Graph proportional relationships and compute slope

Learning goals

Graphs proportional relationships as lines through the origin.
Computes slope from two points.
Interprets slope as the constant of proportionality.
Compares proportional relationships using their slopes.

Difficulty: 1 – 5

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Graph proportional relationships and compute slope 1

The graph shows a proportional relationship between time and distance. What is the constant of proportionality, and what does it mean in this context?

Hints

- Use two clear grid points on the line. - Find the change in distance for one unit of time.

Solution

1. Choose two points on the line, such as $(0, 0)$ and $(1, 2)$. 2. The slope is $\frac{2 - 0}{1 - 0} = 2$. 3. The constant of proportionality is $2$, meaning the distance increases by $2$ miles for each hour.

Answer

The constant of proportionality is $2$. It means a rate of $2\,\text{miles per hour}$.

Graph proportional relationships and compute slope 2

The table shows a proportional relationship. <table><tr><th>$x$</th><th>$y$</th></tr><tr><td>$1$</td><td>$3$</td></tr><tr><td>$2$</td><td>$6$</td></tr><tr><td>$4$</td><td>$12$</td></tr></table> Find the constant of proportionality and write an equation relating $y$ to $x$.

Hints

- Compare $y$ to $x$ in more than one row. - A proportional relationship has the form $y = kx$.

Solution

1. For each row, compute $\frac{y}{x}$: the ratio is constant at $3$. 2. Therefore, the constant of proportionality is $3$. 3. The proportional equation is $y = 3x$.

Answer

Constant of proportionality: $3$ Equation: $y = 3x$

Graph proportional relationships and compute slope 3

At a copy shop, price is proportional to the number of pages. Printing $40$ pages costs $\$3.20$. a) Find the price per page. b) Find the total price for $150$ pages. c) Describe the graph of total price versus number of pages, with number of pages on the x-axis.

Hints

- Find the cost of one page. - What is the price when no pages are printed? - How are page count and price related?

Solution

1. The price per page is $3.20\div40=\$0.08$. 2. Printing $150$ pages costs $150\cdot0.08=\$12.00$. 3. Because the relationship is proportional, the graph is a straight line through the origin. Its slope is $\$0.08$ per page.

Answer

a) $\$0.08$ per page. b) $\$12.00$. c) A line through the origin with slope $\$0.08$ per page.

Graph proportional relationships and compute slope 4

A hiker travels at a constant pace. The graph shows distance $s$, in miles, as a function of time $t$, in hours. a) Find the hiking speed and write the function. b) How far has the hiker traveled after $4.5$ hours? c) How long does it take the hiker to travel $15$ miles?

Hints

- Use a clear point on the graph to find distance divided by time. - What does the slope represent in this context? - Substitute a time to find distance, or solve the equation for time.

Solution

1. The graph passes through $(5,15)$, so the speed is $15\div5=3\,\text{mph}$. Because the graph passes through the origin, the function is $s(t)=3t$. 2. Evaluate $s(4.5)=3\cdot4.5=13.5$, so the hiker travels $13.5$ miles. 3. Solve $15=3t$ to get $t=5$ hours.

Answer

a) $3\,\text{mph}$; $s(t)=3t$ b) $13.5$ miles c) $5$ hours

Graph proportional relationships and compute slope 5

The graph shows a side view of a planned bicycle ramp. Each grid unit represents $8$ inches in both directions. The top edge of the ramp begins at $(0, 0)$. Find the slope of the ramp and write an equation for the line.

Hints

- Count the vertical and horizontal grid changes. - Slope is rise divided by run. - A line beginning at the origin has a simple equation form.

Solution

1. The graph rises $3$ units while running $12$ units to the right. The slope is $m=3 \div 12=0.25$. 2. The line passes through the origin, so its equation is $y=0.25x$. The physical scale does not change the slope because both axes use the same scale.

Answer

The slope is $0.25$, and the equation is $y=0.25x$.

Graph proportional relationships and compute slope 6

A conveyor belt at a gravel plant carries material upward. The graph shows a side view of the belt, with horizontal distance on the x-axis and height on the y-axis, both in meters. Find the slope of the conveyor belt.

Hints

- Choose two points on the line that lie exactly on grid intersections. - Compare the vertical change with the horizontal change. - Use rise over run.

Solution

1. Use two clear points on the line, such as $(0, 0)$ and $(5, 3)$. 2. Calculate the slope: $m = \frac{3 - 0}{5 - 0} = \frac{3}{5}$. 3. Therefore, the conveyor belt has slope $\frac{3}{5}$, or $0.6$.

Answer

$m = \frac{3}{5}$, or $0.6$

Graph proportional relationships and compute slope 7

A pool fills at a constant rate. Find its fill rate in gallons per minute from the graph.

Hints

- Read each marked point from the axes. - Compare the change in volume to the change in time between the points. - State your slope with units of gallons per minute.

Solution

1. Read two marked points: $P_1(2,10)$ and $P_2(6,30)$. 2. The slope is $m=\frac{30-10}{6-2}=\frac{20}{4}=5$. 3. The pool fills at $5\,\text{gal/min}$.

Answer

$5\,\text{gal/min}$.

Graph proportional relationships and compute slope 8

Relationship A is given by $y = 2.5x$. Relationship B is shown in the table. <table><tr><th>$x$</th><th>$y$</th></tr><tr><td>$2$</td><td>$6$</td></tr><tr><td>$4$</td><td>$12$</td></tr><tr><td>$6$</td><td>$18$</td></tr></table> Which relationship has the greater constant of proportionality, and by how much?

Hints

- In an equation of the form $y=kx$, where is the constant of proportionality shown? - For the table, compare the ratio $\frac{y}{x}$ across the rows. - After finding both constants, compare them directly.

