Applications of Slope and Intercept

1.A young sumo wrestler decided to go on a special high-protein diet to gain weight rapidly. When he started his diet, he weighed $79.5$. He gained weight at a rate of $5.5$ kilograms per month.

Let $y$ represent the sumo wrestler's weight (in kilograms) after $x$ months.

Which of the following could be the graph of the relationship?

Choose 1 answer:

A.

B.

C.

D.

2. Conrad read a $180$-page book cover to cover in a single session, at a constant rate. It took him $6$ hours.

Let $y$ represent the number of pages left to read after $t$ hours.

Which of the following information about the graph of the relationship is given?

Which of the following information about the graph of the relationship is given?

Choose 1 answer:

A. Slope and $x$-intercept

B. Slope and $y$-intercept

C. Slope and a point that is not an intercept

D. $x$-intercept and $y$-intercept

E. $y$-intercept and a point that is not an intercept

F. Two points that are not intercepts

3. Nour drove from the Dead Sea up to Amman, and her altitude changed at a constant rate. When she began driving, her altitude was $400$ meters below sea level. When she arrived in Amman $2$ hours later, her altitude was $1000$ meters above sea level.

Let $y$ represent Nour's altitude (in meters) relative to sea level after $x$ hours.

Which of the following could be the graph of the relationship?

Choose 1 answer:

A.

B.

C.

D.

4. Hiro painted his room. After $3$ hours of painting at a rate of $8$ square meters per hour, he had $28$ square meters left to paint.

Let $y$ represent the area (in square meters) left to paint after $x$ hours.

Which of the following information about the graph of the relationship is given?

Choose 1 answer:

A. Slope and $x$-intercept

B. Slope and $y$-intercept

C. Slope and a point that is not an intercept

D. $x$-intercept and $y$-intercept

E. $y$-intercept and a point that is not an intercept

F. Two points that are not intercepts

5. A charity organization had a fundraiser where each ticket was sold for a fixed price. After selling $200$ tickets, they had a net profit of $$12,000$. They had to sell a few tickets just to cover necessary production costs of $$1,200$.

Let $y$ represent the net profit (in dollars) when they have sold $x$ tickets.

Which of the following could be the graph of the relationship?

Choose 1 answer:

A.

B.

C.

D.

6. A charity organization had to sell $18$ tickets to their fundraiser just to cover necessary production costs. They sold each ticket for $$45$.

Let $y$ represent the net profit (in dollars) when they have sold $x$ tickets.

Which of the following information about the graph of the relationship is given?

Choose 1 answer:

A. Slope and $x$-intercept

B. Slope and $y$-intercept

C. Slope and a point that is not an intercept

D. $x$-intercept and $y$-intercept

E. $y$-intercept and a point that is not an intercept

F. Two points that are not intercepts

7. Mr. Mole left his burrow that lies below the ground and started digging his way at a constant rate deeper into the ground. Let $y$ represent Mr. Mole's altitude (in meters) relative to the ground after $x$ minutes.

Which of the following could be the graph of the relationship?

Choose 1 answer:

A.

B.

C.

D.

1. A.

The wrestler gained weight each month, so we expect to have a positive slope.

The wrestler weighed $79.5$ when he started his diet, so we expect a positive $y$-intercept.

Based on this description, we expect a positive slope and a positive $y$-intercept.

2. D. $x$-intercept and $y$-intercept

To describe a point on the graph, we need a corresponding $(x,y)$.

In our case, these are corresponding values of hours of reading and number of pages left to read.

$\text { hours reading}$ $\text { pages left}$

                       ↓    ↓

                     $(x,  y)$

To describe the slope of the graph, we need information about the rate of change of the relationship.

In our case, this is the rate at which the number of pages left to read changes over time (in pages per hour).

Conrad has $180$ pages left when he has read for $0$ hours. This corresponds to the point $(0,180)$ and is a $y$-intercept.

It took Conrad $6$ hours to have $0$ pages left to read. This corresponds to the point $(6, 0)$ and is an $x$-intercept.

The $x$-intercept and $y$-intercept are given.

