Aimathic – Graph relationships from tables

Graph relationships from tables

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Difficulty: 1 – 5

5497426

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.

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

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

5497686

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 $3.6\,\text{mi}$.

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.

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 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

$18\,\text{mi}$

5497506

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

$15$

5497516

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 $6^\circ\text{C}$

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 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 $s=9r$.

5497606

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.

5497616

A student claims the bottle relationship shown on the graph is $L=b+0.75$, where $b$ is the number of bottles and $L$ is the number of 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 the student'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 the student'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

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

5497626

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.

5497706

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 $800\,\text{m}$.

5497716

The graph describes practice sessions and songs ready. Read the plotted point whose x-coordinate is $6$. Interpret that ordered pair in context. Then explain why reversing its coordinates would describe a different situation.

Hints

- Read the axis labels before interpreting either coordinate. - Find the point at x-coordinate $6$. - Coordinate order follows horizontal quantity first, vertical quantity second.

Solution

1. The plotted point at x-coordinate $6$ is $(6,5)$. 2. The horizontal axis is labeled practice sessions, so the first coordinate represents $6$ sessions. 3. The vertical axis is labeled songs ready, so the second coordinate represents $5$ songs. 4. Reversing the coordinates would mean $5$ practice sessions and $6$ songs ready, which is a different statement.

Answer

The point is $(6,5)$: $6$ practice sessions correspond to $5$ songs ready. Reversing the coordinates would instead mean $5$ sessions and $6$ songs ready.

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 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

- Use the vertical change and horizontal change between two plotted points. - The graph's points are two cuts apart, so divide the ribbon change by the change in cuts. - Use the point at zero cuts for the starting value.

Solution

1. The graph shows $(0,50)$, so the starting length is $50$ inches. 2. From $(0,50)$ to $(2,42)$, the ribbon decreases by $8$ inches 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

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

5497586

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) $H=18-1.5t$

5497596

Jordan writes $P=5g$ for the arcade relationship shown on the graph. Use the graph to find $P$ when $g=0$, then use the points at $g=2$ and $g=4$ to determine the change per game. Explain what Jordan omitted and write the correct equation.

Hints

- Read the graph at zero games first. - The points used for the rate are two games apart, so account for the horizontal change. - A linear equation here needs both the per-game change and the starting value.

Solution

1. The graph shows $(0,7)$, so the relationship starts at $P=7$. 2. From $(2,17)$ to $(4,27)$, the output increases by $10$ over $2$ games. 3. The increase per game is $10\div2=5$. 4. Jordan included the rate but omitted the starting value, so the correct equation is $P=5g+7$.

Answer

Jordan omitted the starting value $7$. The correct equation is $P=5g+7$.

5497636

The graph shows the area $A$, in square units, of a square with side length $s$, in units. <table> <thead> <tr><th>Side length $s$</th><th>Area $A$</th></tr> </thead> <tbody> <tr><td>$2$</td><td>$?$</td></tr> <tr><td>$5$</td><td>$?$</td></tr> </tbody> </table> Use the graph to complete the table. Then compare the change in area from $s=1$ to $s=2$ with the change from $s=4$ to $s=5$. Is the relationship linear? Explain.

Hints

- Find the y-value on the graph for each given x-value in the table. - Then compare how much the area increases over each one-unit increase in side length. - A linear relationship has the same output change for equal input changes.

Solution

1. Read the graph at $s=2$. The corresponding area is $4$, so the first missing value is $4$. 2. Read the graph at $s=5$. The corresponding area is $25$, so the second missing value is $25$. 3. From $s=1$ to $s=2$, the area changes from $1$ to $4$, so the increase is $3$. 4. From $s=4$ to $s=5$, the area changes from $16$ to $25$, so the increase is $9$. 5. Equal input changes do not produce equal output changes, so the relationship is not linear.

Answer

$A=4$ when $s=2$, and $A=25$ when $s=5$. The change from $s=1$ to $s=2$ is $+3$, but the change from $s=4$ to $s=5$ is $+9$. Since the output change is not constant, the relationship is not linear.

5497646

The graph shows the travel time $T$, in hours, for a $24$-mile bike ride at speed $v$, in miles per hour. <table> <thead> <tr><th>Speed $v$</th><th>Time $T$</th></tr> </thead> <tbody> <tr><td>$3$</td><td>$?$</td></tr> <tr><td>$6$</td><td>$?$</td></tr> </tbody> </table> Use the graph to complete the table. Then explain why the graph does not represent a linear relationship.

Hints

- Read the y-value from the graph at each listed speed. - Look at the overall shape of the graph. - For a linear relationship, equal x-changes give equal y-changes and the graph is a straight line.

Solution

1. Read the graph at $v=3$. The corresponding travel time is $8$ hours. 2. Read the graph at $v=6$. The corresponding travel time is $4$ hours. 3. The graph is curved rather than straight, and the output does not change by a constant amount for equal changes in the input. 4. Therefore, the relationship is not linear.

Answer

$T=8$ when $v=3$, and $T=4$ when $v=6$. The relationship is not linear because the graph is not a straight line and the output change is not constant for equal input changes.