Solution

1. In Relationship A, the coefficient of $x$ is the constant of proportionality, so the constant is $2.5$. 2. In Relationship B, each ratio $\frac{y}{x}$ equals $3$, so its constant of proportionality is $3$. 3. Relationship B has the greater constant, and the difference is $3-2.5=0.5$.

Answer

Relationship B has the greater constant of proportionality by $0.5$.

Graph proportional relationships and compute slope 9

Relation A is a proportional relationship shown in the graph. Relation B is shown in the table. <table> <tr><th>$x$</th><th>$y$</th></tr> <tr><td>$2$</td><td>$5$</td></tr> <tr><td>$4$</td><td>$10$</td></tr> </table> Find the constant of proportionality for each relation. Which relation has the greater constant of proportionality, and by how much?

Hints

- For a proportional relationship, compare $\frac{y}{x}$. - Read one exact point on Relation A from the graph. - Check whether the table gives the same ratio in both rows before comparing the relations.

Solution

1. From the graph of Relation A, a convenient point is $(4,8)$, so its constant of proportionality is $\frac{8}{4}=2$. 2. From Relation B's table, $\frac{5}{2}=2.5$, and the second row confirms $\frac{10}{4}=2.5$. 3. Relation B has the greater constant of proportionality. 4. The difference is $2.5-2=0.5$.

Answer

Relation A: $2$ Relation B: $2.5$ Relation B is greater by $0.5$.

Graph proportional relationships and compute slope 10

A proportional relationship has the points $(1,4)$ and $(2,8)$. Find two more points on its graph: one where $x=0$ and one where $x=3$. Explain how you found them.

Hints

- What is the value of $y/x$ at either given point? - What must the output be when a proportional relationship has input zero? - Use the same rule for $x=3$.

Solution

1. A proportional graph has the form $y=kx$. Since $(1,4)$ is on the graph, $k=4$. 2. At $x=0$, $y=0$, and at $x=3$, $y=12$. 3. The two new points are $(0,0)$ and $(3,12)$; they lie on the same straight line as the given points.

Answer

$(0,0)$ and $(3,12)$. The rule is $y=4x$.

Graph proportional relationships and compute slope 11

A bike rental costs $\$3$ per hour, including fractions of an hour. You want to graph cost $y$ against time $x$ for up to $4$ hours. Choose useful tick intervals on both axes, give two points, and say how the graph connects them.

Hints

- What is the cost after one hour? After no time has passed? - Choose tick spacing so all four hours and their costs fit clearly. - Can a rental last half an hour, or are only whole hours possible?

Solution

1. The cost is $y=3x$ for $0\le x\le4$. 2. One useful scale is $1$ hour per x-axis tick and $3$ dollars per y-axis tick. 3. Two points are $(0,0)$ and $(4,12)$. Connect them with a straight line segment because a rental can last part of an hour.

Answer

For example: $1$ hour per x-tick, $\$3$ per y-tick; points $(0,0)$ and $(4,12)$; connect them with a straight line segment.

Graph proportional relationships and compute slope 12

To graph $y=-2x$, give three points that include one with a negative $x$-coordinate and one with a positive $x$-coordinate. Explain how to draw the graph through them.

Hints

- Pick x-values on both sides of zero. - Multiply each chosen input by the negative rate. - Does the line rise or fall as $x$ increases?

Solution

1. Choose $x=-1$, $x=0$, and $x=2$, giving $y=2$, $0$, and $-4$. 2. Plot $(-1,2)$, $(0,0)$, and $(2,-4)$. 3. Draw one straight line through all three points; it falls from left to right and crosses the origin.

Answer

One choice is $(-1,2)$, $(0,0)$, and $(2,-4)$. Draw the straight line through the points.

Graph proportional relationships and compute slope 13

A student drew line $g$ for the plotted dots. Does the line fit all the dots? Explain, then find the equation of the line that would fit them all.

Hints

- Check whether the drawn line passes through every dot. - What ratio of $y$ to $x$ do the dots share? - Does a proportional line pass through the origin?

Solution

1. From the dots $(1,3)$, $(2,6)$, and $(3,9)$, the constant ratio is $3$. 2. The correct line is $y=3x$. The student's line $g$ is $y=2x+1$, which passes through $(1,3)$ but misses $(2,6)$ and $(3,9)$. 3. Replace $g$ with the line through the three dots and the origin.

Answer

No. For example, $g$ gives $y=5$ at $x=2$, but the dot is at $(2,6)$. The correct line is $y=3x$.

Graph proportional relationships and compute slope 14

Read the water amounts at $1$ minute and $3$ minutes from line $g$. Is the amount proportional to time? Compare the two ratios $y/x$.

Hints

- Read the height of the line at each requested time. - Divide the amount by the time in each case. - What must be true of these ratios in a proportional relationship?

Solution

1. The graph shows $(1,3)$ and $(3,5)$. 2. The ratios are $3/1=3$ and $5/3$, which are different. 3. The amount is not proportional to time. The line starts at $2$ liters, not at zero.

Answer

At $x=1$, $y=3$ liters; at $x=3$, $y=5$ liters. No: $3/1\ne5/3$, so the relationship is not proportional.

Graph proportional relationships and compute slope 15

Find the constant of proportionality of the graphed line. Use two exact grid points to explain the rate of change.

Hints

- Read the values from the axes; the tick intervals are not identical. - Which two easy points on the line can you use? - Find the vertical change per one unit of horizontal change.

Solution

1. The line passes through $(0,0)$ and $(4,6)$. 2. Divide vertical change by horizontal change: $6\div4=1.5$. 3. The constant of proportionality is $1.5$, so $y=1.5x$.

