Graph relationships from tables

Learning goals

Plots table values as ordered pairs on coordinate axes.
Chooses appropriate axis scales and labels.
Uses the graph to describe the relationship between variables.
Checks plotted points against the table.

Difficulty: 1 – 5

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Graph relationships from tables 1

A packing station records the total number of pencils after different numbers of boxes have been added. <table> <thead> <tr><th>Boxes added $x$</th><th>Total pencils $y$</th></tr> </thead> <tbody> <tr><td>$7$</td><td>?</td></tr> </tbody> </table> Use the graph to complete the row. Then explain what the ordered pair means in this situation.

Hints

- Use the table's known input to choose the correct plotted point. - Read the vertical coordinate from the graph scale. - Interpret the two coordinates using the axis labels, in order.

Solution

1. Locate $x=7$ on the horizontal axis. 2. The plotted point at that input has y-coordinate $44$, so the ordered pair is $(7,44)$. 3. In context, $(7,44)$ means that after $7$ boxes have been added, the total is $44$ pencils.

Answer

The completed row is $(7,44)$. It means that after $7$ boxes have been added, the total is $44$ pencils.

Graph relationships from tables 2

A fair charges an entry fee and then a cost per ride. List all ordered pairs produced by the table, using rides first and total cost second. <table> <thead> <tr> <th>Rides $r$</th> <th>Total cost $C$</th> </tr> </thead> <tbody> <tr> <td>$0$</td> <td>$3$</td> </tr> <tr> <td>$1$</td> <td>$5$</td> </tr> <tr> <td>$2$</td> <td>$7$</td> </tr> <tr> <td>$3$</td> <td>$9$</td> </tr> </tbody> </table>

Hints

- Use the table headings to determine coordinate order. - Keep the two values from each row together as one ordered pair.

Solution

1. Read each row in the order $(r, C)$. 2. The ordered pairs are $(0, 3)$, $(1, 5)$, $(2, 7)$, and $(3, 9)$.

Answer

$(0, 3)$, $(1, 5)$, $(2, 7)$, $(3, 9)$

Graph relationships from tables 3

A printer log records pages completed at selected times. One row is missing from the table. <table> <thead> <tr><th>Minutes $m$</th><th>Pages $p$</th></tr> </thead> <tbody> <tr><td>$30$</td><td>?</td></tr> </tbody> </table> Use the graph to supply the missing output.

Hints

- The horizontal axis uses minutes in tens, not single minutes. - Find the plotted point whose x-coordinate matches the table. - Read its y-coordinate from the pages scale.

Solution

1. Find $m=30$ on the horizontal axis. 2. The plotted point at $m=30$ has y-coordinate $19$. 3. Therefore, the missing table value is $19$ pages.

Answer

$19$ pages

Graph relationships from tables 4

The graph shows a walker's distance over time. Use the graph to complete the statement: “At $45$ minutes, the point on the graph is ______, meaning ______.”

Hints

- Locate $45$ on the horizontal time axis. - Read the corresponding vertical coordinate from the line. - Interpret the first coordinate as time and the second as distance.

Solution

1. At $45$ minutes, the graph has y-coordinate $3.6$. 2. The ordered pair is $(45,3.6)$. 3. It means that after $45$ minutes, the walker has traveled $3.6$ miles.

Answer

$(45,3.6)$, meaning that after $45$ minutes the walker has traveled $3.6\,\text{mi}$.

Graph relationships from tables 5

A bakery records dozens of cookies. <table> <thead> <tr> <th>Dozens $d$</th> <th>Cookies $c$</th> </tr> </thead> <tbody> <tr> <td>$0$</td> <td>$0$</td> </tr> <tr> <td>$1$</td> <td>$12$</td> </tr> <tr> <td>$2$</td> <td>$24$</td> </tr> <tr> <td>$3$</td> <td>$36$</td> </tr> </tbody> </table> Which quantity is shown on the x-axis, which quantity is shown on the y-axis, and which quantity is the input?

Hints

- Match the first and second table columns to the axis order. - Identify which quantity is chosen before the other quantity is determined.

Solution

1. The first table column matches the x-axis, so the x-axis shows dozens. 2. The second table column matches the y-axis, so the y-axis shows cookies. 3. The number of dozens is the input, and the number of cookies depends on it.

Answer

The x-axis shows dozens, the y-axis shows cookies, and dozens is the input quantity.

Graph relationships from tables 6

The table gives $x$ and $y$ values. <table><tr><th>$x$</th><td>$0$</td><td>$1$</td><td>$2$</td></tr><tr><th>$y$</th><td>$2$</td><td>$4$</td><td>$6$</td></tr></table> Write the ordered pairs you would plot, in order.

Hints

- An ordered pair writes the x-coordinate first. - Keep each x-value with the y-value from the same column.

Solution

1. Match each x-value with the y-value directly below it. 2. The pairs are $(0, 2)$, $(1, 4)$, and $(2, 6)$.

Answer

$(0, 2)$, $(1, 4)$, $(2, 6)$

Graph relationships from tables 7

A table gives an input of $2$ and an output of $7$. Kai plots the point $(7, 2)$. Is his point correct? If not, write the point he should plot.

