=k

immediately. Then solve for the missing variable. That pattern will carry you straight to the rational function.

Question 13

A researcher plots $(\ln x,\ln y)$ for a data set with $x>0$ and finds the points lie on a line with slope $3$ and intercept $-1$ . Which function family is indicated?

Power: $y=e^{-1}x^3$ (correct answer)
Exponential: $y=e^{-1}e^{3x}$
Logarithmic: $y=-1+3\ln x$
Quadratic: $y=3x^2-1$

Explanation: The correct model is power. A linear plot of

\ln y

against

\ln x

means

\ln y=-1+3\ln x

, so

y=e^{-1}x^3

. The exponential option would follow from a linear plot of

\ln y

against

x

; the logarithmic option confuses the variables; the quadratic option has the wrong structure.

Question 14

A cup of coffee at $90^\circ$ C is placed in a $20^\circ$ C room. Every minute, the temperature difference between the coffee and the room decreases by 8% of its current value. Which function family best models the coffee's temperature $T(t)$ over time?

Linear: $T(t)=90-5.6t$
Quadratic: $T(t)=90-5.6t+0.2t^2$
Logarithmic: $T(t)=20+70\ln(t+1)$
Exponential decay: $T(t)=20+70(0.92)^t$ (correct answer)

Explanation: The correct model is exponential decay. The difference

T(t)-20

is multiplied by

0.92

each minute, so

T(t)-20=70(0.92)^t

. The linear option assumes a constant decrease of

5.6^\circ

per minute; the quadratic option adds a changing rate without the multiplicative decay; the logarithmic option grows without bound and does not match the stated percent decrease.

Question 15

A manufacturer records the average cost per unit $C(n)$ for producing $n$ items: $C(100)=8.00$ , $C(200)=5.50$ , $C(400)=4.25$ , $C(800)=3.625$ . The values continue to decrease and appear to approach $3.00 as $n$ grows. Which function family best models $C(n)$ ?

Exponential decay, because each doubling of $n$ reduces the cost by a smaller amount.
Power, because the graph is decreasing and concave up for all large $n$ .
Rational, because a fixed setup cost is spread over more units: $C(n)=3+\frac{500}{n}$ . (correct answer)
Logarithmic, because the cost decreases slowly and approaches a horizontal asymptote.

Explanation: The correct model is rational. The values match

C(n)=3+\frac{500}{n}

: at

n=100

C=8

; at

n=200

C=5.5

; at

n=400

C=4.25

; at

n=800

C=3.625

. This is the form of a rational function with a horizontal asymptote at

3

. The exponential option would require a constant percent change over equal steps in

n

; the power and logarithmic options do not produce the exact pattern.

Question 16

A student fits a linear model to the height of a ball versus time and then examines the residuals. As time increases, the residuals are negative, then positive, then negative. Which conclusion is most justified?

The linear model is appropriate because the residuals are not all the same sign.
The data should be modeled by a logarithmic function because residuals change sign.
The data should be modeled by a quadratic function because the linear fit misses the curvature. (correct answer)
The data should be modeled by an exponential function because the residuals form a curve.

Explanation: The correct model is quadratic. The context is a ball's height over time, which has a curved rise-and-fall shape, and the residual pattern from a linear fit shows that the line misses that curvature. The linear option mistakes a patterned residual plot for randomness; the logarithmic and exponential options do not match a rise-and-fall height model.

Question 17

A student makes four claims about function models. Which claim is correct?

A car's value that decreases by $2,000 each year is linear, because the dollar change is constant. (correct answer)
A population that grows by 4% each year is linear, because the same percent is added each year.
A bouncing ball that rebounds to 80% of its previous height is linear, because equal time intervals are used.
A bacteria culture that doubles every hour is linear, because the increase doubles each hour.

Explanation: The correct claim is the car with a constant $2,000 yearly loss. A constant dollar change is the signature of a linear model. Constant percent change and constant multipliers are signatures of exponential models, so the population, bouncing ball, and bacteria claims are exponential, not linear.

Question 18

A yeast population in a closed container grows by about 30% per hour for the first several hours, but then the growth rate steadily slows and the population levels off near 5000. Which function family best models the population over a 24-hour period?

Exponential growth, because the initial growth rate is 30% per hour for the first several hours.
Quadratic growth, because the growth rate decreases after the population is large.
Logarithmic growth, because the population increases slowly and approaches a limit.
Logistic growth, because the population grows rapidly at first and then approaches a carrying capacity. (correct answer)

Explanation: The correct model is logistic growth. Logistic growth has an early period of rapid, nearly exponential increase, then an inflection point, and finally levels off at a carrying capacity. The exponential option ignores the leveling off; the quadratic option does not have a carrying capacity; the logarithmic option increases without bound and does not have the early rapid-growth phase.

Question 19

A student records a quantity $y$ at equally spaced $x$ -values and computes the first differences: $4, 6, 8, 10$ . The student claims the data are exponential because the first differences are increasing. Which statement best evaluates this claim?

The claim is correct; increasing first differences are the signature of exponential growth.
The claim is incorrect; the second differences are constant, so a quadratic model is more appropriate. (correct answer)
The claim is incorrect; increasing first differences indicate a linear model with positive slope.
The claim is incorrect; the data are best modeled by a logarithmic function because the increases are getting larger.

Explanation: The correct model is quadratic. The first differences

4,6,8,10

increase by

2

each time, so the second differences are constant. Constant second differences are the finite-difference signature of a quadratic function. Exponential growth would require a roughly constant multiplicative ratio between consecutive

y

-values, not merely increasing differences. The linear option is wrong because first differences would be constant; the logarithmic option is wrong because the increases are getting larger, not smaller.

Question 20

For a data set $(x,y)$ , a student plots $(x,\ln y)$ and finds that the points lie almost exactly on a line with slope $0.12$ and intercept $2.1$ . Which function family best models $y$ as a function of $x$ ?

Linear: $y=0.12x+2.1$
Power: $y=e^{2.1}x^{0.12}$
Exponential: $y=e^{2.1}e^{0.12x}$ (correct answer)
Logarithmic: $y=2.1+0.12\ln x$

Explanation: The correct model is exponential. A linear plot of

\ln y

against

x

means

\ln y=2.1+0.12x

. Exponentiating gives

y=e^{2.1}e^{0.12x}

, which is an exponential function. The linear option ignores the logarithm; the power option would require a linear plot of

\ln y

against

\ln x

; the logarithmic option would require

y

itself to be linear in

\ln x