3. B.

Nour's altitude increases after driving awhile, so we expect to have a positive slope.

Nour starts below sea level, so we expect a negative $y$-intercept.

Based on this description, we expect a positive slope and a negative $y$-intercept.

4. C. Slope and a point that is not an intercept

To describe a point on the graph, we need a corresponding $(x,y)$ pair.

In our case, these are corresponding values of hours of painting and area left to paint.

$\text { hours of painting}$ $\text { area left}$

                              ↓    ↓

                            $(x,  y)$

To describe the slope of the graph, we need information about the rate of change of the relationship.

In our case, this is the rate at which the area left to paint changes over time (in square meters per hour).

Hiro painted his room at a rate of $8$ square meters per hour. This corresponds to a slope of $-8$.

After $3$ hours of painting, Hiro had $28$ square meters left to paint. This corresponds to the point $(3, 28)$ and is not an intercept.

The slope and a point that is not an intercept are given.

5. B.

The profit increases with each ticket, so we expect to have a positive slope.

They had to sell a few tickets just to cover production costs (to bring their profit from a negative value up to $0$), so we expect a positive xxx-intercept.

Based on this description, we expect a positive slope and a positive $x$-intercept.

6. A. Slope and $x$-intercept

To describe a point on the graph, we need a corresponding $(x,y)$ pair.

In our case, these are corresponding values of number of tickets sold and net profit.

$\text { tickets sold}$ $\text { net profit}$

                   ↓    ↓

                $(x,  y)$

To describe the slope of the graph, we need information about the rate of change of the relationship.

In our case, this is the rate at which the net profit changes per ticket sold (in dollars per ticket).

The organization had to sell $18$ tickets to cover their production costs and bring their net profit from a negative value up to $$0$. This corresponds to the point $(18,0)$ and is an $x$-intercept.

They sold each ticket for $$45$. This corresponds to a slope of $45$.

The slope and $x$-intercept are given.

7. D.

Mr. Mole descends each minute, so we expect to have a negative slope.

Mr. Mole starts below the ground, so we expect a negative $y$-intercept.

Based on this description, we expect a negative slope and a negative $y$-intercept.

1. Beatrice graphed the relationship between the time (in seconds) since she sent a print job to the printer and the number of pages printed.

What does the $x$-intercept represent in this context?

Choose 1 answer:

A. The number of pages already printed when Beatrice sent the job

B. The number of seconds that passed before the printer started printing pages

C. The number of pages printed per second

D. None of the above

2. Nirmala graphed the relationship between the duration (in hours) of using an oil lamp and the volume (in milliliters) of oil remaining.

What feature of the graph represents how long Nirmala can use the lamp before it runs out of oil?

Choose 1 answer:

A. Slope

B. $x$-intercept

C. $y$-intercept

D. None of the above

3. Coleman graphed the relationship between the number of rotations of his front bike tire and the distance he traveled past a tree. Negative distance means the tree is in front of Coleman and positive distance means the tree is behind Coleman.

What does the $y$-intercept represent in this context?

Choose 1 answer:

A. The distance Coleman traveled per tire rotation

B. The number of tire rotations it took to be beside the tree

C. The number of tire rotations it took to travel $1$ meter

D. None of the above

4.Reagan graphed the relationship between the number of bars of music he played and the total number of times his metronome clicked.

What feature of the graph represents how many times the metronome clicks per bar?

Choose 1 answer:

A. Slope

B. $x$-intercept

C. $y$-intercept

D. None of the above

1. B. The number of seconds that passed before the printer started printing pages

What does the $x$-intercept represent?

The $x$-intercept tells us the value of the $x$-variable when the $y$-variable equals $0$.

In this context, $x$ represents the seconds since sending the print job, and $y$ represents how many pages have printed. So the $x$-intercept $(5, 0)$ says that when $5$ seconds had passed, there were $0$ pages printed.

The Answer

The $x$-intercept represents the number of seconds that passed before the printer started printing pages.

2. B. $x$-intercept

What does each feature tell us?