Hints

- Which quantity goes along the horizontal axis? - In what order are coordinates written?

Solution

1. An ordered pair lists the input (x-coordinate) first. 2. Kai reversed the input and output. The correct point is $(2, 7)$.

Answer

No. Kai reversed the coordinates. He should plot $(2, 7)$.

Graph relationships from tables 8

The table records the height of a seedling. The graph contains one y-coordinate error. <table> <thead> <tr> <th>Days $d$</th> <th>Height $h$ in cm</th> </tr> </thead> <tbody> <tr> <td>$0$</td> <td>$5$</td> </tr> <tr> <td>$2$</td> <td>$7$</td> </tr> <tr> <td>$4$</td> <td>$9$</td> </tr> <tr> <td>$6$</td> <td>$11$</td> </tr> </tbody> </table> Find the error and state how far the plotted value is from the table value.

Hints

- Compare the output for each displayed day. - After finding the mismatch, subtract the two vertical values.

Solution

1. At $d=4$, the table gives $h=9$. 2. The graph plots $(4, 8)$. 3. The plotted height is $1$ centimeter below the table value.

Answer

The error is $(4, 8)$; it should be $(4, 9)$. The plotted value is $1\,\text{cm}$ too low.

Graph relationships from tables 9

The graph should include one point for every row of the table. <table> <thead> <tr> <th>Cartons $c$</th> <th>Eggs $e$</th> </tr> </thead> <tbody> <tr> <td>$1$</td> <td>$12$</td> </tr> <tr> <td>$2$</td> <td>$24$</td> </tr> <tr> <td>$3$</td> <td>$36$</td> </tr> <tr> <td>$4$</td> <td>$48$</td> </tr> </tbody> </table> Which ordered pair is missing from the graph?

Hints

- List the table’s ordered pairs before inspecting the plotted set. - Find the row whose input does not appear on the graph.

Solution

1. The table gives $(1, 12)$, $(2, 24)$, $(3, 36)$, and $(4, 48)$. 2. The graph shows the first, second, and fourth pairs. 3. The missing point is $(3, 36)$.

Answer

$(3, 36)$

Graph relationships from tables 10

Compare the recipe table with the displayed graph. <table> <thead> <tr> <th>Batches $b$</th> <th>Flour $f$ in cups</th> </tr> </thead> <tbody> <tr> <td>$0$</td> <td>$0$</td> </tr> <tr> <td>$1$</td> <td>$1.5$</td> </tr> <tr> <td>$2$</td> <td>$3$</td> </tr> <tr> <td>$3$</td> <td>$4.5$</td> </tr> </tbody> </table> The graph has one point at the wrong height. Identify the point and correct it.

Hints

- Use the table headings to preserve coordinate order. - Check exact decimal heights rather than only the overall visual pattern.

Solution

1. The table pairs are $(0, 0)$, $(1, 1.5)$, $(2, 3)$, and $(3, 4.5)$. 2. The graph shows $(1, 2)$ instead of $(1, 1.5)$. 3. The corrected point is $(1, 1.5)$.

Answer

Replace $(1, 2)$ with $(1, 1.5)$.

Graph relationships from tables 11

The table and graph model tickets remaining. <table> <thead> <tr> <th>Tickets sold $s$</th> <th>Tickets left $r$</th> </tr> </thead> <tbody> <tr> <td>$0$</td> <td>$60$</td> </tr> <tr> <td>$10$</td> <td>$50$</td> </tr> <tr> <td>$20$</td> <td>$40$</td> </tr> <tr> <td>$30$</td> <td>$30$</td> </tr> </tbody> </table> One graph point breaks the table’s relationship. Identify it and explain the direction of the error.

Hints

- Compare the point at each listed sales value. - Describe whether the incorrect y-coordinate is above or below the required one.

Solution

1. At $s=20$, the table gives $r=40$. 2. The graph shows $(20, 35)$. 3. The graphed value is $5$ tickets too low.

Answer

$(20, 35)$ is incorrect; it should be $(20, 40)$, so the graph is $5$ tickets too low.

Graph relationships from tables 12

The graph shows distance during a continuous trip. <table> <thead> <tr><th>Time $t$ in min</th><th>Distance $d$ in mi</th></tr> </thead> <tbody> <tr><td>$45$</td><td>?</td></tr> </tbody> </table> Use the line on the graph to complete the table.

Hints

- Locate $45$ minutes on the horizontal axis. - Read vertically from that input until you reach the plotted line. - Use the distance scale to read the matching y-value.

Solution

1. Locate $45$ minutes on the x-axis. 2. Follow that x-value to the plotted line. 3. The corresponding y-coordinate is $18$, so the trip distance at $45$ minutes is $18$ miles.

Answer

$18\,\text{mi}$

Graph relationships from tables 13

A bead pattern is plotted only for selected figure numbers. One input is missing from the table. <table> <thead> <tr><th>Figure $f$</th><th>Beads $b$</th></tr> </thead> <tbody> <tr><td>?</td><td>$50$</td></tr> </tbody> </table> Use the graph to find the missing figure number.

Hints

- This time the known table entry is an output, so begin with the y-coordinate. - Find the plotted point at that height. - Read its horizontal coordinate using the figure-number scale.