5497676

The graph shows a relationship between an input $x$ and an output $y$. <table> <thead> <tr><th>$x$</th><th>$y$</th></tr> </thead> <tbody> <tr><td>$1$</td><td>$?$</td></tr> <tr><td>$4$</td><td>$?$</td></tr> </tbody> </table> Use the graph to complete the table. Then decide whether the relationship is linear. Explain your reasoning by comparing the output change from $x=1$ to $x=2$ with the output change from $x=3$ to $x=4$.

Hints

- Read the graph values at $x=1$ and $x=4$ first. - Then compare how much the output changes over the two one-unit intervals. - A linear relationship has the same output change for equal input changes.

Solution

1. Read the graph at $x=1$. The output is $3$. 2. Read the graph at $x=4$. The output is $18$. 3. From $x=1$ to $x=2$, the output changes from $3$ to $6$, so the increase is $3$. 4. From $x=3$ to $x=4$, the output changes from $11$ to $18$, so the increase is $7$. 5. The output changes are not equal, so the relationship is not linear.

Answer

$y=3$ when $x=1$, and $y=18$ when $x=4$. The relationship is not linear because the output change from $x=1$ to $x=2$ is $+3$, but from $x=3$ to $x=4$ it is $+7$.

5497756

Panels a) and b) show two relationships at the same four inputs. a) At which plotted input do the outputs match? b) At the largest plotted input, which relationship has the greater output, and by how much?

Hints

- Compare points only when they have the same x-coordinate. - Look for a shared y-coordinate across the two panels. - For part b), use the rightmost plotted input in both panels.

Solution

1. Compare the panels at matching x-values $10,20,30,40$. 2. At $x=30$, both panels have y-coordinate $24$, so the outputs match there. 3. At $x=40$, panel a) shows $30$ and panel b) shows $33$. 4. Panel b) is greater by $33-30=3$.

Answer

a) $x=30$ b) Panel b) is greater by $3$.

5497766

Panels a) and b) show two relationships for negative and positive inputs. At which plotted input are the outputs equal? At the next plotted input to the right, which panel has the greater output?

Hints

- The x-axis includes negative values, so compare the panels from left to right carefully. - Find the input where the y-coordinates match. - Then move to the next plotted x-value to the right in both panels.

Solution

1. Compare the two panels at $x=-2,-1,0,1,2$. 2. At $x=0$, both panels have output $5$, so the outputs are equal there. 3. The next plotted input to the right is $x=1$. 4. At $x=1$, panel a) has output $3$ and panel b) has output $7$, so panel b) is greater.

Answer

The outputs are equal at $x=0$. At $x=1$, panel b) has the greater output.

5497796

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 $E=6h+9$.

5497806

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. A student 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 the student 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 $3$ minutes, 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 $6$ minutes, so the rate is $6\div6=1\,\text{cm/min}$. 3. The rates are different, so the student 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.

5540856

The graph shows a relationship between an input $s$ and an output $A$; no rule is given. <table> <thead> <tr><th>$s$</th><th>$A$</th></tr> </thead> <tbody> <tr><td>$3$</td><td>?</td></tr> <tr><td>$4$</td><td>$16$</td></tr> </tbody> </table> Use the graph to complete the table. Then compare the output change from $s=1$ to $s=2$ with the change from $s=3$ to $s=4$. Based on those changes, decide whether the relationship is linear and explain your reasoning.

Hints

- Read the output at $s=3$ directly from the graph. - Compare the vertical changes over the two one-unit horizontal intervals named in the question. - Ask whether equal horizontal changes produce equal vertical changes.

Solution

1. At $s=3$, the graph has y-coordinate $9$, so the missing output is $9$. 2. From $s=1$ to $s=2$, the output changes from $1$ to $4$, an increase of $3$. 3. From $s=3$ to $s=4$, the output changes from $9$ to $16$, an increase of $7$. 4. Equal one-unit input increases produce different output changes, so the relationship is not linear.

Answer

The missing output is $9$. The two output increases are $3$ and $7$. Because equal input increases do not produce equal output changes, the relationship is not linear.

5540866

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.

5540876

The graph shows a relationship for both negative and positive x-values. a) Read the y-values at $x=-2$ and $x=2$. b) Compare the y-changes from $x=-2$ to $-1$, from $-1$ to $0$, and from $0$ to $1$. Explain why the relationship is not linear.

Hints

- Read the graph on both sides of the y-axis. - For part b), compare consecutive points that are one x-unit apart. - A straight-line relationship would have one constant y-change for every equal x-step.

Solution

1. The graph shows $(-2,6)$ and $(2,6)$, so both requested y-values are $6$. 2. From $x=-2$ to $-1$, y changes from $6$ to $3$, a change of $-3$. 3. From $x=-1$ to $0$, y changes from $3$ to $2$, a change of $-1$. 4. From $x=0$ to $1$, y changes from $2$ to $3$, a change of $+1$. 5. Equal one-unit changes in x produce different y-changes, so the relationship is not linear.

Answer

a) Both y-values are $6$. b) The successive y-changes are $-3$, $-1$, and $+1$, so the relationship is not linear.

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