Aimathic – Graph relationships from tables

5497426

The table and graph show the same relationship. <table> <thead> <tr> <th>Pencil packs $x$</th> <th>Pencils $y$</th> </tr> </thead> <tbody> <tr> <td>$0$</td> <td>$0$</td> </tr> <tr> <td>$1$</td> <td>$2$</td> </tr> <tr> <td>$2$</td> <td>$4$</td> </tr> <tr> <td>$3$</td> <td>$6$</td> </tr> </tbody> </table> Which plotted ordered pair represents $2$ pencil packs? Explain what the point means.

Hints

- Use the first coordinate to find the requested input. - Match the second coordinate to the output in the same table row.

Solution

1. Find the table row with $x=2$. 2. That row has $y=4$, so the ordered pair is $(2, 4)$. 3. The point means that $2$ pencil packs contain $4$ pencils.

Answer

$(2, 4)$. It means that $2$ pencil packs contain $4$ pencils.

5497436

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

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

5497486

Use the graph to complete the missing table entry. <table> <thead> <tr> <th>Hours $h$</th> <th>Pages printed $p$</th> </tr> </thead> <tbody> <tr> <td>$0$</td> <td>$1$</td> </tr> <tr> <td>$2$</td> <td>$5$</td> </tr> <tr> <td>$4$</td> <td>?</td> </tr> <tr> <td>$6$</td> <td>$13$</td> </tr> </tbody> </table>

Hints

- Find the graph point whose x-coordinate matches the missing row’s input. - Read the y-coordinate using the labeled scale.

Solution

1. Locate the plotted point with x-coordinate $4$. 2. Its y-coordinate is $9$. 3. The missing table entry is $p=9$.

Answer

$9$ pages

5497516

The graph shows a cooling pattern. Complete the missing table value. <table> <thead> <tr> <th>Hours $h$</th> <th>Temperature $T$ in °C</th> </tr> </thead> <tbody> <tr> <td>$0$</td> <td>$20$</td> </tr> <tr> <td>$1$</td> <td>$18$</td> </tr> <tr> <td>$2$</td> <td>?</td> </tr> <tr> <td>$3$</td> <td>$14$</td> </tr> </tbody> </table>

Hints

- Use the input value from the incomplete row. - Follow the graph to the matching vertical value and include the temperature unit.

Solution

1. At $h=2$, the graphed temperature is $16^\circ\text{C}$. 2. Therefore, the missing table value is $16^\circ\text{C}$.

Answer

$16^\circ\text{C}$

5497686

The table and graph show a walking relationship. <table> <thead> <tr> <th>Hours $h$</th> <th>Miles $d$</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> <tr> <td>$3$</td> <td>$12$</td> </tr> </tbody> </table> What does the point $(3, 12)$ mean in this situation?

Hints

- Match each coordinate to the corresponding table heading. - State the relationship as a complete sentence with units.

Solution

1. The first coordinate represents hours. 2. The second coordinate represents miles. 3. The point means that after $3$ hours, the walker has traveled $12$ miles.

Answer

After $3$ hours, the walker has traveled $12\,\text{mi}$.

5497696

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.

5497706

The table and graph describe complete laps around a $400$-meter track. <table> <thead> <tr> <th>Laps $l$</th> <th>Distance $d$ in m</th> </tr> </thead> <tbody> <tr> <td>$0$</td> <td>$0$</td> </tr> <tr> <td>$1$</td> <td>$400$</td> </tr> <tr> <td>$2$</td> <td>$800$</td> </tr> <tr> <td>$3$</td> <td>$1200$</td> </tr> </tbody> </table> How much does the y-coordinate increase for each additional lap? Use the points $(1, 400)$ and $(2, 800)$ to explain.

Hints

- Compare two consecutive points whose x-coordinates differ by $1$. - Subtract the earlier y-coordinate from the later y-coordinate.

Solution

1. The x-coordinate increases from $1$ lap to $2$ laps, an increase of $1$. 2. The y-coordinate increases from $400$ to $800$: $800-400=400$. 3. Therefore, the distance increases by $400$ meters for each additional lap.