Solution

1. Start at $b=50$ on the vertical scale. 2. The plotted point at that height has x-coordinate $15$. 3. Therefore, the missing figure number is $15$.

Answer

$15$

Graph relationships from tables 14

The graph shows a cooling drink over time. a) Read the temperature at $30$ minutes. b) How much does the temperature decrease from $10$ minutes to $40$ minutes?

Hints

- Read the y-coordinate directly above $30$ minutes for part a). - For part b), read two different graph values before subtracting. - Report the change as a decrease with temperature units.

Solution

1. At $30$ minutes, the graph has y-coordinate $20$, so the temperature is $20^\circ\text{C}$. 2. At $10$ minutes the graph shows $24^\circ\text{C}$, and at $40$ minutes it shows $18^\circ\text{C}$. 3. The decrease is $24-18=6^\circ\text{C}$.

Answer

a) $20^\circ\text{C}$ b) A decrease of $6^\circ\text{C}$

Graph relationships from tables 15

Which graph panel represents each table? Table A <table> <thead> <tr> <th>$x$</th> <th>$y$</th> </tr> </thead> <tbody> <tr> <td>$0$</td> <td>$10$</td> </tr> <tr> <td>$1$</td> <td>$8$</td> </tr> <tr> <td>$2$</td> <td>$6$</td> </tr> </tbody> </table> Table B <table> <thead> <tr> <th>$x$</th> <th>$y$</th> </tr> </thead> <tbody> <tr> <td>$0$</td> <td>$1$</td> </tr> <tr> <td>$1$</td> <td>$3$</td> </tr> <tr> <td>$2$</td> <td>$5$</td> </tr> </tbody> </table>

Hints

- First decide whether each table increases or decreases. - Use the output at zero to distinguish the starting heights.

Solution

1. Table A decreases from $10$ by $2$ per input step, matching panel a). 2. Table B increases from $1$ by $2$ per input step, matching panel b).

Answer

Table A → a), the decreasing set of points. Table B → b), the increasing set of points.

Graph relationships from tables 16

Two equations are proposed for the relationship in the graph: $s=9r$ and $s=r+9$. Use the graph value at $r=6$ to decide which equation matches the plotted relationship. State the graph-derived value before choosing.

Hints

- Read the graph before evaluating either proposed rule. - Use the same input, $r=6$, in both equations. - The matching rule must reproduce the plotted output.

Solution

1. The plotted point with x-coordinate $6$ is $(6,54)$, so the graph-derived output is $s=54$. 2. The rule $s=9r$ gives $9\times6=54$. 3. The rule $s=r+9$ gives $6+9=15$. 4. Therefore, $s=9r$ matches the graph.

Answer

The graph gives $s=54$ when $r=6$, so the matching equation is $s=9r$.

Graph relationships from tables 17

The graph shows a ferry relationship of the form $d=kt$, where $t$ is time in minutes and $d$ is distance in miles. Use two plotted points to determine $k$, write the equation, and explain what $k$ means.

Hints

- This graph does not give a convenient one-minute point. - Compare distance to time for one plotted point, then check another. - Interpret the multiplier using the units on the axes.

Solution

1. The graph includes $(15,4.5)$ and $(30,9)$. 2. The ratio $d/t$ is $4.5\div15=0.3$, and $9\div30=0.3$, so $k=0.3$. 3. The equation is $d=0.3t$. 4. The value $0.3$ means the ferry travels $0.3$ mile per minute.

Answer

$k=0.3$, so $d=0.3t$. The value $0.3$ represents $0.3\,\text{mi}$ per minute.

Graph relationships from tables 18

Nerea claims the bottle relationship shown on the graph is $L=b+0.75$, where $b$ is the number of bottles and $L$ is the total volume in liters. Use the graph at $b=0$ and $b=8$ to test the claim. Then write the equation that actually matches the graph.

Hints

- Check Nerea's equation against the graph at zero bottles. - Use the point at $8$ bottles to determine liters per bottle. - Decide whether $0.75$ belongs as a constant term or as a coefficient.

Solution

1. The graph shows $(0,0)$, while Nerea's equation gives $L=0.75$ at $b=0$, so the claim fails immediately. 2. The graph also shows $(8,6)$. Dividing gives $6\div8=0.75$ liter per bottle. 3. The quantity $0.75$ is therefore a multiplier, not a one-time addition. 4. The correct equation is $L=0.75b$.

Answer

Nerea's claim is incorrect. The graph matches $L=0.75b$.

Graph relationships from tables 19

Two rules, $T=2n+5$ and $T=4n+1$, both give $9$ when $n=2$. The graph shows the actual relationship. Use the graph value at $n=0$ to decide which rule is correct, and explain why the shared value at $n=2$ is not enough by itself.

Hints

- Read the graph at the input where the proposed rules disagree. - Evaluate each rule at $n=0$. - One matching point does not prove that two rules describe the same relationship.

Solution

1. The graph shows $(0,5)$, so the actual output at $n=0$ is $5$. 2. The rule $T=2n+5$ gives $5$ when $n=0$. 3. The rule $T=4n+1$ gives $1$ when $n=0$. 4. Therefore, $T=2n+5$ matches the graph. The input $n=2$ cannot distinguish the rules because both happen to give $9$ there.