• The $x$-intercept tells us the value of the $x$-variable when the $y$ variable equals $0$.
• The $y$-intercept tells us the value of the $y$-variable when the $x$ variable equals $0$.
• The slope tells us how much the $y$-variable changes for each $1$-unit increase in the $x$-variable.

What feature do we need?

We want to know how long Nirmala can use the lamp before it runs out of oil. Since $y$ represents the volume of oil, we are looking for the value of $x$ (the duration) when $y = 0$ (when the lamp has no oil).

Answer

The $x$-intercept represents how long Nirmala can use the lamp before it runs out of oil.

3. D. None of the above

What does the $y$-intercept represent?

The $y$-intercept tells us the value of the $y$-variable when the $x$-variable equals $0$.

In this context, $x$ represents the number of tire rotations, and $y$ represents the distance past the tree. So the $y$-intercept at $(0, -15)$ says after $0$ tire rotations, Coleman was $15$ meters away from the tree and the tree was in front of him.

The Answer

The $y$-intercept represents the distance Coleman was away the tree when he started traveling.

Since that was not an option, the answer is none of the above.

4. A. Slope

What does each feature tell us?

• The $x$-intercept tells us the value of the $x$-variable when the $y$ variable equals $0$.
• The $y$-intercept tells us the value of the $y$-variable when the $x$ variable equals $0$.
• The slope tells us how much the $y$-variable changes for each $1$-unit increase in the $x$-variable.

What feature do we need?

We want to know how many times the metronome clicks per bar. Since $x$ represents the number of bars of music he played, we are looking for how much $y$ (metronome clicks) changes when $x$ (bars played) increases by $1$.

The Answer

Slope represents how many times the metronome clicks per bar.

1. Suraj took a slice of pizza from the freezer and put it in the oven. The pizza was heated at a constant rate.

The table compares the pizza's temperature (in degrees Celsius) and the time since Suraj started heating it (in minutes).

How fast was the pizza heated?

2. Suraj took a slice of pizza from the freezer and put it in the oven. The pizza was heated at a constant rate.

The table compares the pizza's temperature (in degrees Celsius) and the time since Suraj started heating it (in minutes).

How long did it take the pizza to reach $100$ degrees Celsius?

3. Julia got a delivery order. She took some time to get ready, and then she rode her bicycle at a constant speed.

The table compares the distance Julia made (in kilometers) and the time since she got the order (in minutes).

How long did it take Julia to ride $1$ kilometer?

4. Julia got a delivery order. She took some time to get ready, and then she rode her bicycle at a constant speed.

The table compares the distance Julia made (in kilometers) and the time since she got the order (in minutes).

How long did it take Julia to get ready for a delivery?

1. $7.5$ degrees Celsius per minute.

Since the pizza was heated at a constant rate, the table describes a linear relationship.

Moreover, the rate of change of this relationship corresponds to the rate at which the pizza was heated.

The table of values shows that for each increase of $6$ minutes in Time, Temperature increased by $45$ degrees Celsius. The rate at which the pizza was heated is the ratio of those corresponding differences:

$\frac{\Delta \text { Temperature }}{\Delta \text { Time }}=\frac{45}{6}=7.5$

In conclusion, the pizza was heated at a rate of $7.5$ degrees Celsius per minute.

2. $14$ minutes.

Since the pizza was heated at a constant rate, the table describes a linear relationship.

Moreover, the time it took the pizza to reach $100$ degrees Celsius corresponds to the case where the temperature was $100$ degrees Celsius.

The table of values shows that for each increase of $2$ minutes in the time, the temperature increased by $15$ degrees celsius.

Let's extend the table to get to $100$ degrees Celsius.

In conclusion, the pizza reached $100$ degrees Celsius after $14$ minutes.

3. $6$ minutes

Since Julia rode at a constant speed, the table describes a linear relationship.

Moreover, the rate of change of this relationship corresponds to the time it takes Julia to ride $1$ kilometer.

The table of values shows that for each increase of $2$ kilometers in Distance, Time increased by $12$ minutes. The time it takes Julia to ride $1$ kilometer is the ratio of those corresponding differences:

In conclusion, Julia rode $1$ kilometer in $6$ minutes.

