6th Grade Math Quiz: Analyze Dependent And Independent Variables

20 questions · exam conditions

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Analyze Dependent And Independent VariablesQuestion 1 of 20

A student earns $4 for each lawn they mow. Let $n $be the number of lawns mowed and let$ m $be the money earned (in dollars). Which equation shows$ m $in terms of$ n$?

m = 4n

n = 4m

m = 4 + n

m = n + 4

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6th Grade Math Quiz

6th Grade Math Quiz: Analyze Dependent And Independent Variables

Practice Analyze Dependent And Independent Variables in 6th Grade Math 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 Analyze Dependent And Independent Variables, giving you a quick way to practice the rules, question types, and explanations that matter most for 6th Grade Math.

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 student earns $4 for each lawn they mow. Let $n $be the number of lawns mowed and let$ m $be the money earned (in dollars). Which equation shows$ m $in terms of$ n$?

$m = 4n$ (correct answer)
$n = 4m$
$m = 4 + n$
$m = n + 4$

n

mowed, items we buy—we control), dependent is output/depends on independent (money

m

depends on lawns, cost depends on items—results from independent). Equation: express dependent in terms of independent (

m=4n

: money equals 4 times lawns, dependent

m

on left, independent

n

in expression). Table: list independent values (

n

: 0,1,2,3), calculate dependent using equation (if

n=1

m=4 \times 1=4

;

n=2

m=8

; etc.). Graph: independent on x-axis (horizontal: lawns), dependent on y-axis (vertical: money), plot ordered pairs (

(1,4), (2,8), \dots

), proportional

m=kn

graphs through origin. For this lawn mowing example, independent=number of lawns

n

, dependent=money

m

, equation

m=4n

, table

n

:0,1,2

m

:0,4,8, graph x=lawns, y=money, line through (0,0) with slope 4. The correct equation is

m=4n

as in choice C, showing dependent

m

in terms of independent

n

; errors like B reverse variables, A/D use addition instead of multiplication.

Question 2

A gym membership costs $10 to sign up plus $5 each month. Let $m $be the number of months and$ C$ be the total cost. Which description correctly connects the equation, table, and graph for this situation?

Equation $C=10m+5$ ; table increases by 10 each month; graph has y-intercept 5.
Equation $C=5m+10$ ; table starts at $C=10$ when $m=0$ and increases by 5 each month; graph is a line with y-intercept 10. (correct answer)
Equation $C=m+15$ ; table increases by 15 each month; graph is a curve.
Equation $C=5m$ ; table starts at $C=0$ when $m=0$ ; graph is a line through the origin with slope 5.

Explanation: Tests identifying independent (input) and dependent (output) variables, writing equations relating them, creating tables, graphing with proper axes, and connecting equation-table-graph representations. Variables: independent is input/chosen first (months m subscribed, periods we track—we control), dependent is output/depends on independent (cost C depends on months, result depends on period—results from independent). Equation: express dependent in terms of independent (C=10+5m: cost equals 10 plus 5 times months, dependent C on left, independent m in expression). Table: list independent values (m: 0,1,2,3), calculate dependent using equation (if m=1, C=10+5×1=15; m=2, C=20; etc.). Graph: independent on x-axis (horizontal: months), dependent on y-axis (vertical: cost), plot ordered pairs ((1,15), (2,20),...), line with y-intercept 10. For this gym membership, independent=months m, dependent=cost C, equation C=5m+10, table m:0,1,2 C:10,15,20 increasing by 5, graph line with slope 5 and y-intercept 10. The correct description connects equation C=5m+10, table starting at 10 and increasing by 5, graph with y-intercept 10 as in C; errors like A omit intercept, B/D swap or misstate values.

Question 3

A student earns $5 for each hour they help at an event. Let $h$ be the number of hours worked and let $m$ be the money earned (in dollars). Which ordered pairs should be graphed to represent this relationship for $h=0,1,2,3$ ?