Answer

The matching rule is $T=2n+5$. The graph gives $T=5$ at $n=0$, which distinguishes the rules.

Graph relationships from tables 20

The graph gives total distance after several complete laps around a track. Use the graph to read the distances at $4$ laps and $6$ laps. Then find the vertical change between those two plotted points.

Hints

- Read the y-coordinate at each of the two requested x-values. - The graph's horizontal spacing is two laps between plotted points. - Subtract the first graph-derived distance from the second.

Solution

1. The point at $4$ laps is $(4,1600)$. 2. The point at $6$ laps is $(6,2400)$. 3. The vertical change is $2400-1600=800$ meters.

Answer

At $4$ laps the distance is $1600\,\text{m}$; at $6$ laps it is $2400\,\text{m}$. The vertical change is $800\,\text{m}$.

Graph relationships from tables 21

Find the point on the graph whose x-coordinate is $6$. Interpret its coordinates using the axis labels. Explain how the meaning changes if the coordinates are reversed.

Hints

- Check which quantity is on each axis. - Find the mark above the requested x-coordinate. - Read the horizontal coordinate before the vertical one.

Solution

1. The plotted point is $(6, 5)$. 2. Its coordinates mean that after $6$ practice sessions, $5$ songs are ready. 3. Reversing them gives $(5, 6)$, meaning $5$ sessions and $6$ songs ready. That is a different situation.

Answer

$(6, 5)$ means $6$ sessions and $5$ songs ready. $(5, 6)$ would mean $5$ sessions and $6$ songs ready.

Graph relationships from tables 22

The graph shows total cubes in complete boxes, with $6$ cubes per box. <table> <thead> <tr> <th>Boxes $b$</th> <th>Cubes $c$</th> </tr> </thead> <tbody> <tr> <td>$0$</td> <td>$0$</td> </tr> <tr> <td>$1$</td> <td>$6$</td> </tr> <tr> <td>$2$</td> <td>$12$</td> </tr> <tr> <td>$3$</td> <td>$18$</td> </tr> </tbody> </table> Does the point $(1.5, 9)$ belong to this relationship? Explain.

Hints

- Check whether the proposed x-coordinate is allowed by the context. - Separate a numerical pattern from the domain values the model permits.

Solution

1. The x-coordinate counts complete boxes. 2. A value of $1.5$ does not represent a whole number of complete boxes in this model. 3. Therefore, $(1.5, 9)$ is not one of the allowed points, even though $1.5 \times 6=9$.

Answer

No. The input must be a whole number of complete boxes, so $(1.5, 9)$ is not an allowed point in this relationship.

Graph relationships from tables 23

The table gives water in a tank while it fills steadily. <table> <thead> <tr> <th>Minutes $m$</th> <th>Water $W$ in L</th> </tr> </thead> <tbody> <tr> <td>$0$</td> <td>$0$</td> </tr> <tr> <td>$1$</td> <td>$5$</td> </tr> <tr> <td>$2$</td> <td>$10$</td> </tr> <tr> <td>$3$</td> <td>$15$</td> </tr> </tbody> </table> Why is the connected graph appropriate here?

Hints

- Ask whether fractional input values are meaningful. - Decide whether the output changes only at separate moments or throughout the interval.

Solution

1. Time can take values between the whole minutes, such as $1.5$ minutes. 2. The amount of water also changes throughout those intervals. 3. A connected graph represents those meaningful intermediate values.

Answer

The graph is connected because both time and water amount can vary continuously between the listed rows.

Graph relationships from tables 24

A cyclist moves continuously at a steady speed. <table> <thead> <tr> <th>Time $t$ in h</th> <th>Distance $d$ in mi</th> </tr> </thead> <tbody> <tr> <td>$0$</td> <td>$0$</td> </tr> <tr> <td>$1$</td> <td>$12$</td> </tr> <tr> <td>$2$</td> <td>$24$</td> </tr> <tr> <td>$3$</td> <td>$36$</td> </tr> </tbody> </table> What does the connected segment between $(1, 12)$ and $(2, 24)$ represent?

Hints

- Interpret the interval of x-values first. - A connected segment includes all intermediate input-output pairs.

Solution

1. The segment includes times between $1$ and $2$ hours. 2. It shows the corresponding distances traveled during that interval. 3. Points on the segment represent the cyclist’s distance at fractional times, not only at whole hours.

Answer

It represents the cyclist’s distance at every time between $1$ and $2$ hours.

Graph relationships from tables 25

a) At which plotted input do the outputs match? b) At the largest plotted input, which panel has the greater output, and by how much?

Hints

- Compare only points with the same x-coordinate. - Look for a height shared across the two panels. - Use the rightmost points for part b).

Solution

1. Compare both panels at the same x-values: $10,20,30,40$. 2. Both panels show an output of $24$ at $x=30$. 3. At $x=40$, panel a) shows $30$ and panel b) shows $33$, so panel b) is greater by $3$.

Answer

a) $x=30$ b) Panel b), by $3$.

Graph relationships from tables 26

Movie tickets cost $\$9$ each. The graph shows some possible purchases. Zuzana wants to connect the graph points because they form a straight pattern. Is that appropriate for ticket purchases?