Answer

The y-coordinate increases by $400$, representing $400\,\text{m}$ per additional lap.

5497786

Use the displayed graph to complete the price table. <table> <thead> <tr> <th>Smoothies $s$</th> <th>Cost $C$</th> </tr> </thead> <tbody> <tr> <td>$1$</td> <td>$\$2.50$</td> </tr> <tr> <td>$2$</td> <td>$\$5.00$</td> </tr> <tr> <td>$3$</td> <td>?</td> </tr> <tr> <td>$4$</td> <td>$\$10.00$</td> </tr> </tbody> </table>

Hints

- Find the input from the incomplete row on the x-axis. - Read the y-coordinate and write it using currency notation.

Solution

1. Locate the point with x-coordinate $3$. 2. Its y-coordinate is $7.5$. 3. Written as currency, the missing cost is $\$7.50$.

Answer

$\$7.50$

5497446

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.

5497456

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

$(3, 36)$

5497466

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 $(1, 1.5)$.

5497476

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.

5497496

The graph shows a constant-speed trip. Complete the table. <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>$45$</td> </tr> <tr> <td>$2$</td> <td>?</td> </tr> <tr> <td>$3$</td> <td>$135$</td> </tr> </tbody> </table>

Hints

- Match the missing time to the x-coordinate. - Use the y-axis scale carefully because each grid step represents more than one mile.

Solution

1. Find the point with x-coordinate $2$. 2. The graph gives a y-coordinate of $90$. 3. The missing pair is $(2, 90)$.

Answer

$90\,\text{mi}$

5497506

One input is missing from the pattern table. Use the graph to find it. <table> <thead> <tr> <th>Figure number $f$</th> <th>Beads $b$</th> </tr> </thead> <tbody> <tr> <td>$1$</td> <td>$5$</td> </tr> <tr> <td>$2$</td> <td>$8$</td> </tr> <tr> <td>?</td> <td>$11$</td> </tr> <tr> <td>$4$</td> <td>$14$</td> </tr> </tbody> </table>

Hints

- This time the output is known, so locate that y-coordinate first. - Read the corresponding x-coordinate from the point.

Solution

1. Find the graphed point whose y-coordinate is $11$. 2. Its x-coordinate is $3$. 3. The missing figure number is $3$.

Answer

$3$

5497536

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.

5497576

Two equations are proposed for the seating relationship: $s=8r$ and $s=r+8$. Which equation matches the table and graph? Support the choice with one nonzero row. <table> <thead> <tr> <th>Rows $r$</th> <th>Seats $s$</th> </tr> </thead> <tbody> <tr> <td>$0$</td> <td>$0$</td> </tr> <tr> <td>$1$</td> <td>$8$</td> </tr> <tr> <td>$2$</td> <td>$16$</td> </tr> <tr> <td>$3$</td> <td>$24$</td> </tr> </tbody> </table>

Hints

- Choose a displayed input that makes the two proposed rules give different outputs. - Compare each result with the corresponding plotted point.

Solution

1. Use the row $(2, 16)$. 2. The equation $s=8r$ gives $s=8\times2=16$. 3. The equation $s=r+8$ gives $s=2+8=10$, so it does not match. 4. Therefore, $s=8r$.

Answer

$s=8r$. For example, when $r=2$, it gives $s=16$, matching the table and graph.

5497606

The ferry relationship has the form $d=kt$. Use the table and graph to determine $k$, write the equation, and state what $k$ means in context. <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>

Hints

- Use a row where the time input is one unit. - Interpret the multiplier by comparing the units on the two axes.

Solution

1. One hour corresponds to $12$ miles, so $k=12$. 2. The equation is $d=12t$. 3. The value $12$ means the ferry travels $12$ miles per hour.

Answer

$k=12$, so $d=12t$. The value $12$ represents $12\,\text{mi}$ per hour.