$(0,5),(1,10),(2,15),(3,20)$
$(0,0),(5,1),(10,2),(15,3)$
$(5,0),(10,1),(15,2),(20,3)$
$(0,0),(1,5),(2,10),(3,15)$ (correct answer)

Explanation: This question tests identifying independent (input) and dependent (output) variables, writing equations relating them, creating tables, graphing with proper axes, and connecting equation-table-graph representations. Variables: independent is input/chosen first (hours h worked—we control), dependent is output/depends on independent (money m depends on hours—results from independent). Equation: express dependent in terms of independent (m=5h: money equals 5 times hours, dependent m on left, independent h in expression). Table: list independent values (h: 0,1,2,3), calculate dependent using equation (if h=0, m=0; h=1, m=5; etc.). Graph: independent on x-axis (horizontal: hours), dependent on y-axis (vertical: money), plot ordered pairs ((0,0), (1,5),...), proportional through origin. For example, earning $4 per chore, independent=chores c, dependent=earnings e, equation e=4c, table c:0,1,2 e:0,4,8, graph: x-chores, y-earnings, points (0,0),(1,4),(2,8) line through origin slope 4. The correct ordered pairs are (0,0),(1,5),(2,10),(3,15), with (h,m) format. Errors like reversing pairs ((5,1) instead of (1,5)) or starting without (0,0) for proportional. Analyzing: (1) identify relationship (money depends on hours at $5 each), (2) determine independent (h) and dependent (m), (3) write equation (m=5h), (4) create table (h:0,1,2,3; m:0,5,10,15), (5) graph (x=h, y=m, plot pairs), (6) connect (pairs match equation: 5*1=5✓). Axes convention: independent horizontal (x-axis: hours), dependent vertical (y-axis: money).

Question 4

A movie theater charges a $12 ticket plus $2 per snack. Let $s$ be the number of snacks and let $c$ be the total cost in dollars. Which choice correctly identifies the independent and dependent variables?

Independent: $12$ ; Dependent: $2$
Independent: $c$ (total cost); Dependent: $s$ (snacks)
Independent: $s$ (snacks); Dependent: $c$ (total cost) (correct answer)
Independent: $c$ (total cost); Dependent: $12$ (ticket price)

Explanation: This question tests identifying independent (input) and dependent (output) variables, writing equations relating them, creating tables, graphing with proper axes, and connecting equation-table-graph representations. Variables: independent is input/chosen first (snacks s we buy—we control), dependent is output/depends on independent (total cost c depends on snacks plus fixed ticket—results from independent). Equation: express dependent in terms of independent (c=12+2s: cost equals 12 plus 2 times snacks, dependent c on left, independent s in expression). For example, similar to movie with $10 ticket plus $3 drink, independent=drinks d, dependent=cost c, equation c=10+3d, table d:0,1,2 c:10,13,16, graph: x-drinks, y-cost, points (0,10),(1,13) line with y-intercept 10, slope 3. The correct identification is independent s (snacks), dependent c (total cost), as we choose snacks and cost results. Errors include reversing (cost independent, snacks dependent—backward) or misidentifying constants as variables. Analyzing: (1) identify relationship (cost depends on snacks with fixed $12 plus $2 each), (2) determine independent (s: we choose number) and dependent (c: result), (3) write equation (c=12+2s), (4) create table (s:0,1,2; c:12,14,16), (5) graph (x=s, y=c, plot pairs, line through (0,12)), (6) connect (equation matches table✓, slope=2✓). Axes convention: independent horizontal (x-axis: snacks), dependent vertical (y-axis: cost).

Question 5

A school store sells notebooks for $3 each. Let $n $be the number of notebooks bought and let$ c$ be the total cost (in dollars). Which statement correctly identifies the independent and dependent variables and gives an equation for the relationship?

Independent: $c$ ; Dependent: $n$ ; Equation: $n=3c$
Independent: $n$ ; Dependent: $c$ ; Equation: $c=3n$ (correct answer)
Independent: $n$ ; Dependent: $c$ ; Equation: $c=n+3$
Independent: $c$ ; Dependent: $n$ ; Equation: $c=3+n$

Explanation: This question tests identifying independent (input) and dependent (output) variables, writing equations relating them, creating tables, graphing with proper axes, and connecting equation-table-graph representations. Variables: independent is input/chosen first (number of notebooks n we buy—we control), dependent is output/depends on independent (cost c depends on notebooks—results from independent). Equation: express dependent in terms of independent (c=3n: cost equals 3 times notebooks, dependent c on left, independent n in expression). For example, in a similar cost scenario like buying apples at $2 each, independent=number a (apples bought), dependent=cost c (dollars spent), equation c=2a, table a:0,1,2,3 c:0,2,4,6 (each from 2×a), graph: x-axis apples, y-axis cost, points (0,0),(1,2),(2,4),(3,6) forming line through (0,0) with slope 2 matching equation coefficient. The correct choice identifies independent as n (notebooks), dependent as c (cost), with equation c=3n, as we choose notebooks and cost results proportionally. Errors include reversing variables (cost independent, notebooks dependent—backward), or wrong equation (additive like c=n+3 when should be multiplicative c=3n). To analyze: (1) identify relationship (cost depends on notebooks at $3 each), (2) determine independent (n: we choose number bought) and dependent (c: result of buying that many), (3) write equation (c=3n), (4) connect to table or graph (e.g., for n=1, c=3; n=2, c=6, plotting (1,3),(2,6) with slope 3). Axes convention: independent horizontal (x-axis: notebooks), dependent vertical (y-axis: cost).