Hints

- Separate the visual pattern from the values allowed by the context. - Consider what a point halfway between two ticket counts would mean.

Solution

1. The points do follow a straight pattern. 2. However, the input is a count of whole tickets in this situation. 3. Connecting the points would include fractional-ticket purchases, so the points should remain discrete.

Answer

No. The straight pattern does not make the context continuous; whole ticket counts should be shown as separate points.

Graph relationships from tables 27

A reading log has these totals. <table><tr><th>Day</th><td>$0$</td><td>$2$</td><td>$4$</td><td>$6$</td></tr><tr><th>Pages read</th><td>$0$</td><td>$10$</td><td>$20$</td><td>$30$</td></tr></table> You have room for at most $8$ tick intervals on each axis, starting at $0$. Choose a tick size for each axis that lets you plot every table entry exactly on a tick. Give the axis labels and list the four points.

Hints

- Check the largest number on each axis and how many tick intervals you have. - Which tick sizes put all the listed numbers at whole ticks? - Pair each day with the total on the same table column.

Solution

1. Label the x-axis day and the y-axis pages read. 2. One choice is $2$ days per tick and $10$ pages per tick. All values fit within three intervals on each axis and land on labeled ticks. Other scales satisfying the two rules are possible. 3. The ordered pairs are $(0, 0)$, $(2, 10)$, $(4, 20)$, and $(6, 30)$.

Answer

One valid plan: x-axis = day, $2$ days per tick; y-axis = pages read, $10$ pages per tick. Points: $(0, 0)$, $(2, 10)$, $(4, 20)$, $(6, 30)$. Other choices meeting both scale rules are correct.

Graph relationships from tables 28

A walking model gives these values. <table><tr><th>Time (minutes)</th><td>$0$</td><td>$5$</td><td>$10$</td></tr><tr><th>Distance (meters)</th><td>$0$</td><td>$20$</td><td>$40$</td></tr></table> Plan a graph: name both axes with units, choose a useful tick interval for each, and list all the points to plot.

Hints

- Which variable is the input and belongs on the x-axis? - Choose scales that fit the largest values and place all table values at clear positions. - Read each column as one ordered pair.

Solution

1. Put time in minutes on the x-axis and distance in meters on the y-axis. 2. One useful choice is x-axis ticks every $5$ minutes and y-axis ticks every $10$ meters. 3. Plot $(0, 0)$, $(5, 20)$, and $(10, 40)$.

Answer

x-axis: time (minutes), ticks every $5$. y-axis: distance (meters), ticks every $10$. Points: $(0, 0)$, $(5, 20)$, $(10, 40)$. Other clear scales are possible.

Graph relationships from tables 29

Compare the table with the plotted points. Which point was plotted incorrectly? Write the correct ordered pair. <table><tr><th>$x$</th><td>$1$</td><td>$2$</td><td>$3$</td></tr><tr><th>$y$</th><td>$2$</td><td>$5$</td><td>$7$</td></tr></table>

Hints

- Compare each table column with the point at that x-coordinate. - Use the labeled grid to check the y-coordinate.

Solution

1. The point for $x=1$ is $(1, 2)$, which matches. 2. The point for $x=2$ is plotted at $(2, 4)$, but the table requires $(2, 5)$. 3. The point for $x=3$ matches $(3, 7)$.

Answer

The point at $(2, 4)$ is wrong. It should be $(2, 5)$.

Graph relationships from tables 30

a) Read all four plotted points and list them in order of x-coordinate. b) Between which two consecutive x-values does the y-value decrease?

Hints

- Read the x-coordinate before the y-coordinate at each plotted point. - Compare the output heights from one input to the next.

Solution

1. The plotted pairs are $(0, 3)$, $(1, 5)$, $(2, 4)$, and $(3, 6)$. 2. The output drops from $5$ to $4$ between $x=1$ and $x=2$.

Answer

a) $(0, 3)$, $(1, 5)$, $(2, 4)$, $(3, 6)$ b) Between $x=1$ and $x=2$

Graph relationships from tables 31

Match each table to graph a) or b). Table A <table> <thead> <tr> <th>$x$</th> <th>$y$</th> </tr> </thead> <tbody> <tr> <td>$0$</td> <td>$0$</td> </tr> <tr> <td>$1$</td> <td>$3$</td> </tr> <tr> <td>$2$</td> <td>$6$</td> </tr> </tbody> </table> Table B <table> <thead> <tr> <th>$x$</th> <th>$y$</th> </tr> </thead> <tbody> <tr> <td>$0$</td> <td>$4$</td> </tr> <tr> <td>$1$</td> <td>$5$</td> </tr> <tr> <td>$2$</td> <td>$6$</td> </tr> </tbody> </table>

Hints

- Compare a distinctive point such as the one with x-coordinate $0$. - Then confirm the match using the y-change between consecutive rows.

Solution

1. Table A includes $(0, 0)$ and has a y-change of $3$ for each increase of $1$ in $x$; this matches graph b). 2. Table B includes $(0, 4)$ and has a y-change of $1$; this matches graph a).