Answer

$k=1.5$; the line rises $6$ units for a run of $4$ units.

Graph proportional relationships and compute slope 16

Two tables represent proportional relationships. **Table A:** <table> <tr><td>$x$</td><td>$2$</td><td>$4$</td><td>$6$</td></tr> <tr><td>$y$</td><td>$5$</td><td>$10$</td><td>$15$</td></tr> </table> **Table B:** <table> <tr><td>$x$</td><td>$3$</td><td>$6$</td><td>$9$</td></tr> <tr><td>$y$</td><td>$6$</td><td>$12$</td><td>$18$</td></tr> </table> a) Find the constant of proportionality $k$ for each table and write each equation in the form $y=kx$. b) Describe one feature the two graphs have in common. c) Which graph is steeper? Justify your answer using the constants of proportionality.

Hints

- Use one ordered pair to calculate $\frac{y}{x}$. - What feature do all graphs of proportional relationships share? - Compare the change in $y$ for the same change in $x$. - How does the coefficient of $x$ affect a line's steepness?

Solution

1. For Table A, $k=5\div2=2.5$, so its equation is $y=2.5x$. For Table B, $k=6\div3=2$, so its equation is $y=2x$. 2. Both proportional relationships graph as straight lines through $(0,0)$. 3. Table A's graph is steeper because its slope, $2.5$, is greater than Table B's slope, $2$.

Answer

a) Table A: $k=2.5$, $y=2.5x$. Table B: $k=2$, $y=2x$. b) Both graphs are lines through the origin. c) Table A's graph is steeper because $2.5>2$.

Graph proportional relationships and compute slope 17

Two school copiers work at different constant rates. Copier A prints $120$ pages in $3$ minutes. Copier B prints $150$ pages in $5$ minutes. For each copier, time $x$, in minutes, and pages printed $y$ form a proportional relationship. a) Write an equation for each copier. b) If both relationships are graphed, which graph is steeper? Explain what steepness means in this context. c) Both copiers start at the same time. How many pages has Copier A printed when Copier B finishes $360$ pages?

Hints

- What does the constant of proportionality represent here? - Find how many pages each copier prints in one minute. - What does a steeper graph mean for pages printed per minute? - First find how long Copier B takes to print $360$ pages.

Solution

1. Copier A prints $120\div3=40$ pages per minute, so $y_A=40x$. Copier B prints $150\div5=30$ pages per minute, so $y_B=30x$. 2. Copier A's graph is steeper because $40>30$. In context, greater steepness means more pages printed per minute. 3. Copier B needs $360\div30=12$ minutes to print $360$ pages. 4. In $12$ minutes, Copier A prints $40\cdot12=480$ pages.

Answer

a) Copier A: $y=40x$. Copier B: $y=30x$. b) Copier A's graph is steeper; slope represents pages per minute. c) Copier A has printed $480$ pages.

Graph proportional relationships and compute slope 18

The graph of a proportional relationship passes through $P(4, 14)$. a) Find the constant of proportionality $k$ and write the equation of the relationship. b) Use calculation to determine whether $Q(7, 24.5)$ also lies on the graph. c) Nikola claims, “If I double the $x$-value, the $y$-value always doubles.” Is Nikola correct? Justify your answer generally using $y=kx$.

Hints

- Use the known point with the proportional form $y=kx$. - Test a point by checking whether its coordinates satisfy the equation. - For the general claim, examine what happens to $kx$ when the input is doubled.

Solution

1. Substitute $(4,14)$ into $y=kx$: $14=4k$, so $k=3.5$ and the equation is $y=3.5x$. 2. For $x=7$, the equation gives $y=3.5\cdot7=24.5$, so $Q$ lies on the graph. 3. Replacing $x$ by $2x$ gives $y_{\text{new}}=k(2x)=2(kx)=2y$. Therefore, doubling $x$ doubles $y$ in any proportional relationship of the form $y=kx$.

Answer

a) $k=3.5$, so $y=3.5x$. b) Yes, because $3.5\cdot7=24.5$. c) Yes. Since $k(2x)=2(kx)$, doubling the input doubles the output.

Graph proportional relationships and compute slope 19

A cyclist travels at a constant speed. a) Explain what the marked point $P$ means. b) Find the slope and explain what it tells you about the cyclist. c) How far does the cyclist travel in $40$ minutes? d) How long does the cyclist take to travel $18$ miles?

Hints

- Read the coordinates of $P$ using the labeled axes. - Find the distance traveled for each minute. - Use that rate to find the distance for a different time. - How many minutes are needed to cover $18$ miles at this rate?

Solution

1. The marked point is $P(15,3)$: after $15$ minutes, the cyclist has traveled $3$ miles. 2. The slope is $3\div15=0.2$ mile per minute, the cyclist's speed. 3. In $40$ minutes, the cyclist travels $0.2\cdot40=8$ miles. 4. Traveling $18$ miles takes $18\div0.2=90$ minutes, or $1$ hour $30$ minutes.

Answer

a) The cyclist travels $3$ miles in $15$ minutes. b) $0.2$ mile per minute. c) $8$ miles. d) $90$ minutes.

Graph proportional relationships and compute slope 20

Two mobile-data plans charge in proportion to the amount of data used. Plan A: $\$1.50$ for $500\,\text{MB}$. Plan B: $\$2.40$ for $1000\,\text{MB}$. a) Which plan has the lower cost per megabyte? Justify your answer with calculations. b) If cost is graphed against data used, with data on the x-axis, which plan's graph is steeper? Explain the connection between cost per megabyte and slope.

