Math 1 Quiz: Constructing Functions From Context

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Constructing Functions From ContextQuestion 1 of 20

A population of bacteria doubles every 3 hours. If there are initially 200 bacteria, which function $P(t)$ models the population after $t$ hours?

P(t) = 200 + 2^{t/3}

P(t) = 200 \cdot 2^{3t}

P(t) = 200 \cdot 3^{t/2}

P(t) = 200 \cdot 2^{t/3}

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Math 1 Quiz

Math 1 Quiz: Constructing Functions From Context

Practice Constructing Functions From Context in Math 1 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 Constructing Functions From Context, giving you a quick way to practice the rules, question types, and explanations that matter most for Math 1.

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.

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Question 1

A population of bacteria doubles every 3 hours. If there are initially 200 bacteria, which function $P(t)$ models the population after $t$ hours?

$P(t) = 200 + 2^{t/3}$
$P(t) = 200 \cdot 2^{3t}$
$P(t) = 200 \cdot 3^{t/2}$
$P(t) = 200 \cdot 2^{t/3}$ (correct answer)

Explanation: This question tests exponential growth models, where a quantity changes by a constant factor over regular time intervals. When you see "doubles every [time period]," you're dealing with exponential growth with base 2. The general form for exponential growth is

P(t) = P_0 \cdot b^{t/k}

, where

P_0

is the initial amount,

b

is the growth factor, and

k

is the time period for one complete growth cycle. Here, the bacteria population starts at 200, doubles (factor of 2), and completes this doubling every 3 hours. So we need

P(t) = 200 \cdot 2^{t/3}

. Let's verify: at

t = 0

P(0) = 200 \cdot 2^0 = 200

. At

t = 3

P(3) = 200 \cdot 2^{3/3} = 200 \cdot 2^1 = 400

. The population correctly doubles from 200 to 400 after 3 hours. Choice A (

200 + 2^{t/3}

) represents linear growth plus an exponential term, not pure exponential growth. Choice B (

200 \cdot 2^{3t}

) has the exponent backwards—this would mean the population doubles every

\frac{1}{3}

hour instead of every 3 hours, causing explosive growth. Choice C (

200 \cdot 3^{t/2}

) uses the wrong base (3 instead of 2) and wrong time period (2 instead of 3). The correct answer is D. Study tip: For exponential growth problems, always check your function at

t = 0

(initial value) and at one complete growth period. This quick verification catches most common mistakes with the base and exponent structure.

Question 2

A population of bacteria doubles every 4 hours. If there are initially 250 bacteria, which function represents the population $P(t)$ after $t$ hours?

$P(t) = 250 \cdot 4^{t/2}$
$P(t) = 250 \cdot 2^{4t}$
$P(t) = 250 + 2t$
$P(t) = 250 \cdot 2^{t/4}$ (correct answer)

Explanation: When you encounter exponential growth problems, you're dealing with situations where a quantity multiplies by a constant factor at regular intervals. The key is translating the given information into the correct exponential function form:

P(t) = P_0 \cdot b^{t/d}

, where

P_0

is the initial amount,

b

is the growth factor, and

d

is the time period for one complete cycle. Here, the bacteria population doubles (multiplies by 2) every 4 hours, starting with 250 bacteria. Since doubling means the growth factor is 2, and this happens every 4 hours, you need

P(t) = 250 \cdot 2^{t/4}

. The exponent

t/4

ensures that when

t = 4

, you get

2^1 = 2

, doubling the population exactly once. Choice A (

250 \cdot 4^{t/2}

) incorrectly uses 4 as the base instead of 2, and has the wrong time scaling. Choice B (

250 \cdot 2^{4t}

) has the reciprocal time relationship—this would mean the population doubles every

1/4

hour instead of every 4 hours, creating explosive growth. Choice C (

250 + 2t

) represents linear growth, not exponential growth, where the population would only increase by 2 bacteria per hour. For exponential growth problems, always identify three things: the initial value, the growth factor, and the time period. Then construct your function as

\text{initial} \cdot \text{factor}^{\text{time}/\text{period}}

. Double-check by substituting the given time period to verify you get exactly one growth cycle.