5497616

A student claims the bottle relationship is $L=b+0.75$. Use the row with zero bottles to evaluate the claim, then write the correct equation. <table> <thead> <tr> <th>Bottles $b$</th> <th>Liters $L$</th> </tr> </thead> <tbody> <tr> <td>$0$</td> <td>$0$</td> </tr> <tr> <td>$1$</td> <td>$0.75$</td> </tr> <tr> <td>$2$</td> <td>$1.5$</td> </tr> <tr> <td>$3$</td> <td>$2.25$</td> </tr> </tbody> </table>

Hints

- Test the claimed rule at the input where no bottles are used. - Decide whether $0.75$ is an amount added once or an amount contributed by each bottle.

Solution

1. The proposed rule gives $L=0+0.75=0.75$ when $b=0$, but the table gives $L=0$. 2. Each bottle contributes $0.75$ liter. 3. The correct equation is $L=0.75b$.

Answer

The claim is incorrect because zero bottles should give zero liters. The correct equation is $L=0.75b$.

5497626

Two rules, $T=3n+1$ and $T=4n$, both give $4$ tiles when $n=1$. Which rule matches the entire table and graph? Explain how the row with $n=0$ settles the question. <table> <thead> <tr> <th>Figure $n$</th> <th>Tiles $T$</th> </tr> </thead> <tbody> <tr> <td>$0$</td> <td>$1$</td> </tr> <tr> <td>$1$</td> <td>$4$</td> </tr> <tr> <td>$2$</td> <td>$7$</td> </tr> <tr> <td>$3$</td> <td>$10$</td> </tr> </tbody> </table>

Hints

- A rule that matches one point may still fail elsewhere. - Use the displayed input that makes the two proposed rules easiest to distinguish.

Solution

1. At $n=0$, the rule $T=3n+1$ gives $T=1$. 2. At $n=0$, the rule $T=4n$ gives $T=0$. 3. The table and graph contain $(0, 1)$, so $T=3n+1$ is the matching rule.

Answer

$T=3n+1$. At $n=0$, it gives $1$, while $T=4n$ gives $0$.

5497716

A music practice chart is shown. <table> <thead> <tr> <th>Practice sessions $s$</th> <th>Songs ready $r$</th> </tr> </thead> <tbody> <tr> <td>$0$</td> <td>$1$</td> </tr> <tr> <td>$1$</td> <td>$4$</td> </tr> <tr> <td>$2$</td> <td>$7$</td> </tr> <tr> <td>$3$</td> <td>$10$</td> </tr> </tbody> </table> Interpret $(2, 7)$, including why the point does not mean $7$ sessions and $2$ songs.

Hints

- Read the x-axis quantity before the y-axis quantity. - Use the table headings to reject the reversed interpretation.

Solution

1. Coordinates follow x-coordinate, then y-coordinate order. 2. The x-axis represents sessions, so the first coordinate is $2$ sessions. 3. The y-axis represents songs ready, so the second coordinate is $7$ songs. 4. Reversing the meanings would ignore the axis order.

Answer

The point means $2$ practice sessions correspond to $7$ songs ready. The coordinates cannot be reversed because the axes fix their order.

5497726

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.

5497736

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.

5497746

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.

5497816

Movie tickets cost $\$9$ each. <table> <thead> <tr> <th>Tickets $t$</th> <th>Cost $C$</th> </tr> </thead> <tbody> <tr> <td>$0$</td> <td>$\$0$</td> </tr> <tr> <td>$1$</td> <td>$\$9$</td> </tr> <tr> <td>$2$</td> <td>$\$18$</td> </tr> <tr> <td>$3$</td> <td>$\$27$</td> </tr> </tbody> </table> A student 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.

5497526

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 $1$.

5497546

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.

5497556

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.

5497566

A student claims the ribbon relationship is $r=24-2c$, where $r$ is the remaining length in inches and $c$ is the number of pieces cut. Use the table and graph to test the claim. If it is incorrect, write the correct equation. <table> <thead> <tr> <th>Pieces cut $c$</th> <th>Remaining length $r$ (in.)</th> </tr> </thead> <tbody> <tr> <td>$0$</td> <td>$24$</td> </tr> <tr> <td>$1$</td> <td>$21$</td> </tr> <tr> <td>$2$</td> <td>$18$</td> </tr> <tr> <td>$3$</td> <td>$15$</td> </tr> </tbody> </table>

Hints

- Test the proposed rule with one input other than zero. - Use the repeated change in the remaining length to repair the coefficient and keep track of the units.