Hints

- Divide each price by the matching number of megabytes. - What does a greater slope mean in a cost graph? - Compare the costs for the same amount of data.

Solution

1. Plan A costs $1.50\div500=\$0.003$ per megabyte. Plan B costs $2.40\div1000=\$0.0024$ per megabyte. 2. Since $0.0024<0.003$, Plan B has the lower cost per megabyte. 3. In a cost-versus-data graph, slope equals cost per megabyte. Plan A's graph is steeper because $0.003>0.0024$.

Answer

a) Plan B is cheaper at $\$0.0024$ per megabyte; Plan A costs $\$0.003$ per megabyte. b) Plan A's graph is steeper because its cost per megabyte is greater.

Graph proportional relationships and compute slope 21

Two pumps fill a water tank. Pump A moves $1200$ gallons in $4$ hours. Pump B moves $1500$ gallons in $6$ hours. For each pump, time and volume form a proportional relationship. a) Find each pump's rate in gallons per hour. b) When both pumps run together, their rates add. How much water enters the tank in $5$ hours? c) Describe the graphs for Pump A, Pump B, and their combined rate, with time on the x-axis and volume on the y-axis. Compare the steepness of the three lines and explain.

Hints

- Find how much each pump moves in one hour. - Add the rates when both pumps work at the same time. - How does a rate in gallons per hour relate to graph slope?

Solution

1. Pump A's rate is $1200\div4=300$ gallons per hour. Pump B's rate is $1500\div6=250$ gallons per hour. 2. Their combined rate is $300+250=550$ gallons per hour. In $5$ hours, they move $5\cdot550=2750$ gallons. 3. The three graphs are lines through $(0,0)$: $V_A=300t$, $V_B=250t$, and $V_C=550t$. 4. Slope represents pumping rate, so the combined graph is steepest, followed by Pump A and then Pump B because $550>300>250$.

Answer

a) Pump A: $300$ gallons per hour. Pump B: $250$ gallons per hour. b) $2750$ gallons. c) The graphs are the lines $V_A=300t$, $V_B=250t$, and $V_C=550t$, all through the origin. The combined graph is steepest, Pump A is next, and Pump B is least steep.

Graph proportional relationships and compute slope 22

Three points have coordinates $P_1(1.2, 3.6)$, $P_2(2.5, 7.5)$, and $P_3(5, 15.5)$. Determine whether all three points can lie on the same line through the origin. Justify your answer by testing proportionality.

Hints

- What distinguishes a line through the origin from other lines? - What ratio is constant for points satisfying $y=kx$? - Test the ratio for every point.

Solution

1. Points on a line through the origin satisfy $y=kx$, so $y\div x$ must be constant. 2. For $P_1$, $3.6\div 1.2=3$. 3. For $P_2$, $7.5\div 2.5=3$. 4. For $P_3$, $15.5\div 5=3.1$. 5. Since the third ratio differs, the points do not all lie on one line through the origin.

Answer

No. The first two ratios $y\div x$ equal $3$, but the third equals $3.1$.

Graph proportional relationships and compute slope 23

A line through the origin passes through $P(4.5, 1.8)$. a) Find the equation of the line. b) Determine whether $Q(7, 2.8)$ also lies on the line. c) A point $R(a, 4.4)$ lies on the line. Find $a$.

Hints

- What form does the equation of a line through the origin have? - Use the known point to find the slope. - Test a point by substituting its coordinates. - Solve the equation backward when the y-coordinate is known.

Solution

1. A line through the origin has equation $y=kx$. Using $P$, $k=1.8\div4.5=0.4$, so the equation is $y=0.4x$. 2. For $x=7$, the equation gives $y=0.4\cdot7=2.8$, so $Q$ lies on the line. 3. For $R(a,4.4)$, solve $4.4=0.4a$. Thus $a=4.4\div0.4=11$.

Answer

a) $y=0.4x$. b) Yes, because $0.4\cdot7=2.8$. c) $a=11$.

Graph proportional relationships and compute slope 24

Two runners, Anna and Ben, train for a charity run at constant speeds. - Anna runs $200$ meters in $40$ seconds. - Ben runs $300$ meters in $75$ seconds. a) Find each runner's speed in meters per second. Interpret each speed as the constant of proportionality for time $\to$ distance. b) Who runs faster? c) If both motions are graphed with time on the x-axis and distance on the y-axis, whose graph lies above the other for $t>0$? Justify your answer without drawing.

Hints

- Find how far each runner travels in one second. - What does slope represent on a distance-time graph? - For two lines through the origin, which is higher when its slope is greater?

Solution

1. Anna's speed is $200\div40=5\,\text{m/s}$. Ben's speed is $300\div75=4\,\text{m/s}$. Their equations are $s_A(t)=5t$ and $s_B(t)=4t$. 2. Anna is faster because $5>4$. 3. Anna's graph lies above Ben's for $t>0$ because it has the greater slope and therefore gives a greater distance at every positive time.

Answer

a) Anna: $5\,\text{m/s}$. Ben: $4\,\text{m/s}$. b) Anna. c) Anna's graph lies above Ben's because its slope is greater.

Graph proportional relationships and compute slope 25

Two liquids, A and B, have proportional relationships between volume $V$, in $\text{cm}^3$, and mass $m$, in grams. For Liquid A, $20\,\text{cm}^3$ has a mass of $16\,\text{g}$. For Liquid B, $15\,\text{cm}^3$ has a mass of $13.5\,\text{g}$. 1) Find the density of each liquid in $\text{g/cm}^3$. 2) If both relationships are graphed with volume on the x-axis and mass on the y-axis, which line is steeper? Explain using the densities. 3) Find the mass of $50\,\text{cm}^3$ of Liquid B. 4) Find the volume of Liquid A that has a mass of $64\,\text{g}$.