Question 3

A rectangular garden has a perimeter of 100 feet. If the length is $x$ feet, which function $A(x)$ represents the area of the garden in terms of its length?

$A(x) = x(100 - x)$
$A(x) = x(50 - x)$ (correct answer)
$A(x) = x(100 - 2x)$
$A(x) = 2x(50 - x)$

Explanation: Given perimeter = 100, we have

2x + 2w = 100

, so

w = 50 - x

. The area is

A(x) = x \cdot w = x(50 - x)

. Choice A incorrectly uses the full perimeter instead of solving for width. Choice C uses

100 - 2x

which would be correct for width if the perimeter equation were

x + w = 100

. Choice D includes an extra factor of 2, confusing area with perimeter calculations.

Question 4

A store offers a discount structure where customers get 5% off their total if they spend at least $100, and an additional 3% off the discounted amount if they spend at least $200. Which function $ $D(x)$ $gives the amount a customer pays for a purchase of$ $x$ $ dollars?

(correct answer)

Explanation: For

x \geq 200

: First apply 5% discount:

0.95x

. Then apply additional 3% discount to the already discounted amount:

0.95x \times 0.97 = 0.9215x

. Choice A incorrectly calculates the second discount as 8% total instead of applying 3% to the discounted amount. Choice C uses fixed dollar amounts instead of percentages. Choice D shows the calculation incorrectly as

0.95x - 0.03x = 0.92x

, applying the 3% to the original amount rather than the discounted amount.

Question 5

A spring's length depends on the weight attached to it. With no weight, the spring is 8 inches long. Each pound of weight stretches the spring an additional 0.5 inches, but the spring cannot stretch beyond 15 inches total. Which function $L(w)$ models the spring's length for weight $w$ pounds?

(correct answer)
$L(w) = \min(8 + 0.5w, 15)$ for $w \geq 0$

Explanation: When you encounter a piecewise function modeling a real-world constraint, you need to identify the natural behavior and where limitations kick in. This spring follows a linear relationship until it hits a physical maximum. The spring starts at 8 inches with no weight, then stretches 0.5 inches per pound:

L = 8 + 0.5w

. However, it can't exceed 15 inches total length. To find when this limit is reached, solve:

8 + 0.5w = 15

, which gives

w = 14

pounds. So the spring follows the linear formula

8 + 0.5w

when

0 \leq w \leq 14

, and stays constant at 15 inches when

w > 14

. This matches choice C perfectly. Choice A has the right idea but poor execution—the condition "if

8 + 0.5w > 15

" in the second piece should specify the weight range, not repeat the length formula. Choice B incorrectly uses

w \leq 15

and

w > 15

as the boundary conditions. The weight limit isn't 15 pounds; that's the maximum length in inches. The actual weight limit is 14 pounds. Choice D uses a minimum function, but

\min(8 + 0.5w, 15)

would give the smaller value, which doesn't model this situation correctly—we want the spring length, not the minimum of length and 15. Study tip: For piecewise functions with physical constraints, always solve for the exact point where the limitation kicks in. Don't assume the constraint value appears directly as a boundary in the domain.

Question 6

A water tank starts with 500 gallons and drains at a rate of 8 gallons per minute for the first 30 minutes, then drains at 5 gallons per minute thereafter. Which function $f(t)$ correctly models the amount of water in the tank after $t$ minutes?

(correct answer)

Explanation: For

t \leq 30

: The tank loses 8 gallons per minute, so

f(t) = 500 - 8t

. After 30 minutes, the tank has

500 - 8(30) = 260

gallons. For

t > 30

: The tank continues draining at 5 gallons per minute from the 260-gallon level, so we subtract

5(t-30)

additional gallons:

f(t) = 260 - 5(t-30) = 500 - 240 - 5t + 150 = 500 - 8(30) - 5(t-30)

. Choice B ignores the water already drained in the first 30 minutes. Choice C uses the wrong starting amount for the second phase. Choice D incorrectly adds the rates together.

Question 7

A taxi company charges a base fare plus a rate per mile. The total cost for a 5-mile trip is $18, and the total cost for a 12-mile trip is $32.