4. $10$ minutes

Since Julia rode at a constant speed, the table describes a linear relationship.

Moreover, the time it took Julia to get ready for the delivery corresponds to the case where the distance $0$ kilometers.

The table of values shows that for each increase of $2$ kilometers in the distance, the time increases by $12$ minutes.

Let's extend the table backwards to get to $0$ kilometers.

In conclusion, it took Julia $10$ minutes to get ready for the delivery.

1. A battery is charged.

The percentage of the battery's capacity that is charged as a function of time (in minutes) is graphed.

At what rate is the battery charged?

Choose 1 answer:

A. $20$ percent per minute

B. $1$ percent per minute

C. $10$ percent per minute

D. $2$ percent per minute

2. Suraj took a slice of pizza from the freezer and put it in the oven.

The pizza's temperature (in degrees Celsius) as a function of time (in minutes) is graphed.

What was the pizza's temperature after $2$ minutes?

3. Archimedes drained the water in his tub.

The amount of water left in the tub (in liters) as a function of time (in minutes) is graphed.

How long did it take each time $18$ liters of water were drained?

Choose 1 answer:

A. $\frac {1}{3}$ minute

B. $\frac {1}{4}$ minute

C. $\frac {1}{2}$ minute

D. $1$ minute

4. Karl set out to Alaska in his truck.

The amount of fuel remaining in the truck's tank (in liters) as a function of distance driven (in kilometers) is graphed.

How fast did the truck consume its fuel?

Choose 1 answer:

A. $60$ liters per kilometer

B. $0.40$ liters per kilometer

C. $40$ liters per kilometer

D. $0.60$ per kilometer

1. D. $2$ percent per minute

The rate at which the battery was charged is equivalent to the rate of change of this relationship. In linear relationships, the rate of change is represented by the& slope of the line. We can calculate this slope from any two points on the line.

Two points whose coordinates are clearly visible from the graph are $(0, 40)$ and $(5, 50)$.

Now, to find the slope, let's take the ratio of the corresponding differences in the $y$-values and the $x$-values:

$\frac{50-40}{5-0}=\frac{10}{5}=2$

The slope of the line is $2$, which means the battery was charged at a rate of $2$ percent per minute.

2. $10$ degrees Celsius

To find the temperature that corresponds to $2$ minutes, we need to look for the point on the graph where Time is $2$.

The point we are looking for is $(2,10)$, which means that after $2$ minutes, the pizza's temperature was $10$ degrees Celsius.

3. B. $\frac{1}{4}$ minute

To find the duration that corresponds to a draining of $18$ liters of water, we need to find the relationship's rate of change. In linear relationships, the rate of change is represented by the slope of the line. We can calculate this slope from any two points on the line.

Two points whose coordinates are clearly visible from the graph are $(0,360)$ and $(2.5,180)$.

Now, to find the slope, let's take the ratio of the corresponding differences in the $y$-values and the $x$-values:

$\frac{180-360}{2.5-0}=\frac{-180}{2.5}=-72$

The slope of the line is $-72$, which means the rate of change is $72$ liters per minute. So $18$ liters of water were drained every $\frac{18}{72}=\frac{1}{4}$ minutes.

4. D. $0.6$ per kilometer

The rate at which Karl's truck consumed fuel is equivalent to the rate of change of this relationship. In linear relationships, the rate of change is represented by the slope of the line. We can calculate this slope from any two points on the line.

Two points whose coordinates are clearly visible from the graph are $(250, 350)$ and $(500, 200)$.

Now, to find the slope, let's take the ratio of the corresponding differences in the $y$-values and the $x$-values:

$\frac{200-350}{500-250}=\frac{-150}{250}=-0.6$

The slope of the line is $-0.6$, which means Karl's truck consumed its fuel at a rate of $0.60$ liters per kilometer.

1. Suraj took a slice of pizza from the freezer and put it in the oven. The oven heated the pizza at a rate of $7.5^{\circ}$ degrees Celsius per minute, and it reached the desired temperature of $80^{\circ}$ degrees Celsius after $12$ minutes.