Hints

- What does mass divided by volume represent? - How is the constant of proportionality related to the slope? - Use the density to find mass from volume. - Rearrange the mass equation to solve for volume.

Solution

1. For Liquid A, $\rho_A=16\div20=0.8\,\text{g/cm}^3$. For Liquid B, $\rho_B=13.5\div15=0.9\,\text{g/cm}^3$. 2. Density is the slope of each mass-versus-volume line. Since $0.9>0.8$, the line for Liquid B is steeper. 3. For Liquid B, $m=0.9\cdot50=45\,\text{g}$. 4. For Liquid A, solve $64=0.8V$. Thus $V=64\div0.8=80\,\text{cm}^3$.

Answer

1) $\rho_A=0.8\,\text{g/cm}^3$; $\rho_B=0.9\,\text{g/cm}^3$ 2) Liquid B has the steeper line because $0.9>0.8$. 3) $45\,\text{g}$ 4) $80\,\text{cm}^3$

Graph proportional relationships and compute slope 26

At a farmers market, the price $P$, in dollars, of organic apples is proportional to the weight $w$, in pounds. A $2.5$-pound bag costs $\$6.25$. 1. Find the price per pound and write an equation for price as a function of weight. 2. Make a value table for $1$, $2$, $3$, and $4$ pounds. 3. The graph below shows the proportional relationship using $1$ grid unit per pound on the x-axis and $2$ dollars per grid unit on the y-axis. Name at least three points on the line. 4. Use the graph's linear pattern to estimate the cost of $3.5$ pounds and the weight that costs $\$5.00$. Then verify both values by calculation.

Hints

- First find the price of one pound. - A proportional relationship graphs as a line through the origin. - Multiply each weight by the unit price to complete the table. - Use horizontal and vertical guide lines when reading values from the graph.

Solution

1. The unit price is $6.25\div2.5=\$2.50$ per pound, so $P(w)=2.5w$. 2. The prices for $1$, $2$, $3$, and $4$ pounds are $\$2.50$, $\$5.00$, $\$7.50$, and $\$10.00$, respectively. 3. Three points on the displayed line are $(0,0)$, $(2,5)$, and $(4,10)$. 4. From the graph, $3.5$ pounds costs about $\$8.75$, and $\$5.00$ buys about $2$ pounds. Calculation verifies these values: $2.5\cdot3.5=8.75$ and $5.00\div2.5=2$.

Answer

1. $\$2.50$ per pound; $P(w)=2.5w$. 2. <table> <tr><th>Weight</th><td>$1\,\text{lb}$</td><td>$2\,\text{lb}$</td><td>$3\,\text{lb}$</td><td>$4\,\text{lb}$</td></tr> <tr><th>Price</th><td>$\$2.50$</td><td>$\$5.00$</td><td>$\$7.50$</td><td>$\$10.00$</td></tr> </table> 3. One valid set is $(0,0)$, $(2,5)$, and $(4,10)$. 4. The graph gives approximately $\$8.75$ for $3.5$ pounds and approximately $2$ pounds for $\$5.00$; calculation verifies both values exactly.

Graph proportional relationships and compute slope 27

Two candles burn at constant rates. The height lost $h$, in centimeters, is proportional to burning time $t$, in hours. - Candle A loses $3\,\text{cm}$ in $2$ hours. - Candle B loses $3\,\text{cm}$ in $3$ hours. 1. Write equations $h_A(t)$ and $h_B(t)$ for the height lost by each candle. 2. Which candle burns down faster? Explain using the slopes. 3. Find the height lost by each candle after $4.5$ hours. 4. After how many hours is the combined height loss $10\,\text{cm}$?

Hints

- A slope gives the height lost in one hour. - The candle with the greater height loss per hour burns down faster. - Add the two hourly rates to model their combined height loss.

Solution

1. Candle A's rate is $3\div2=1.5\,\text{cm/h}$, and Candle B's rate is $3\div3=1\,\text{cm/h}$. Thus $h_A(t)=1.5t$ and $h_B(t)=t$. 2. Candle A burns down faster because its slope is greater: $1.5>1$. 3. After $4.5$ hours, $h_A(4.5)=1.5\cdot4.5=6.75\,\text{cm}$, and $h_B(4.5)=4.5\,\text{cm}$. 4. The combined height loss is $1.5t+t=2.5t$. Solving $2.5t=10$ gives $t=4$ hours.

Answer

1. $h_A(t)=1.5t$; $h_B(t)=t$ 2. Candle A burns down faster because $1.5>1$. 3. Candle A: $6.75\,\text{cm}$; Candle B: $4.5\,\text{cm}$ 4. $4$ hours

Graph proportional relationships and compute slope 28

A farm stand sells blueberries for $\$1.20$ per $0.1$ pound. 1) Write an equation for the price $y$, in dollars, of $x$ pounds of blueberries. 2) A second stand charges $\$10.00$ per pound. Describe how the graphs of the two offers would differ. 3) Explain mathematically why the first relationship is proportional. 4) Find the price of $0.35$ pound of blueberries at the first stand.

Hints

- First find the price of one full pound. - What does slope represent in this context? - What happens to price when weight is multiplied by a factor? - Use the weight unit in the equation consistently.

Solution

1. The price per pound is $\$1.20\cdot10=\$12.00$, so $y=12x$. 2. The first graph has slope $12$, and the second has slope $10$. The second graph is less steep because its unit price is lower. 3. The ratio $\frac{y}{x}$ is always $12$, so the first relationship is proportional. 4. For $0.35$ pound, $y=12\cdot0.35=\$4.20$.