Which function $C(m)$ represents the total cost for a trip of $m$ miles?

$C(m) = 2m + 8$ (correct answer)
$C(m) = 3m + 3$
$C(m) = 2.67m + 4.67$
$C(m) = 8m + 2$

Explanation: Let

C(m) = b + rm

where

b

is base fare and

r

is rate per mile. From the given information:

b + 5r = 18

and

b + 12r = 32

. Subtracting:

7r = 14

, so

r = 2

. Substituting back:

b + 5(2) = 18

, so

b = 8

. Therefore

C(m) = 8 + 2m = 2m + 8

. Choice B gives costs of $18 and $39 for the given trips. Choice C uses incorrect decimal approximations. Choice D reverses the base fare and rate per mile.

Question 8

A projectile is launched from the ground with initial velocity $v_0$ feet per second. Its height after $t$ seconds is $h(t) = v_0 t - 16t^2$ . If the projectile reaches its maximum height of 64 feet, which function correctly models this specific projectile's height?

$h(t) = 128t - 16t^2$
$h(t) = 32t - 16t^2$
$h(t) = 64t - 16t^2$ (correct answer)
$h(t) = 16t - 16t^2$

Explanation: When you encounter projectile motion problems, you're working with quadratic functions where the maximum or minimum occurs at the vertex. The general form

h(t) = v_0 t - 16t^2

represents height over time, where

v_0

is the initial velocity and

-16t^2

accounts for gravity's effect. To find the maximum height, you need to locate the vertex of this parabola. For a quadratic

h(t) = at^2 + bt + c

, the vertex occurs at

t = -\frac{b}{2a}

. In our case,

h(t) = -16t^2 + v_0 t

, so

a = -16

and

b = v_0

. The time at maximum height is

t = \frac{v_0}{32}

. Substituting this back into the height equation:

h_{max} = v_0 \cdot \frac{v_0}{32} - 16 \cdot \left(\frac{v_0}{32}\right)^2 = \frac{v_0^2}{32} - \frac{v_0^2}{64} = \frac{v_0^2}{64}

Since the maximum height is 64 feet:

\frac{v_0^2}{64} = 64

, which gives us

v_0^2 = 4096

, so

v_0 = 64

. Therefore, the correct function is

h(t) = 64t - 16t^2

, which is choice C. Choice A (

v_0 = 128

) would give a maximum height of 256 feet. Choice B (

v_0 = 32

) yields a maximum of 16 feet. Choice D (

v_0 = 16

) produces a maximum of only 4 feet. Remember: for projectile problems, always find the vertex to determine maximum height, then work backward to find the initial velocity that produces your given maximum.

Question 9

A cylindrical water tank has radius 4 feet. Water is pumped in at 8 cubic feet per minute. Which function $h(t)$ represents the height of water in the tank after $t$ minutes, assuming the tank starts empty?

$h(t) = \frac{2t}{\pi}$
$h(t) = \frac{8t}{4\pi}$
$h(t) = \frac{8t}{8\pi}$
$h(t) = \frac{8t}{16\pi}$ (correct answer)

Explanation: This is a related rates problem involving the volume of a cylinder. When you see questions about filling containers, think about the relationship between volume, rate of change, and the container's dimensions. To find the height function, start with the volume formula for a cylinder:

V = \pi r^2 h

. Since the radius is 4 feet, we have

V = \pi (4^2) h = 16\pi h

. Water enters at 8 cubic feet per minute, so after

t

minutes, the total volume is

V = 8t

. Setting these equal:

8t = 16\pi h

. Solving for height:

h = \frac{8t}{16\pi}

, which matches answer D. Let's examine why the other answers are incorrect. Answer A,

h(t) = \frac{2t}{\pi}

, appears to use an incorrect radius calculation—possibly confusing radius with diameter or making an arithmetic error with the area formula. Answer B,

h(t) = \frac{8t}{4\pi}

, uses

4\pi

in the denominator, suggesting the student used

r = 2

instead of

r = 4

when calculating the area. Answer C,

h(t) = \frac{8t}{8\pi}

, uses

8\pi

in the denominator, which might result from using

r^2 = 8

instead of

r^2 = 16

. Study tip: For cylinder filling problems, always write out

V = \pi r^2 h

first, then substitute the given radius carefully. Double-check that

r^2

calculation—it's the most common source of error in these problems.