Question 6

A streaming service costs $12 per month plus a one-time sign-up fee of $5. Let $m $be the number of months and$ c $be the total cost. Which choice correctly identifies the independent variable and writes an equation for$ c $in terms of$ m$?

Independent: $m$ ; equation: $c = 5m + 12$
Independent: $c$ ; equation: $m = 12c + 5$
Independent: $m$ ; equation: $c = 12m + 5$ (correct answer)
Independent: $12$ ; equation: $c = m + 17$

Question 7

A student buys notebooks that cost $4 each. Let $n $be the number of notebooks and let$ c $be the total cost in dollars. Which equation shows$ c $as a function of$ n$?

$c = n + 4$
$n = 4c$
$c = 4n$ (correct answer)
$c = 4 - n$

Explanation: This question tests identifying independent (input) and dependent (output) variables, writing equations relating them, creating tables, graphing with proper axes, and connecting equation-table-graph representations. Variables: independent is input/chosen first (number of notebooks n we buy—we control), dependent is output/depends on independent (total cost c depends on notebooks—results from independent). Equation: express dependent in terms of independent (c=4n: cost equals 4 times notebooks, dependent c on left, independent n in expression). Table: list independent values (n: 0,1,2,3), calculate dependent using equation (if n=1, c=4×1=4; n=2, c=8; etc.). Graph: independent on x-axis (horizontal: notebooks), dependent on y-axis (vertical: cost), plot ordered pairs ((1,4), (2,8),...), proportional c=4n graphs through origin. In this notebook purchase example, the correct equation is c=4n, as in choice C, showing cost as a function of notebooks. Errors include reversing variables (like B: n=4c, making notebooks depend on cost) or wrong operations (like A: addition instead of multiplication, or D: subtraction). Analyzing: (1) identify relationship (cost depends on number bought at constant price), (2) independent=notebooks n (chosen), dependent=cost c, (3) write equation dependent=rate×independent (c=4n), (4) ensure equation solves for dependent in terms of independent.

Question 8

A plant is 8 cm tall when it is measured at week 0, and it grows 3 cm each week. Let $w$ be the number of weeks since week 0, and let $h$ be the height in centimeters. Which table matches the relationship between $w$ and $h$ ?

$w: 0,1,2,3$ and $h: 0,3,6,9$
$w: 0,1,2,3$ and $h: 8,11,14,17$ (correct answer)
$w: 0,1,2,3$ and $h: 8,10,12,14$
$w: 1,2,3,4$ and $h: 8,11,14,17$

Explanation: This question tests identifying independent (input) and dependent (output) variables, writing equations relating them, creating tables, graphing with proper axes, and connecting equation-table-graph representations. Variables: independent is input/chosen first (weeks w since week 0—we observe over time), dependent is output/depends on independent (height h depends on weeks—results from independent). Equation: express dependent in terms of independent (h=8+3w: height equals 8 plus 3 times weeks, dependent h on left, independent w in expression). Table: list independent values (w: 0,1,2,3), calculate dependent using equation (if w=0, h=8; w=1, h=11; etc.). Graph: independent on x-axis (horizontal: weeks), dependent on y-axis (vertical: height), plot ordered pairs ((0,8), (1,11),...), line through (0,8) with slope 3. For example, similar to plant growth starting at 5 cm growing 2 cm/week, independent=weeks w, dependent=height h, equation h=5+2w, table w:0,1,2,3 h:5,7,9,11, graph: x-weeks, y-height, points (0,5),(1,7) forming line with y-intercept 5 and slope 2. The correct table is w:0,1,2,3 and h:8,11,14,17, matching initial 8 cm plus 3 cm per week. Errors include wrong calculations (like h:8,10,12,14 assuming +2 instead of +3) or starting w at 1 instead of 0. Analyzing: (1) identify relationship (height depends on weeks with initial height and growth rate), (2) determine independent (w) and dependent (h), (3) write equation (h=8+3w), (4) create table (w:0,1,2,3; h=8+30=8, etc.), (5) graph (x=w, y=h, plot pairs), (6) connect (table matches equation: 8+31=11✓). Axes convention: independent horizontal (x-axis: weeks), dependent vertical (y-axis: height).