Answer

Table A → b), the point set containing $(0, 0)$ with a y-change of $3$. Table B → a), the point set containing $(0, 4)$ with a y-change of $1$.

Graph relationships from tables 32

Both tables begin at $(0, 0)$. Table A <table> <thead> <tr> <th>$x$</th> <th>$y$</th> </tr> </thead> <tbody> <tr> <td>$0$</td> <td>$0$</td> </tr> <tr> <td>$2$</td> <td>$1$</td> </tr> <tr> <td>$4$</td> <td>$2$</td> </tr> </tbody> </table> Table B <table> <thead> <tr> <th>$x$</th> <th>$y$</th> </tr> </thead> <tbody> <tr> <td>$0$</td> <td>$0$</td> </tr> <tr> <td>$2$</td> <td>$4$</td> </tr> <tr> <td>$4$</td> <td>$8$</td> </tr> </tbody> </table> 1) Match each table to graph panel a) or b). 2) At $x=4$, how much greater is Table B’s output than Table A’s output?

Hints

- Because the tables share the point $(0, 0)$, compare a different input. - Use the points with x-coordinate $4$ to match the panels. - Compare the two y-coordinates at that input.

Solution

1. Both tables contain $(0, 0)$, so use another shared input to distinguish them. 2. Table A contains $(4, 2)$, which appears in panel a). 3. Table B contains $(4, 8)$, which appears in panel b). 4. At $x=4$, the difference between the outputs is $8-2=6$.

Answer

1) Table A → panel a); Table B → panel b). 2) Table B’s output is $6$ greater.

Graph relationships from tables 33

Match each table with graph a) or b). Explain which point makes the match fastest. Table A <table> <thead> <tr> <th>$x$</th> <th>$y$</th> </tr> </thead> <tbody> <tr> <td>$0$</td> <td>$2$</td> </tr> <tr> <td>$1$</td> <td>$6$</td> </tr> <tr> <td>$2$</td> <td>$10$</td> </tr> </tbody> </table> Table B <table> <thead> <tr> <th>$x$</th> <th>$y$</th> </tr> </thead> <tbody> <tr> <td>$0$</td> <td>$0$</td> </tr> <tr> <td>$1$</td> <td>$4$</td> </tr> <tr> <td>$2$</td> <td>$8$</td> </tr> </tbody> </table>

Hints

- Look for an input value shared by both tables that produces different outputs. - Compare the points where $x=0$.

Solution

1. At $x=0$, Table A has $y=2$, matching panel b). 2. At $x=0$, Table B has $y=0$, matching panel a). 3. The points with x-coordinate $0$ distinguish the panels immediately.

Answer

Table A → b) Table B → a) The points with x-coordinate $0$ give the quickest match.

Graph relationships from tables 34

Inês claims the ribbon relationship shown on the graph is $r=50-3c$, where $c$ is the number of pieces cut and $r$ is the remaining ribbon length in inches. Use two plotted points to test the claimed rate. If the equation is incorrect, write the equation that matches the graph.

Hints

- Compare the vertical change and horizontal change between two plotted points. - Compare the change in ribbon length with the corresponding change in the number of cuts. - Use the point at zero cuts to determine the starting value.

Solution

1. The graph shows $(0,50)$, so the starting length is $50\,\text{in.}$. 2. From $(0,50)$ to $(2,42)$, the ribbon decreases by $8\,\text{in.}$ over $2$ cuts. 3. The decrease per cut is $8\div2=4$ inches, not $3$. 4. Therefore, the matching equation is $r=50-4c$.

Answer

Inês's claim is incorrect. The graph shows a decrease of $4\,\text{in.}$ per cut, so $r=50-4c$.

Graph relationships from tables 35

The graph shows a candle's height $H$, in centimeters, after $t$ hours. a) Read the initial height. b) Determine the change in height per hour from the graph. c) Write an equation for $H$ in terms of $t$.

Hints

- The point where the graph meets $t=0$ gives the starting height. - Use two graph points and divide vertical change by horizontal change. - The equation needs both the starting value and the hourly change.

Solution

1. The graph begins at $(0,18)$, so the initial height is $18$ centimeters. 2. From $(0,18)$ to $(2,15)$, the height changes by $-3$ centimeters over $2$ hours. 3. The hourly change is $-3\div2=-1.5$ centimeters per hour. 4. Combining the initial value and rate gives $H=18-1.5t$.

Answer

a) $18\,\text{cm}$ b) A decrease of $1.5\,\text{cm}$ per hour c) $H=18-1.5t$

Graph relationships from tables 36

<table><tr><th>Side length $s$</th><th>Area $A$</th></tr><tr><td>$2$</td><td>?</td></tr><tr><td>$5$</td><td>?</td></tr></table> Complete the table using the graph. By how much does the area increase from $s=1$ to $s=2$, and from $s=4$ to $s=5$?

Hints

- Find the plotted area above each given side length. - For each change, subtract the earlier area from the later area. - Compare the two increases.

Solution

1. At $s=2$, the area is $4$. At $s=5$, it is $25$. 2. From $s=1$ to $s=2$, the area increases from $1$ to $4$, a change of $3$. 3. From $s=4$ to $s=5$, it increases from $16$ to $25$, a change of $9$. The two increases are different.