Question 10

A company's revenue $R$ is related to the number of employees $n$ by $R = 50000n - 1000n^2$ . The company's costs are $30000 per employee plus $200000 in fixed costs. Which function $ $P(n)$ $ represents the company's profit?

$P(n) = 50000n - 1000n^2 - 30000n + 200000$
$P(n) = -1000n^2 + 80000n - 200000$
$P(n) = -1000n^2 + 50000n - 230000$
$P(n) = -1000n^2 + 20000n - 200000$ (correct answer)

Explanation: When you encounter profit problems, remember that profit equals revenue minus total costs. You need to identify all revenue sources and all cost components, then set up the subtraction correctly. The company's revenue is given as

R = 50000n - 1000n^2

. For costs, you have two parts: variable costs of $30,000 per employee (which equals $

30000n

) and fixed costs of $200,000. So total costs = $

30000n + 200000$$. Setting up the profit function:

P(n) = \text{Revenue} - \text{Total Costs} = (50000n - $1000n^2$) - (30000n + 200000)

Distributing the negative sign:

P(n) = 50000n - 1000n^2 - 30000n - 200000

Combining like terms:

P(n) = -1000n^2 + (50000n - 30000n) - 200000 = -1000n^2 + 20000n - 200000

This matches answer choice D. Let's examine why the other options are incorrect: A)

P(n) = 50000n - 1000n^2 - 30000n + 200000

adds the fixed costs instead of subtracting them, treating costs as revenue. B)

P(n) = -1000n^2 + 80000n - 200000

incorrectly adds the variable costs to revenue (

50000 + 30000 = 80000

) instead of subtracting them. C)

P(n) = -1000n^2 + 50000n - 230000

fails to account for variable costs at all, only subtracting fixed costs from revenue. Study tip: In profit problems, always write out "Profit = Revenue - Costs" first, then carefully identify and subtract each cost component. Double-check your signs when distributing negatives.

Question 11

A manufacturer's cost to produce $x$ items includes a setup cost of $500 plus $12 per item. Each item sells for $20. However, if more than 100 items are produced, the per-item production cost decreases to $10 for all items. Which function $ $P(x)$ $represents the profit from producing and selling$ $x$ $ items?

(correct answer)

Explanation: When you encounter profit problems with changing cost structures, remember that profit equals revenue minus total cost, and you need to account for different pricing tiers separately. Let's build the profit function step by step. Revenue is always

20x

since each item sells for $20. For costs, you have a $500 setup cost plus variable costs that change at 100 items. For $

0 \leq x \leq 100

: Total cost =

500 + 12x

, so profit =

20x - (500 + 12x) = 8x - 500

$ For

x > 100

: Total cost =

500 + 10x

(the $10 per item applies to ALL items when production exceeds 100), so profit = $$20x - (500 + 10x) = 10x - 500$$ This matches answer choice C. Answer A incorrectly calculates the second piece. It uses

8(100) + 10(x-100) - 500

, which assumes the first 100 items still cost $12 each and only additional items cost $10. The problem states that when production exceeds 100, the per-item cost "decreases to $10 for all items." Answer B shows the correct setup ($$20x - 12x - 500

and

20x - 10x - 500

) but fails to simplify, leaving it as separate terms rather than the simplified forms

8x - 500

and

10x - 500$$. Answer D uses the same logic as the correct answer but presents it in unsimplified form with parentheses around

(20-12)

and

(20-10)

, making it unnecessarily complex. Study tip: In piecewise cost problems, pay careful attention to whether cost changes apply to all units or only additional units beyond the threshold.

Question 12

A ball is thrown upward from a building. Its height in feet after $t$ seconds is given by $h(t) = -16t^2 + 64t + 80$ . A second ball is thrown from ground level 2 seconds later with initial velocity 48 ft/s. Which function $d(t)$ represents the distance between the two balls for $t \geq 2$ ?