Question 9

A car rental company charges $30 per day plus $0.25 per mile driven. Two students write different equations: Student A writes $ $C = 30d + 0.25m$ $and Student B writes$ $C = 30 + 0.25m$ $ (for one day). In terms of independent and dependent variables, which analysis is most accurate?

Student A is correct because both days and miles are independent variables that determine cost
Student B is correct because only miles should be independent when analyzing single-day rentals
Both students correctly identify cost as dependent, but Student A's equation handles multiple independent variables (correct answer)
Both equations are wrong because cost should be independent since customers choose their budget first

Explanation: Both students correctly identify cost

C

as the dependent variable. Student A's equation

C = 30d + 0.25m

accounts for multiple independent variables (days

d

and miles

m

). Student B's equation

C = 30 + 0.25m

is valid for single-day analysis with miles as the independent variable. Choice C recognizes that equations can have multiple independent variables.

Question 10

A science experiment measures how temperature affects the volume of a gas. Students record that when temperature increases by 10°C, volume increases by 15 mL. If $T$ represents temperature and $V$ represents volume, and the initial volume is 200 mL at 20°C, which analysis correctly identifies the relationship?

$V = 200 + 1.5T$ with $T$ independent because temperature can be controlled by researchers
$V = 200 + 1.5(T - 20)$ with $T$ independent because temperature determines the gas volume (correct answer)
$T = 200 + 1.5V$ with $V$ independent because volume changes are what scientists measure
$T = 20 + \frac{2}{3}(V - 200)$ with $V$ independent because gas expansion drives temperature changes

Explanation: Temperature is independent (researchers control it), volume is dependent (responds to temperature). Rate is 15 mL per 10°C = 1.5 mL per °C. Since volume is 200 mL at 20°C, the equation is

V = 200 + 1.5(T - 20)

. Choice A ignores the reference temperature. Choices C and D incorrectly make volume independent, which contradicts experimental control.

Question 11

A plant grows according to the equation $h = 15 + 2.5d$ where $d$ is days since planting and $h$ is height in centimeters. If a student wants to predict when the plant will reach 35 cm, which approach correctly identifies the variables for solving this problem?

Rewrite as $d = 15 + 2.5h$ because days now depends on the target height
Substitute $d = 35$ and solve for $h$ because days is given and height is unknown
Use $h = 15 + 2.5(35)$ because both variables are independent in this context
Substitute $h = 35$ and solve for $d$ because height is given and days is unknown (correct answer)

Explanation: When working with equations that model real-world relationships, you need to identify what information you're given and what you're trying to find. This determines which variable you'll substitute and which one you'll solve for. In this plant growth problem, the equation

h = 15 + 2.5d

tells you that height depends on the number of days. You're asked when the plant will reach 35 cm, meaning you know the target height (35 cm) and want to find the corresponding time in days. The correct approach is D: substitute

h = 35

and solve for

d

. Since you know the height and need to find the days, you'd write

35 = 15 + 2.5d

, then solve:

20 = 2.5d

, so

d = 8

days. A is incorrect because you don't rewrite the original equation's structure. The relationship

h = 15 + 2.5d

correctly shows height depending on days, which matches your problem setup. B reverses the roles of the variables. You're not given days (35) to find height—you're given a target height to find the days needed. C treats both variables as if they're given values, but the question asks "when will the plant reach 35 cm?" This means 35 is your target height, not an input for calculating something else. Study tip: In word problems involving equations, always identify "what do I know?" versus "what am I looking for?" The known value gets substituted, and you solve for the unknown variable.

Question 12

A delivery truck travels at 45 mph on a highway. The relationship between time $t$ (in hours) and distance $d$ (in miles) is shown in the graph. Which statement correctly analyzes both the mathematical relationship and the real-world constraints?