Solution

1. The proposed rule gives $r=24-2\times1=22$, or $22\,\text{in.}$, when $c=1$, but the table gives $21\,\text{in.}$. 2. The remaining length decreases by $3\,\text{in.}$ for each additional piece cut. 3. Starting from $24\,\text{in.}$, the correct equation is $r=24-3c$.

Answer

The claim is incorrect. The correct equation is $r=24-3c$, where $r$ is measured in inches.

5497586

The candle-height relationship is shown. <table> <thead> <tr> <th>Hours $t$</th> <th>Height $H$ in cm</th> </tr> </thead> <tbody> <tr> <td>$0$</td> <td>$15$</td> </tr> <tr> <td>$2$</td> <td>$12$</td> </tr> <tr> <td>$4$</td> <td>$9$</td> </tr> <tr> <td>$6$</td> <td>$6$</td> </tr> </tbody> </table> a) What is the candle’s initial height? b) What is the change in height per hour? c) Write height $H$ in terms of hours $t$.

Hints

- Read the output paired with an input of zero. - Compare two rows and divide the loss in height by the elapsed time. - Represent a repeated decrease by subtraction in the equation.

Solution

1. At $t=0$, the height is $15$ centimeters. 2. From $t=0$ to $t=2$, the candle loses $3$ centimeters. Dividing by $2$ hours gives a decrease of $1.5$ centimeters per hour. 3. Starting at $15$ centimeters and subtracting $1.5$ centimeters each hour gives $H=15-1.5t$.

Answer

a) $15\,\text{cm}$ b) The height decreases by $1.5\,\text{cm}$ per hour. c) $H=15-1.5t$

5497596

Jordan writes $P=10g$ for the arcade relationship. Identify what Jordan omitted and write the correct equation. <table> <thead> <tr> <th>Games $g$</th> <th>Points $P$</th> </tr> </thead> <tbody> <tr> <td>$0$</td> <td>$5$</td> </tr> <tr> <td>$1$</td> <td>$15$</td> </tr> <tr> <td>$2$</td> <td>$25$</td> </tr> <tr> <td>$3$</td> <td>$35$</td> </tr> </tbody> </table>

Hints

- Check whether the proposed rule produces the displayed output at zero games. - Keep the per-game increase and the starting amount as separate parts of the rule.

Solution

1. The outputs increase by $10$ points per game, so the coefficient of $g$ is $10$. 2. The graph and table show $P=5$ when $g=0$, so there is a starting bonus of $5$ points. 3. Jordan omitted the starting bonus; the correct equation is $P=10g+5$.

Answer

Jordan omitted the $5$-point starting bonus. The correct equation is $P=10g+5$.

5497636

Which point correctly extends the cyclist relationship to $4$ hours: $(4, 100)$, $(4, 120)$, or $(120, 4)$? Explain your choice. <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>$30$</td> </tr> <tr> <td>$2$</td> <td>$60$</td> </tr> <tr> <td>$3$</td> <td>$90$</td> </tr> </tbody> </table>

Hints

- Extend the constant hourly increase by one more input step. - Use the axis order to eliminate the option with reversed coordinates.

Solution

1. The distance increases by $30$ miles each hour. 2. At $4$ hours, the distance is $4 \times 30=120\,\text{mi}$. 3. Ordered pairs list time first and distance second, so the correct point is $(4, 120)$.

Answer

$(4, 120)$

5497646

Extend the cooler relationship to hour $5$. Give the new ordered pair and state how much the temperature has changed from hour $3$ to hour $5$. <table> <thead> <tr> <th>Hours $h$</th> <th>Temperature $T$ in °C</th> </tr> </thead> <tbody> <tr> <td>$0$</td> <td>$18$</td> </tr> <tr> <td>$1$</td> <td>$16$</td> </tr> <tr> <td>$2$</td> <td>$14$</td> </tr> <tr> <td>$3$</td> <td>$12$</td> </tr> </tbody> </table>

Hints

- Count how many one-hour steps separate the two requested times. - Find the total decrease over those steps. - Subtract that decrease from the temperature at hour $3$ before writing the ordered pair.