$d(t) = |-16t^2 + 64t + 80 - (-16(t-2)^2 + 48(t-2))|$ (correct answer)
$d(t) = |-16t^2 + 64t + 80 - (-16t^2 + 48t)|$
$d(t) = |(-16t^2 + 64t + 80) + (-16(t-2)^2 + 48(t-2))|$
$d(t) = |-16t^2 + 16t + 80 - (-16(t-2)^2 + 48(t-2))|$

Explanation: The first ball's height is

h_1(t) = -16t^2 + 64t + 80

. The second ball starts at

t = 2

from ground level with velocity 48 ft/s, so its height is

h_2(t) = -16(t-2)^2 + 48(t-2)

for

t \geq 2

. The distance between them is

d(t) = |h_1(t) - h_2(t)|

. Choice B ignores the 2-second delay. Choice C adds the heights instead of finding their difference. Choice D has an error in the first ball's velocity coefficient.

Question 13

A company's profit depends on the number of units sold. The company breaks even (zero profit) when selling 20 units or 80 units. The maximum profit occurs when selling 50 units, where the profit is $9000. Assuming the profit function is quadratic, which function models the company's profit $ $P(x)$ $based on$ $x$ $ units sold?

$P(x) = -6(x - 50)^2 + 9000$
$P(x) = -10(x - 50)^2 + 9000$ (correct answer)
$P(x) = -6(x - 20)(x - 80)$
$P(x) = 6(x - 20)(x - 80)$

Explanation: The quadratic has zeros at

x = 20

and

x = 80

, with vertex at

(50, 9000)

. Using vertex form

P(x) = a(x - 50)^2 + 9000

, we find

a

using a zero:

0 = a(20 - 50)^2 + 9000 = 900a + 9000

, so

a = -10

. Therefore

P(x) = -10(x - 50)^2 + 9000

. Choice A uses the wrong coefficient. Choice C has the correct zeros but wrong vertex value (it gives

P(50) = -5400

). Choice D opens upward, contradicting the maximum at the vertex.

Question 14

A rectangular garden has a perimeter of 80 feet. If the width is $w$ feet, which function represents the area $A(w)$ of the garden in terms of its width?

$A(w) = w(80 - w)$
$A(w) = w(40 - w)$ (correct answer)
$A(w) = 2w(40 - w)$
$A(w) = w(80 - 2w)$

Explanation: With perimeter 80, we have 2w + 2l = 80, so l = 40 - w. The area is A = wl = w(40 - w). Choice A uses the full perimeter instead of half. Choice C includes an extra factor of 2. Choice D uses 80 - 2w, which would give l = 80 - 2w, making the perimeter 2w + 2(80 - 2w) = 160, not 80.

Question 15

A phone plan charges a monthly fee plus a rate per gigabyte of data used. Sarah's bill was $45 for using 3 GB and $65 for using 8 GB. Which function represents the total cost $ $C(g)$ $for using$ $g$ $ gigabytes of data?

$C(g) = 4g + 33$ (correct answer)
$C(g) = 4g + 45$
$C(g) = 15g - 20$
$C(g) = 20g - 15$

Explanation: The rate per GB is (65 - 45)/(8 - 3) = 20/5 = $4 per GB. Using point-slope form with (3, 45): C(g) - 45 = 4(g - 3), so C(g) = 4g + 33. We can verify: C(3) = 4(3) + 33 = 45 ✓ and C(8) = 4(8) + 33 = 65 ✓. Choice B uses the cost at 3 GB as the y-intercept. Choice C uses the difference in costs (20) as the slope. Choice D reverses the calculation.

Question 16

A chemistry student is studying the relationship between temperature and the volume of a gas sample. The student records that at 300 K, the volume is 2.4 liters, and at 350 K, the volume is 2.8 liters. The relationship follows a linear pattern.

Which function best represents the volume $V(T)$ in liters as a function of temperature $T$ in Kelvin?