$d = 45t$ with $t$ independent; realistic values include negative time for planning purposes
$d = 45t$ with $t$ independent; realistic values are $t ≥ 0$ since negative time is meaningless (correct answer)
$t = 45d$ with $d$ independent; realistic values are $d ≥ 0$ since negative distance is impossible
$t = \frac{d}{45}$ with $d$ independent; realistic values include all positive real numbers without limits

Explanation: Time is the independent variable (you choose how long to drive), distance is dependent. The equation

d = 45t

shows distance equals rate times time. In real-world context,

t ≥ 0

because negative time doesn't make sense for actual travel. Choice A incorrectly accepts negative time. Choices C and D incorrectly identify distance as independent.

Question 13

Based on the table shown, which equation best represents the relationship between the number of workers $w$ and the time $t$ (in hours) needed to complete a job?

$t = \frac{24}{w}$ where $w$ is independent and $t$ is dependent (correct answer)
$t = 24w$ where $w$ is independent and $t$ is dependent
$w = \frac{24}{t}$ where $t$ is independent and $w$ is dependent
$w = 24t$ where $t$ is independent and $w$ is dependent

Explanation: From the table, as workers increase, time decreases in an inverse relationship:

2 \times 12 = 24

3 \times 8 = 24

4 \times 6 = 24

. The number of workers is chosen first (independent), determining the time needed (dependent). The equation is

t = \frac{24}{w}

. Choice B shows direct variation. Choices C and D incorrectly identify the independent variable.

Question 14

Refer to the graph showing the relationship between study time and test scores. A student claims that $s = 40 + 8h$ represents this relationship, where $h$ is hours studied and $s$ is the test score. What error exists in the student's analysis?

The equation is correct, but $h$ should be dependent and $s$ should be independent
The variable identification is correct, but the equation should be $s = 8h + 40$
The equation is correct, but the slope should be 5 points per hour, not 8 (correct answer)
Both variables and equation are correct; there is no error in the analysis

Explanation: Hours studied

h

is correctly independent (student chooses study time), and test score

s

is correctly dependent. However, from the graph points (0,40), (2,50), (4,60), (6,70), the slope is

\frac{50-40}{2-0} = 5

points per hour, not 8. The correct equation is

s = 40 + 5h

. The student used the wrong slope coefficient.

Question 15

A runner moves at a constant speed of 7 meters per second. Let $t$ be time in seconds and $d$ be distance in meters. Which ordered pairs should be graphed for the relationship $d = 7t$ (with $t$ on the x-axis and $d$ on the y-axis)?

$(0,0), (7,1), (14,2), (21,3)$
$(7,1), (14,2), (21,3), (28,4)$
$(0,0), (1,7), (2,14), (3,21)$ (correct answer)
$(0,7), (1,14), (2,21), (3,28)$

Explanation: Tests identifying independent (input) and dependent (output) variables, writing equations relating them, creating tables, graphing with proper axes, and connecting equation-table-graph representations. Variables: independent is input/chosen first (time t in seconds, periods we measure—we control), dependent is output/depends on independent (distance d depends on time, result depends on period—results from independent). Equation: express dependent in terms of independent (d=7t: distance equals 7 times time, dependent d on left, independent t in expression). Table: list independent values (t: 0,1,2,3), calculate dependent using equation (if t=1, d=7×1=7; t=2, d=14; etc.). Graph: independent on x-axis (horizontal: time), dependent on y-axis (vertical: distance), plot ordered pairs ((1,7), (2,14),...), proportional through origin. In this runner example, independent=time t, dependent=distance d, equation d=7t, table t:0,1,2,3 d:0,7,14,21, graph x=time, y=distance, points (0,0),(1,7),(2,14),(3,21) line with slope 7. The correct ordered pairs are (0,0),(1,7),(2,14),(3,21) as in C (t on x, d on y); errors like A reverse coordinates, B starts at (0,7), D mixes values.

Question 16

Examine the coordinate plane shown. If $x$ represents the independent variable and $y$ represents the dependent variable, which statement best describes the relationship between the variables for the first 4 hours?

$y$ increases at a constant rate of 2 units per hour, so $y = 2x + 4$
$y$ decreases at a constant rate of 2 units per hour, so $y = 8 - 2x$ (correct answer)
$y$ increases at a constant rate of 4 units per hour, so $y = 4x + 8$
$y$ decreases at a constant rate of 4 units per hour, so $y = 8 + 4x$

Explanation: Looking at the plotted points: (0,8), (1,6), (2,4), (3,2), (4,0), the

y

-values decrease by 2 units for each 1-unit increase in

x

. Starting at 8 when

x=0

, the equation is

y = 8 - 2x

. Choice A shows increase instead of decrease. Choice C has wrong rate and operation. Choice D has wrong operation sign.