Solution

1. The temperature decreases by $2^\circ\text{C}$ each hour. 2. From hour $3$ to hour $5$ is two hours, so the temperature decreases by $2\times2=4^\circ\text{C}$. 3. Starting from $12^\circ\text{C}$ at hour $3$, the hour-5 temperature is $12-4=8^\circ\text{C}$. 4. The new point is $(5, 8)$.

Answer

The point is $(5, 8)$, and the temperature decreases by $4^\circ\text{C}$ from hour $3$ to hour $5$.

5497656

At what day does the reading relationship reach $80$ pages? Give the corresponding ordered pair. <table> <thead> <tr> <th>Days $d$</th> <th>Pages $p$</th> </tr> </thead> <tbody> <tr> <td>$0$</td> <td>$20$</td> </tr> <tr> <td>$1$</td> <td>$32$</td> </tr> <tr> <td>$2$</td> <td>$44$</td> </tr> <tr> <td>$3$</td> <td>$56$</td> </tr> </tbody> </table>

Hints

- Compare the target page total with the starting total. - Determine how many equal daily increases fit into that difference.

Solution

1. Pages increase by $12$ each day from an initial $20$. 2. Reaching $80$ requires an increase of $80-20=60$ pages. 3. Since $60\div12=5$, the input is day $5$. 4. The ordered pair is $(5, 80)$.

Answer

Day $5$, represented by $(5, 80)$.

5497666

From minute $3$ to minute $4$, how much does the y-coordinate change? Use that change to give the point for minute $4$. <table> <thead> <tr> <th>Minutes $m$</th> <th>Liters $L$</th> </tr> </thead> <tbody> <tr> <td>$0$</td> <td>$0$</td> </tr> <tr> <td>$1$</td> <td>$4.5$</td> </tr> <tr> <td>$2$</td> <td>$9$</td> </tr> <tr> <td>$3$</td> <td>$13.5$</td> </tr> </tbody> </table>

Hints

- Find the repeated difference between consecutive output values. - Apply exactly one more input step to the last displayed point.

Solution

1. The y-coordinate increases by $4.5\,\text{L}$ for each additional minute. 2. From minute $3$, add $4.5$ to $13.5$: $13.5+4.5=18$. 3. The y-change is $+4.5\,\text{L}$, and the new point is $(4, 18)$.

Answer

The y-coordinate increases by $4.5\,\text{L}$, and the new point is $(4, 18)$.

5497676

The hall adds rows with $24$ seats each. How many more seats are there with $7$ rows than with $4$ rows, and what point represents $7$ rows? <table> <thead> <tr> <th>Rows $r$</th> <th>Seats $s$</th> </tr> </thead> <tbody> <tr> <td>$1$</td> <td>$24$</td> </tr> <tr> <td>$2$</td> <td>$48$</td> </tr> <tr> <td>$3$</td> <td>$72$</td> </tr> <tr> <td>$4$</td> <td>$96$</td> </tr> </tbody> </table>

Hints

- Compare the requested row count with the largest row count already listed. - Use the seats-per-row amount both for the increase and for the new total.

Solution

1. Moving from $4$ rows to $7$ rows adds $3$ rows. 2. The added seats are $3 \times 24=72$. 3. Seven rows contain $7 \times 24=168$ seats, so the point is $(7, 168)$.

Answer

There are $72$ more seats, and the point is $(7, 168)$.

5497756

Compare relationships A and B using the tables and graph panels. Table A <table> <thead> <tr> <th>$x$</th> <th>$A$</th> </tr> </thead> <tbody> <tr> <td>$0$</td> <td>$2$</td> </tr> <tr> <td>$1$</td> <td>$5$</td> </tr> <tr> <td>$2$</td> <td>$8$</td> </tr> <tr> <td>$3$</td> <td>$11$</td> </tr> </tbody> </table> Table B <table> <thead> <tr> <th>$x$</th> <th>$B$</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> <tr> <td>$3$</td> <td>$12$</td> </tr> </tbody> </table> a) At which listed input are the outputs equal? b) At $x=3$, which output is greater and by how much?