Answer

1) $y=12x$ 2) The second stand's graph is less steep because its slope is $10$ rather than $12$. 3) The ratio of price to weight is constant. 4) $\$4.20$

Graph proportional relationships and compute slope 29

A cyclist travels at a constant speed. After $15$ minutes, the cyclist has traveled exactly $4$ miles. 1) Find the constant of proportionality for time $t$, in hours, and distance $d$, in miles. What does it mean? 2) How would the graph change if the cyclist doubled the speed? 3) Find the distance traveled in $1.5$ hours. 4) Find the number of minutes needed to travel $10$ miles.

Hints

- Convert minutes to hours before finding miles per hour. - What happens to slope when a rate is doubled? - Rearrange the proportional equation to solve for the unknown. - Convert a decimal number of hours back to minutes.

Solution

1. Convert $15$ minutes to $0.25$ hour. The constant is $k=4\div0.25=16\,\text{mi/h}$, which is the cyclist's speed. 2. Doubling the speed doubles the slope, so the graph would be steeper. 3. In $1.5$ hours, $d=16\cdot1.5=24$ miles. 4. Traveling $10$ miles takes $10\div16=0.625$ hour. Converting gives $0.625\cdot60=37.5$ minutes.

Answer

1) $k=16$; it represents a speed of $16\,\text{mi/h}$. 2) The graph would have twice the slope and would be steeper. 3) $24$ miles 4) $37.5$ minutes

Graph proportional relationships and compute slope 30

Three proportional functions $f$, $g$, and $h$ are represented by lines through the origin. The equations are $f(x)=\frac{3}{4}x$ and $g(x)=1.2x$. The graph of $h$ passes through $(5, 2)$. a) Write the equation of $h$. b) Order the functions from least steep to most steep. c) Determine whether $(8, 6)$ lies on the graph of $f$.

Hints

- How can a point on a line through the origin be used to find its slope? - What does the coefficient of $x$ tell you about steepness? - Substitute the point's x-coordinate into the equation.

Solution

1. Since $h$ is proportional, substitute $(5,2)$ into $y=kx$: $2=5k$, so $k=\frac{2}{5}=0.4$. Thus $h(x)=0.4x$. 2. The slopes are $0.4$, $0.75$, and $1.2$. Since $0.4<0.75<1.2$, the order from least steep to most steep is $h$, $f$, $g$. 3. $f(8)=\frac{3}{4}\cdot8=6$, so $(8,6)$ lies on the graph of $f$.

Answer

a) $h(x)=0.4x$, or $h(x)=\frac{2}{5}x$ b) $h$, $f$, $g$ c) Yes, because $f(8)=6$.

Graph proportional relationships and compute slope 31

a) Find the slope of $p$ and write its equation. b) Without calculating, explain what happens to $y$ when $x$ is tripled. c) Use your equation to find the x-value when $y=10$, which is outside the graph.

Hints

- Count the horizontal and vertical grid steps in the slope triangle. - What ratio gives slope? - Think about how multiplying the input affects a proportional output. - Use the equation instead of trying to extend the graph to the missing point.

Solution

1. The slope triangle has rise $2$ and run $3$, so the slope is $\frac{2}{3}$. The equation is $p(x)=\frac{2}{3}x$. 2. In a proportional relationship, tripling $x$ also triples $y$. 3. Solve $10=\frac{2}{3}x$: $x=15$.

Answer

a) Slope $\frac{2}{3}$; $p(x)=\frac{2}{3}x$. b) $y$ triples. c) $x=15$.

Graph proportional relationships and compute slope 32

A line through the origin also passes through $(5, -2)$. 1. Write its equation in the form $y=mx$. 2. Without graphing, identify the quadrants through which the line passes. Explain. 3. Give the coordinates of another point on the line, other than the origin.

Hints

- What is the equation form of a line through the origin? - Substitute the known point to determine the slope. - How does the sign of the slope affect the quadrants through which the line passes? - Use the equation to generate another point.

Solution

1. Substitute the point into $y=mx$: $-2=5m$, so $m=-0.4$. Thus the equation is $y=-0.4x$. 2. A negative slope gives positive $y$ for negative $x$ and negative $y$ for positive $x$, so the line passes through Quadrants II and IV. 3. For example, if $x=10$, then $y=-0.4\cdot10=-4$, so $(10, -4)$ is another point on the line.

Answer

1. $y=-0.4x$ 2. Quadrants II and IV because the slope is negative. 3. For example, $(10, -4)$

Graph proportional relationships and compute slope 33

Three lines through the origin have equations $f(x)=-1.5x$, $g(x)=-\frac{1}{3}x$, and $h(x)=-4x$. 1. Order the graphs from least steep to most steep. Justify your order using absolute values of the slopes. 2. A point $R$ lies on $f$ and has y-coordinate $6$. Find its x-coordinate. 3. The constant of proportionality for $h$ is multiplied by $-1$. Describe how the new line compares with the original line.

Hints

- Steepness depends on the absolute value of the slope. - A point on a graph must satisfy its equation. - What changes when only the sign of a slope changes? - Which quadrants contain points whose coordinates have the same sign?

Solution

1. The slope magnitudes are $\frac{1}{3}$, $1.5$, and $4$. Therefore, the order from least steep to most steep is $g$, $f$, $h$. 2. Solve $6=-1.5x$: $x=6\div(-1.5)=-4$. 3. The new slope is $(-4)(-1)=4$. The line changes from decreasing to increasing and passes through Quadrants I and III instead of II and IV. Its steepness is unchanged.

Answer

1. $g$, $f$, $h$, because $\left|-\frac{1}{3}\right|<|-1.5|<|-4|$ 2. $x=-4$ 3. The new line is $y=4x$. It increases instead of decreases, but has the same steepness.