Answer

At $s=2$: $A=4$; at $s=5$: $A=25$. The area increases are $3$ and $9$.

Graph relationships from tables 37

A cyclist rides $24$ miles at a steady speed. <table><tr><th>Speed $v$</th><th>Time $T$</th></tr><tr><td>$3$</td><td>?</td></tr><tr><td>$6$</td><td>?</td></tr></table> Complete the table using the graph. What happens to the travel time when the speed doubles from $3$ to $6$ miles per hour?

Hints

- Find the marked times above the two speeds. - Compare the second time with the first. - Think about traveling the same distance at a faster speed.

Solution

1. At $v=3$, the travel time is $8$ hours. 2. At $v=6$, the travel time is $4$ hours. 3. Doubling the speed halves the travel time for the same distance.

Answer

At $v=3$: $T=8$ hours; at $v=6$: $T=4$ hours. The time is cut in half.

Graph relationships from tables 38

Use the graph to fill the table, then compare two changes in $y$. <table><tr><th>$x$</th><th>$y$</th></tr><tr><td>$1$</td><td>?</td></tr><tr><td>$4$</td><td>?</td></tr></table> How much does $y$ increase from $x=1$ to $x=2$, and from $x=3$ to $x=4$?

Hints

- Read the y-values at the x-values in the table. - Find both endpoints of each one-unit interval. - Subtract the earlier output from the later one.

Solution

1. The graph gives $y=3$ at $x=1$ and $y=18$ at $x=4$. 2. From $x=1$ to $x=2$, $y$ increases from $3$ to $6$, a change of $3$. 3. From $x=3$ to $x=4$, $y$ increases from $11$ to $18$, a change of $7$.

Answer

Table values: $3$ and $18$. Increases: $3$ and $7$.

Graph relationships from tables 39

From $x=-2$ to $x=2$, how much does the output change in each panel? Moving from left to right, at which plotted input does panel a) first have a smaller output than panel b)?

Hints

- Read the leftmost and rightmost plotted outputs in each panel. - Is a change from a larger value to a smaller one positive or negative? - When comparing panels, pair points with matching x-coordinates.

Solution

1. In panel a), the output changes from $9$ at $x=-2$ to $1$ at $x=2$, a decrease of $8$. 2. In panel b), the output changes from $1$ to $9$ over the same inputs, an increase of $8$. 3. At $x=0$ both outputs are $5$. The first plotted input to the right where panel a) is smaller is $x=1$, where its output is $3$ versus $7$ in panel b).

Answer

Panel a): decreases by $8$. Panel b): increases by $8$. Panel a) first has the smaller output at $x=1$.

Graph relationships from tables 40

A dog walker charges a base booking fee plus an hourly amount. The graph shows earnings $E$, in dollars, for portions of an hour $h$. Use the graph to determine the base fee and hourly rate, then write an equation for $E$ in terms of $h$.

Hints

- The graph uses half-hour data, so do not mistake the change between neighboring points for an hourly rate. - Read the output at $h=0$ for the fixed fee. - Scale the earnings change to one full hour.

Solution

1. At $h=0$, the graph shows $E=9$, so the base fee is $\$9$. 2. From $(0,9)$ to $(0.5,12)$, earnings increase by $\$3$ in $0.5$ hour. 3. The hourly rate is $3\div0.5=6$ dollars per hour. 4. Therefore, $E=6h+9$.

Answer

The base fee is $\$9$, the hourly rate is $\$6$ per hour, and $E=6h+9$.

Graph relationships from tables 41

The graph shows the water depth $d$, in centimeters, in a tank after $t$ minutes. The labeled points $A$, $B$, and $C$ mark three observations. Dawit says the water rose at the same rate from $A$ to $B$ and from $B$ to $C$ because the depth increased by $6\,\text{cm}$ on each interval. Is Dawit correct? Use the graph to find the rate of change on each interval and explain whether one constant rate describes the whole graph.

Hints

- Read the coordinates of $A$, $B$, and $C$ from the graph. - For each interval, compare the vertical change with the horizontal change. - Equal vertical changes do not imply equal rates when the time intervals have different lengths.

Solution

1. From $A=(0,4)$ to $B=(3,10)$, the depth increases by $6\,\text{cm}$ over $3\,\text{min}$, so the rate is $6\div3=2\,\text{cm/min}$. 2. From $B=(3,10)$ to $C=(9,16)$, the depth also increases by $6\,\text{cm}$, but this takes $6\,\text{min}$, so the rate is $6\div6=1\,\text{cm/min}$. 3. The rates are different, so Dawit is not correct and one constant rate does not describe the whole graph.

Answer

No. From $A$ to $B$, the rate is $2\,\text{cm/min}$. From $B$ to $C$, the rate is $1\,\text{cm/min}$. Because the rates differ, the whole relationship does not have one constant rate.

Graph relationships from tables 42

A greenhouse counts the seedlings that have sprouted by the end of each week. a) How many had sprouted by the end of week $3$? b) How many new seedlings sprouted from week $1$ to week $2$, and from week $3$ to week $4$? c) Which increase was greater, and by how many seedlings?

Hints

- Locate the point for each named week. - Compare the two heights at the ends of each week-to-week interval. - How can you compare the sizes of the two increases?