Hints

- Compare outputs row by row at the same input. - For the second part, subtract the smaller output from the larger one.

Solution

1. At $x=2$, both tables give output $8$, so the outputs are equal there. 2. At $x=3$, $A=11$ and $B=12$. 3. Output B is greater by $1$.

Answer

a) $x=2$ b) B is greater by $1$.

5497766

The tables and graphs compare two relationships. Table A <table> <thead> <tr> <th>$x$</th> <th>$A$</th> </tr> </thead> <tbody> <tr> <td>$0$</td> <td>$6$</td> </tr> <tr> <td>$1$</td> <td>$8$</td> </tr> <tr> <td>$2$</td> <td>$10$</td> </tr> <tr> <td>$3$</td> <td>$12$</td> </tr> <tr> <td>$4$</td> <td>$14$</td> </tr> <tr> <td>$5$</td> <td>$16$</td> </tr> </tbody> </table> Table B <table> <thead> <tr> <th>$x$</th> <th>$B$</th> </tr> </thead> <tbody> <tr> <td>$0$</td> <td>$2$</td> </tr> <tr> <td>$1$</td> <td>$5$</td> </tr> <tr> <td>$2$</td> <td>$8$</td> </tr> <tr> <td>$3$</td> <td>$11$</td> </tr> <tr> <td>$4$</td> <td>$14$</td> </tr> <tr> <td>$5$</td> <td>$17$</td> </tr> </tbody> </table> At what listed input do the outputs first become equal, and at what next listed input does B become greater than A?

Hints

- Compare the two tables at matching inputs in order. - After locating equality, inspect the next listed row rather than jumping ahead.

Solution

1. Compare the outputs in increasing input order. 2. At $x=4$, both outputs are $14$, so this is the first listed equality. 3. At the next listed input, $x=5$, $B=17$ and $A=16$, so B becomes greater.

Answer

The outputs first become equal at $x=4$. At $x=5$, B becomes greater than A.

5497796

A dog walker charges a base booking fee plus an hourly amount. The earnings are shown in the table and graph. Write an equation for earnings $E$ in terms of hours $h$. <table> <thead> <tr> <th>Hours $h$</th> <th>Earnings $E$</th> </tr> </thead> <tbody> <tr> <td>$0$</td> <td>$\$6$</td> </tr> <tr> <td>$1$</td> <td>$\$10$</td> </tr> <tr> <td>$2$</td> <td>$\$14$</td> </tr> <tr> <td>$3$</td> <td>$\$18$</td> </tr> </tbody> </table>

Hints

- Use the earnings at $h=0$ to identify the base booking fee. - Compare neighboring rows to find what one additional hour contributes.

Solution

1. Earnings increase by $\$4$ for each additional hour. 2. At $h=0$, the base booking fee is $\$6$. 3. The equation is $E=4h+6$.

Answer

$E=4h+6$

5497806

A student predicts the point $(6, 12)$ by multiplying $6$ hours by the hourly charge of $\$2$. Explain what the student missed and give the correct point. <table> <thead> <tr> <th>Hours $h$</th> <th>Cost $C$</th> </tr> </thead> <tbody> <tr> <td>$0$</td> <td>$\$7$</td> </tr> <tr> <td>$1$</td> <td>$\$9$</td> </tr> <tr> <td>$2$</td> <td>$\$11$</td> </tr> <tr> <td>$3$</td> <td>$\$13$</td> </tr> </tbody> </table>

Hints

- Separate the one-time charge from the amount that repeats each hour. - Check whether the proposed y-coordinate agrees with the cost at zero hours.

Solution

1. Multiplying $6 \times 2$ accounts for $\$12$ in hourly charges. 2. The relationship also includes the $\$7$ starting fee. 3. The total is $\$12+\$7=\$19$, so the correct point is $(6, 19)$.

Answer

The student missed the $\$7$ starting fee. The correct point is $(6, 19)$.

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