Graph proportional relationships and compute slope 34

A pump fills an empty tank at a constant rate. After $4$ minutes, the tank contains $140$ gallons; after $10$ minutes, it contains $350$ gallons. a) Find an equation $f(x)=kx$, where $x$ is time in minutes and $f(x)$ is volume in gallons. Explain the meaning of $k$. b) A technician records another point, $(15, 520)$. Determine whether it lies on the graph and explain what the result means. c) In general, if $(x_Q, y_Q)$, with $x_Q\ne0$, lies on this graph, what must $\frac{y_Q}{x_Q}$ equal? What is true for a point with nonzero x-coordinate that does not lie on the graph?

Hints

- Find the slope of a line through the origin from a known point. - What does slope mean for a filling process? - Substitute the recorded x-value into the equation and compare outputs. - What ratio is constant in a proportional relationship?

Solution

1. Compute the constant ratios: $140\div4=35$ and $350\div10=35$. Thus $f(x)=35x$, where $35$ is the filling rate in gallons per minute. 2. Evaluate $f(15)=35\cdot15=525$. Since $525\ne520$, the recorded point is not on the graph; the measurement is inconsistent with the constant-rate model. 3. Every point on the graph with nonzero x-coordinate satisfies $\frac{y_Q}{x_Q}=35$. A point with nonzero x-coordinate that is not on the graph has a different ratio.

Answer

a) $f(x)=35x$; $k=35$ is the rate in gallons per minute. b) No. $f(15)=525\ne520$, so the measurement is inconsistent with the model. c) On the graph, $\frac{y_Q}{x_Q}=35$. Off the graph, the ratio is not $35$.

Graph proportional relationships and compute slope 35

Two proportional linear functions $f$ and $g$ have graphs through the origin. The slope of $f$ is $-2$. The graph of $g$ passes through $(4, -6)$. 1. Write equations for $f$ and $g$. 2. Which line decreases more steeply? Explain. 3. Determine whether $(-1.5, 3)$ lies on $f$, on $g$, or on neither graph.

Hints

- How do slope and direction relate for a line? - What does the absolute value of slope tell you about steepness? - Substitute the point's coordinates to test whether it satisfies each equation. - How can one point determine the slope of a line through the origin?

Solution

1. Since $f$ has slope $-2$, $f(x)=-2x$. For $g$, $-6=4m$, so $m=-1.5$ and $g(x)=-1.5x$. 2. Since $|-2|>|-1.5|$, $f$ decreases more steeply. 3. Evaluate $f(-1.5)=-2(-1.5)=3$, so the point lies on $f$. Also, $g(-1.5)=(-1.5)(-1.5)=2.25\ne3$, so it does not lie on $g$.

Answer

1. $f(x)=-2x$; $g(x)=-1.5x$ 2. $f$ decreases more steeply because $|-2|>|-1.5|$. 3. The point lies on $f$ only.

Graph proportional relationships and compute slope 36

The proportional functions are $f(x)=2.5x$, $g(x)=-0.4x$, and $h(x)=\frac{1}{4}x$. a) Which graph is steepest? Justify your answer using the slopes. b) Through which quadrants does the graph of $g$ pass? Explain how the sign of the slope determines this. c) For each function, give one point $(x, y)$ with $x>0$ that lies on its graph.

Hints

- What does $|m|$ tell you about a line? - How do y-values behave for positive x-values when the slope is negative? - Choose an x-value and use each equation to find y.

Solution

1. Steepness is determined by the absolute value of slope. Since $|2.5|>|-0.4|$ and $|2.5|>\left|\frac{1}{4}\right|$, $f$ is steepest. 2. The slope of $g$ is negative, so for positive $x$, $y$ is negative, and for negative $x$, $y$ is positive. Thus the line passes through Quadrants II and IV. 3. Examples are $(2, 5)$ for $f$, $(5, -2)$ for $g$, and $(4, 1)$ for $h$.

Answer

a) $f$ is steepest because $|2.5|$ is the greatest slope magnitude. b) Quadrants II and IV because the slope is negative. c) For example, $f$: $(2, 5)$; $g$: $(5, -2)$; $h$: $(4, 1)$.

Graph proportional relationships and compute slope 37

Two electricity plans have no monthly base fee, so cost $y$, in dollars, is proportional to energy use $x$, in kilowatt-hours. Plan A is modeled by $y=0.40x$. For Plan B, $100\,\text{kWh}$ costs $\$35.00$. a) Write the equation for Plan B. b) Compare the costs for $500\,\text{kWh}$. Which plan is less expensive, and by how much? c) Which graph is steeper? Explain using the constants of proportionality. d) Determine whether $(120, 48)$ lies on the graph of Plan A.

Hints

- Find the price of one kilowatt-hour for Plan B. - What does the coefficient of $x$ tell you about the graph? - Substitute a point's x-coordinate into the equation to test it.

Solution

1. Plan B's unit rate is $\$35.00\div100=\$0.35$ per kilowatt-hour, so its equation is $y=0.35x$. 2. Plan A costs $\$0.40\cdot500=\$200.00$, while Plan B costs $\$0.35\cdot500=\$175.00$. Plan B is $\$25.00$ less expensive. 3. Plan A has the steeper graph because its constant of proportionality, $0.40$, is greater than Plan B's $0.35$. 4. For Plan A, $0.40\cdot120=48$, so $(120,48)$ lies on the graph.

Answer

a) $y=0.35x$ b) Plan B is $\$25.00$ less expensive: $\$175.00$ compared with $\$200.00$. c) Plan A is steeper because $0.40>0.35$. d) Yes, because $0.40\cdot120=48$.