Solution

1. The point at week $3$ shows $9$ seedlings. 2. From week $1$ to week $2$, the number goes from $1$ to $4$, an increase of $3$. 3. From week $3$ to week $4$, it goes from $9$ to $16$, an increase of $7$. 4. The second increase is greater by $7-3=4$ seedlings.

Answer

a) $9$ seedlings b) $3$ seedlings; $7$ seedlings c) The increase from week $3$ to week $4$ was greater by $4$ seedlings.

Graph relationships from tables 43

The graph shows a relationship between an input $v$ and an output $T$. No equation is given. <table> <thead> <tr><th>$v$</th><th>$T$</th></tr> </thead> <tbody> <tr><td>$3$</td><td>$12$</td></tr> <tr><td>$6$</td><td>?</td></tr> </tbody> </table> Use the graph to complete the table. Then compare the graph values at $v=3$, $v=6$, and $v=12$: what happens to $T$ each time $v$ doubles?

Hints

- Read the curve at $v=6$ before looking for a pattern. - Compare the three marked input-output pairs multiplicatively. - Focus on what happens to the output when the input is doubled.

Solution

1. At $v=6$, the graph has y-coordinate $6$, so the missing table value is $T=6$. 2. The graph shows $(3,12)$, $(6,6)$, and $(12,3)$. 3. Each time the input doubles, the output is halved. 4. The curved graph also shows that the relationship does not have a constant additive rate of change.

Answer

The missing value is $6$. When $v$ doubles from $3$ to $6$ and from $6$ to $12$, $T$ is halved.

Graph relationships from tables 44

a) Read the y-values at $x=-2$ and $x=2$. b) How much does y change from $x=-2$ to $-1$, from $-1$ to $0$, and from $0$ to $1$? Are the three changes equal?

Hints

- Find the points on both sides of the y-axis. - Compare the y-values at consecutive whole-number inputs. - Some changes may be decreases rather than increases.

Solution

1. The points $(-2,6)$ and $(2,6)$ have the same output, $6$. 2. From $-2$ to $-1$, the y-value changes from $6$ to $3$, a change of $-3$. 3. From $-1$ to $0$, it changes from $3$ to $2$, a change of $-1$. 4. From $0$ to $1$, it changes from $2$ to $3$, a change of $+1$. The changes are not equal.

Answer

a) $6$ and $6$ b) Changes: $-3$, $-1$, and $+1$. They are not equal.

Graph relationships from tables 45

A game records how many levels a player completes and how many tokens remain. <table><tr><th>Levels completed</th><td>$1$</td><td>$3$</td><td>$5$</td></tr><tr><th>Tokens remaining</th><td>$8$</td><td>$4$</td><td>$0$</td></tr></table> A graph has five grid spaces to the right of $0$ and four above $0$. Choose whole-number step sizes so the largest values reach the right and top edges and every point lands on a grid intersection. Name both axes and list the ordered pairs. Should the points be connected by a line? Explain.

Hints

- Which quantity goes across the graph, and which goes up? - Use the number of grid spaces and the largest table values to test your step sizes. - Think about what the count of completed levels means between two whole numbers.

Solution

1. Label the x-axis levels completed and the y-axis tokens remaining. 2. Five horizontal spaces with a step of $1$ level reach $5$. Four vertical spaces with a step of $2$ tokens reach $8$. 3. The ordered pairs are $(1, 8)$, $(3, 4)$, and $(5, 0)$. Each lands on a grid intersection. 4. Keep the points separate. Completed levels are whole-number counts; the table gives no values between those counts.

Answer

x-axis: levels completed, step $1$. y-axis: tokens remaining, step $2$. Points: $(1, 8)$, $(3, 4)$, $(5, 0)$. Do not join the points; only whole numbers of completed levels are shown.

Graph relationships from tables 46

A neighborhood recycling center records the number of used batteries it collects each week. <table><tr><th>Week</th><td>$1$</td><td>$2$</td><td>$3$</td><td>$4$</td></tr><tr><th>Batteries collected</th><td>$12$</td><td>$18$</td><td>$14$</td><td>$23$</td></tr></table> Plan a graph: give axis labels and tick intervals, list all four points, and decide whether it is better to show separate points or a line through every value. Explain your choice.

Hints

- The week and the number collected need different axis labels. - Find a scale that fits all the counts without making the graph very tall. - Are counts for times between these four weeks actually given?

Solution

1. Label the x-axis week with a tick every week. Label the y-axis batteries collected, with major ticks every $5$ batteries from $0$ to $25$. Split each $5$-battery space into five equal parts to place individual counts exactly. 2. The points are $(1, 12)$, $(2, 18)$, $(3, 14)$, and $(4, 23)$. 3. Separate points are appropriate because the observations are counts taken only at specific weeks; the table gives no values between observations.

Answer

x-axis: week, ticks every $1$. y-axis: batteries collected, major ticks every $5$, with single-battery subdivisions. Points: $(1, 12)$, $(2, 18)$, $(3, 14)$, $(4, 23)$. Use separate points because values between the weeks are not given. Other clear scales are possible.

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Grade 6 math worksheets

Graph relationships from tables

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