Graph the relationship between the pizza's temperature (in degrees Celsius) and time (in minutes).

2. A pool has some initial amount of water in it. Then it starts being filled so the water level rises at a rate of $6$ centimeters per minute. After $20$ minutes, the water level is $220$ centimeters.

Graph the relationship between the pool's water level (in centimeters) and time (in minutes).

3. Sean tried to drink a slushy as fast as he could. He drank $5$ milliliters of slushy each second and finished all the slushy after $50$ seconds.

Graph the relationship between the amount of slushy left in the cup (in milliliters) and time (in seconds).

4. Rip van Winkle fell asleep for a very long time. When he fell asleep, his beard was $8$ millimeters long, and each passing week it grew $2$ additional millimeters.

Graph the relationship between the length of Rip van Winkle's beard (in millimeters) and time (in weeks).

1. The pizza reached the desired temperature of $80^{\circ}$ degrees Celsius after $12$ minutes. This is the same as saying that when Time was $12$, Temperature was $80$.

Therefore, the graph of the relationship should pass through the point $(12, 80)$.

We are also given the rate of change of the relationship: $7.5^{\circ}$ degrees Celsius per minute. This means that when Time increases by $1$, Temperature increases by $7.57$.

Therefore, the graph should also pass through the point $(12+1,80+7.5)$, which is the point $(13,87.5)$. However, this point doesn't appear on the grid! We need to find an equivalent pair of corresponding changes that would help us find a point that we can graph.

Note, for instance, that when Time decreases by $2$, Temperature decreases by $2 \cdot 7.5=15$. Therefore, the graph should pass through the point $( 12-2,80-15)$, which is the point $(10,65)$.

Now we can define the graph of the relationship.

2. The water level reached the desired height of $220$ centimeters after $20$ minutes. This is the same as saying that when Time was $20$, Water level was $220$.

Therefore, the graph of the relationship should pass through the point $(20, 220)$.

We are also given the rate of change of the relationship: $6$ centimeters per minute. This means that when Time increases by $1$, Water level increases by $6$.

Therefore, the graph should also pass through the point $(20+1, 220+6)$, which is the point $(21, 226 )$. However, this point doesn't appear on the grid! We need to find an equivalent pair of corresponding changes that would help us find a point that we can graph.

Note, for instance, that when Time decreases by $5$, Water level decreases by $5 \cdot 6=30$. Therefore, the graph should pass through the point $(20-5, 220-30)$, which is the point $( 15, 190)$.

Now we can define the graph of the relationship.

3. Sean finished all the slushy after $50$ seconds. This is the same as saying that when Time was $50$, Slushy was $0$.

Therefore, the graph of the relationship should pass through the point $(50, 0)$.

We are also given the rate of change of the relationship: $5$ milliliters per second. This means that when Time increases by $1$, Slushy decreases by $5$.

Therefore, the graph should also pass through the point $(50+1, 0-5)$, which is the point $(51, -5)$. However, this point doesn't appear on the grid (and it doesn't make sense either)! We need to find an equivalent pair of corresponding changes that would help us find a point that we can graph.

Note, for instance, that when Time decreases by $5$, Slushy increases by $5 \cdot 5=25$. Therefore, the graph should pass through the point $(50-5,0+25)$, which is the point $(45, 25)$.

Now we can define the graph of the relationship.

4. When Rip van Winkle fell asleep his beard was $8$ millimeters long. This is the same as saying that when Time was $0$, Beard was $8$.

Therefore, the graph of the relationship should pass through the point $(0,8)$.

We are also given the rate of change of the relationship: $2$ millimeters per week. This means that when Time increases by $1$, Beard increases by $2$.

Therefore, the graph should also pass through the point $(0+1,8+2)$, which is the point $(1,10)$. However, this point doesn't appear on the grid! We need to find an equivalent pair of corresponding changes that would help us find a point that we can graph.

Note, for instance, that when Time increases by $2$, Beard increases by $2 \cdot 2=4$. Therefore, the graph should pass through the point $(0+2,8+4)$, which is the point $(2, 12)$.

Now we can define the graph of the relationship.