Question 17

A movie theater sells tickets for $9 each. Let $t $be the number of tickets and$ c$ be the total cost. Which ordered pair correctly represents buying 4 tickets on a graph where the x-axis is tickets and the y-axis is cost?

$(9,4)$
$(36,4)$
$(4,9)$
$(4,36)$ (correct answer)

Question 18

A bike rental shop charges a $3 starting fee plus $2 for each hour you rent the bike. Which statement correctly identifies the independent and dependent variables?

Independent: $2 per hour; Dependent: starting fee
Independent: starting fee; Dependent: hours rented
Independent: hours rented; Dependent: total cost (correct answer)
Independent: total cost; Dependent: hours rented

Explanation: This question tests identifying independent (input) and dependent (output) variables, writing equations relating them, creating tables, graphing with proper axes, and connecting equation-table-graph representations. Variables: independent is input/chosen first (hours rented h we choose), dependent is output/depends on independent (total cost c depends on hours plus starting fee). Equation: express dependent in terms of independent (c=2h+3: cost equals 2 times hours plus 3, dependent c on left, independent h in expression). Table: list independent values (h: 0,1,2,3), calculate dependent using equation (if h=1, c=2×1+3=5; h=2, c=7; etc.). Graph: independent on x-axis (horizontal: hours), dependent on y-axis (vertical: cost), plot ordered pairs ((1,5), (2,7),...), non-proportional c=2h+3 graphs through (0,3). In this bike rental example, independent=hours rented (we control how long we rent), dependent=total cost (results from hours plus fee), so choice B correctly identifies them. Common errors include reversing variables (like A, making cost independent, which is backward) or confusing constants as variables (like C or D). To analyze: (1) identify relationship (cost depends on hours rented with fixed fee), (2) determine independent (hours: we choose) and dependent (cost: result), (3) avoid mistaking fees or rates as variables.

Question 19

A science class warms water. The temperature starts at 18°C and increases by 4°C each minute. Let $t$ be time in minutes and $T$ be temperature in °C. Which statement correctly connects the equation, table, and graph?

Equation $T=4t+18$ generates the table values, and plotting the table as $(T,t)$ gives a straight line with slope 18.
Equation $T=18t+4$ generates the table values, and the graph must pass through the origin.
Equation $t=4T+18$ generates the table values, and plotting the table as $(T,t)$ gives a straight line with y-intercept 18.
Equation $T=4t+18$ generates the table values, and plotting the table as $(t,T)$ gives a straight line with y-intercept 18. (correct answer)

Question 20

A runner jogs at a constant speed of 6 miles per hour. Let $t$ be time in hours and let $d$ be distance in miles. Which equation shows $d$ as a function of $t$ ?

$t=6d$
$d=\dfrac{t}{6}$
$d=6t$ (correct answer)
$d=t+6$

Explanation: This question tests identifying independent (input) and dependent (output) variables, writing equations relating them, creating tables, graphing with proper axes, and connecting equation-table-graph representations. Variables: independent is input/chosen first (time t we jog—we control), dependent is output/depends on independent (distance d depends on time—results from independent). Equation: express dependent in terms of independent (d=6t: distance equals 6 times time, dependent d on left, independent t in expression). Table: list independent values (t: 0,1,2,3), calculate dependent using equation (if t=1, d=6×1=6; t=2, d=12; etc.). Graph: independent on x-axis (horizontal: time), dependent on y-axis (vertical: distance), plot ordered pairs ((1,6), (2,12),...), proportional d=kt graphs through origin. For example, distance-time at 65 mph, independent=time t (hours driven), dependent=distance d (miles traveled), equation d=65t, table t:1,2,3 d:65,130,195 (each from 65×t), graph: x-axis time, y-axis distance, points (1,65),(2,130),(3,195) forming line through (0,0) with slope 65 matching equation coefficient. The correct equation is d=6t, showing distance as a function of time at constant speed. Errors like d=t+6 (additive instead of multiplicative) or d=t/6 (inverse) don't match the proportional relationship. Analyzing: (1) identify relationship (distance depends on time at 6 mph), (2) determine independent (time t) and dependent (distance d), (3) write equation (d=6t), (4) create table (t:0,1,2, d:0,6,12), (5) graph (x=t, y=d, plot pairs, line through origin), (6) connect (equation matches table: 6×1=6✓, slope=6✓). Axes convention: independent horizontal (x-axis: time), dependent vertical (y-axis: distance).