Graph proportional relationships and compute slope 38

Water is poured at a constant rate into a cylindrical container. The graph shows water height $h$, in centimeters, as a function of time $t$, in seconds. a) Use the graph to explain why the relationship is proportional. Find the constant of proportionality with units and explain its meaning. b) Read the water height after $5$ seconds from the graph. c) Calculate when the water height reaches $20\,\text{cm}$.

Hints

- Describe the graph's shape and starting point. - Use a clearly readable point to find height divided by time. - What physical quantity is measured in centimeters per second? - Use the equation based on the graph to solve for the final time.

Solution

1. The graph is a straight line through $(0,0)$, so the relationship is proportional. It contains $(5,8)$, so $k=8\div5=1.6\,\text{cm/s}$. This means the water height rises $1.6\,\text{cm}$ each second. 2. At $t=5$, the graph shows $h=8\,\text{cm}$. 3. Using $h(t)=1.6t$, solve $20=1.6t$: $t=20\div1.6=12.5$ seconds.

Answer

a) The graph is a line through the origin. The constant is $1.6\,\text{cm/s}$, meaning the water height rises $1.6\,\text{cm}$ per second. b) $8\,\text{cm}$ c) $12.5$ seconds

Graph proportional relationships and compute slope 39

Four graphs of functions in the form $y = mx$ are shown. a) Match graphs $p_1$ through $p_4$ with the slopes $2$, $0.5$, $-1$, and $-0.25$. b) What happens to a graph as $|m|$ gets closer to zero? c) What point do all graphs of this form share? What are these relationships called?

Hints

- Use rise over run for each labeled line. - Compare how sharply the lines rise or fall. - Inspect the graph for any point that appears on every line. - Think about a line whose slope is very close to zero.

Solution

1. Graph $p_1$ passes through $(1, 2)$, so its slope is $2$. Graph $p_2$ passes through $(2, 1)$, so its slope is $0.5$. Graph $p_3$ passes through $(1, -1)$, so its slope is $-1$. Graph $p_4$ passes through $(4, -1)$, so its slope is $-0.25$. 2. As $|m|$ approaches zero, the line becomes less steep and gets closer to the x-axis. 3. Every graph passes through $(0, 0)$. These are proportional relationships.

Answer

a) $p_1$: $2$; $p_2$: $0.5$; $p_3$: $-1$; $p_4$: $-0.25$ b) The graph becomes less steep and closer to the x-axis. c) They all contain $(0, 0)$; they are proportional relationships.

Graph proportional relationships and compute slope 40

Line $g$ passes through the origin and the labeled point $A$. 1. State the coordinates of $A$. 2. Write the equation of $g$ in the form $y=mx$. 3. The y-value $-7.5$ is outside the displayed graph. Use your equation from part 2 to find the x-coordinate of a point $(x, -7.5)$ on the line.

Hints

- Pay attention to signs when reading coordinates. - For a line through the origin, slope can be found using $\frac{y}{x}$. - For part 3, use the equation rather than extending the displayed graph.

Solution

1. The graph shows $A=(-2,3)$. 2. The slope is $m=3\div(-2)=-1.5$, so the equation is $y=-1.5x$. 3. Use the equation to solve $-7.5=-1.5x$. Then $x=(-7.5)\div(-1.5)=5$.

Answer

1. $(-2, 3)$ 2. $y=-1.5x$ 3. $x=5$

Graph proportional relationships and compute slope 41

The graph shows the amount of water Tank A receives over time. Tank B receives $21\,\text{gal}$ in $7$ minutes at a constant rate. Which tank has the greater unit rate of water added? State each rate and explain how the graph represents Tank A's rate.

Hints

- On a proportional graph, what does the slope represent? - Read a convenient point on Tank A's line and compare rise to run from the origin. - Convert Tank B's information to a per-minute rate before comparing.

Solution

1. From the graph, Tank A's line passes through the origin and points such as $(1,4)$, so its unit rate is $4\,\text{gal/min}$. 2. Tank B's unit rate is $\frac{21}{7}=3\,\text{gal/min}$. 3. Tank A has the greater unit rate. 4. On Tank A's graph, the slope is the unit rate: the water amount rises $4$ gallons for each $1$-minute increase.

Answer

Tank A: $4\,\text{gal/min}$; Tank B: $3\,\text{gal/min}$. Tank A has the greater unit rate. The slope of Tank A's graph represents its unit rate.

Graph proportional relationships and compute slope 42

Consider all lines through the origin in the form $y=mx$. a) How does a line with $0<m<1$ compare with the line $y=x$? b) The line $p$ has equation $y=4x$. Reflecting it across the y-axis creates a line $q$. Write the equation of $q$ and explain. c) Give an equation of a line that passes through Quadrants I and III and is less steep than $y=0.5x$.

Hints

- Compare the slope with the slope of $y=x$. - What happens to x-coordinates under reflection across the y-axis? - What sign must the slope have to pass through Quadrants I and III? - Less steep means the slope magnitude is closer to zero.

Solution

1. The line $y=x$ has slope $1$. A slope between $0$ and $1$ is positive but smaller, so the line increases more gradually and lies closer to the x-axis. 2. Reflection across the y-axis replaces $x$ with $-x$: $y=4(-x)=-4x$. Thus $q$ has equation $y=-4x$. 3. To pass through Quadrants I and III, the slope must be positive. To be less steep than $y=0.5x$, it must satisfy $0<m<0.5$. One possible equation is $y=0.2x$.

Answer

a) It is less steep than $y=x$ and lies closer to the x-axis. b) $y=-4x$ c) For example, $y=0.2x$

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Graph proportional relationships and compute slope

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