$V(T) = 0.008T + 0$
$V(T) = 0.008T - 0.6$ (correct answer)
$V(T) = 0.8T - 237.6$
$V(T) = 0.08T - 21.6$

Explanation: The slope is (2.8 - 2.4)/(350 - 300) = 0.4/50 = 0.008 L/K. Using point-slope form with (300, 2.4): V(T) - 2.4 = 0.008(T - 300), so V(T) = 0.008T - 2.4 + 2.4 = 0.008T - 0.6. Choice A assumes the line passes through origin. Choice C uses 0.8 instead of 0.008 for slope. Choice D uses 0.08 instead of 0.008 for slope.

Question 17

A company's profit depends on the number of items sold. The profit is $800 when 20 items are sold, and the profit decreases by $15 for each additional item sold beyond 20 due to bulk discounts. Which function represents the profit $ $P(x)$ $when$ $x$ $items are sold, where$ $x ≥ 20$ $?

$P(x) = -15x + 1100$
$P(x) = 15x + 500$
$P(x) = -15(x - 20) + 800$ (correct answer)
$P(x) = -15x + 800$

Explanation: The profit decreases by $15 for each item beyond 20, so the function is P(x) = 800 - 15(x - 20). This can also be written as P(x) = -15(x - 20) + 800. Choice A incorrectly assumes the decrease starts from x = 0. Choice B has the wrong sign and wrong y-intercept. Choice D doesn't account for the baseline of 20 items.

Question 18

A water tank is being filled at a constant rate. After 5 minutes, the tank contains 120 gallons. After 12 minutes, it contains 204 gallons. If the tank was not empty when filling began, which function represents the total amount of water $W(t)$ in gallons after $t$ minutes of filling?

$W(t) = 12t + 60$ (correct answer)
$W(t) = 12t + 48$
$W(t) = 24t + 12$
$W(t) = 84t - 300$

Explanation: The rate of filling is (204 - 120)/(12 - 5) = 84/7 = 12 gallons per minute. Using point-slope form with (5, 120): W(t) - 120 = 12(t - 5), so W(t) = 12t - 60 + 120 = 12t + 60. Choice B uses an incorrect initial amount. Choice C doubles the rate and uses wrong initial amount. Choice D uses the total change (84) as the rate.

Question 19

A taxi company charges a base fare plus a rate per mile. The total cost for a 5-mile trip is $12, and for a 12-mile trip is $23.50. If a customer wants to spend exactly $30, which equation can be used to find the maximum distance $ $d$ $ they can travel?

$30 = 1.5d + 4.5$
$30 = 2.5d + 12$
$30 = \frac{23}{14}d + \frac{17}{14}$ (correct answer)
$30 = 11.5d - 45.5$

Explanation: The rate per mile is (23.50 - 12)/(12 - 5) = 11.5/7 = 23/14 dollars per mile. Using point (5, 12): C(d) - 12 = (23/14)(d - 5), so C(d) = (23/14)d - 115/14 + 12 = (23/14)d + 17/14. To find distance for $30: 30 = (23/14)d + 17/14. Choice A uses an incorrect rate. Choice B uses wrong calculations. Choice D uses 11.5 as the rate instead of 11.5/7.

Question 20

A projectile is launched from the top of a 64-foot building with an initial upward velocity of 48 feet per second. The height of the projectile follows the general form h(t) = -16t² + v₀t + h₀, where v₀ is initial velocity and h₀ is initial height.

Which function represents the height $h(t)$ of the projectile $t$ seconds after launch, and what does the coefficient of $t$ represent in this context?

$h(t) = -16t^2 + 48t + 64$ ; the coefficient 48 represents the acceleration due to gravity
$h(t) = -16t^2 + 48t + 64$ ; the coefficient 48 represents the initial upward velocity (correct answer)
$h(t) = -16t^2 + 64t + 48$ ; the coefficient 64 represents the initial upward velocity
$h(t) = -48t^2 + 16t + 64$ ; the coefficient 16 represents the initial upward velocity

Explanation: Substituting the given values: h(t) = -16t² + 48t + 64. The coefficient of t (which is 48) represents the initial upward velocity in ft/s. Choice A has the correct function but wrong interpretation of the coefficient. Choice C switches the initial velocity and height values. Choice D switches the gravity and velocity coefficients.