Intercept (of Linear) Formula
\begin{equation*}
\text{intercept} = \text{dep} - \text{slope}\ast\text{indep}
\end{equation*}
Section 4.3 Intercepts and direct proportionality
Kaleb runs \(8 \tfrac12\) minute miles, which means it takes him around 8.5 minutes to run each mile. Yesterday he was out for about 30 minutes and ran the 2.8 mile loop by our house. That strikes me as curious because if he ran 2.8 miles at 8.5 minutes per mile that should take
\begin{equation*}
\frac{8.5 \text{ minutes}}{\text{mile}} \ast 2.8 \text{ miles} = 8.5 \times 2.8 = 23.8 \approx 24 \text{ minutes}
\end{equation*}
But Kaleb took 30 minutes. That’s 6 minutes longer than expected. Well, technically 6.2 minutes since
\begin{equation*}
30 - 23.8 = 6.2 \approx 6 \text{ minutes}
\end{equation*}
but let’s work with 6 since the 30 was only approximate to begin with.
The point is, what’s up with that missing 6 minutes? Oh, I bet I know what it is. Ever since Kaleb turned fifty years old, he’s been having trouble with his knees. I bet he’s finally stretching like his doctor ordered. Must be around 6 minutes of stretches after each run.
Since Kaleb’s total time is function of how far he runs, our variables are
\begin{align*}
T \amp= \text{ total time (minutes) } \sim \text{ dep} \\
D \amp= \text{ distance (miles) } \sim \text{ indep}
\end{align*}
Notice that we are determining how the time Kaleb spends running depends on the distance he runs, so the time \(T\) is our dependent variable. Often time is the independent variable, but not so here.
For the sake of this problem, we assume Kaleb runs a steady 8.5 minutes per mile so the rate of change is constant. The equation must be linear and so it fits the template
\begin{equation*}
\text{dep }=\text{ start } + \text{slope} \ast {\text{indep}}
\end{equation*}
The slope is 8.5 minutes per mile. The 6 minutes Kaleb spends stretching is the intercept, even though it’s named “start” in the template and Kaleb is actually stretching at the end of his run. A better name might be “fixed.” Whatever you call it, the equation is
\begin{equation*}
\textbf{Kaleb:}\quad T = 6 + 8.5D
\end{equation*}
As a quick check, for that 2.8 mile run we have \(D=2.8\) and so
\begin{equation*}
T = 6 + 8.5 \ast 2.8 = 6 + 8.5 \times \underline{2.8} = 29.8 \approx 30 \text{ minutes}
\end{equation*}
By the way, there’s a shorter way to find the intercept. The intercept is the “starting value,” or in this case the time spent stretching. So we take the total time and then subtract out the time spent running
\begin{equation*}
\text{intercept }=30 - 8.5 \times 2.8 = 6.2 \approx 6 \text{ minutes}
\end{equation*}
In general,
Kaleb’s daughter Muna runs considerably faster, 7 minute miles, and she’s not into stretching at all. For her to run the 2.8 mile loop by our house, it would take
\begin{equation*}
\frac{7 \text{ minutes}}{\text{mile}} \ast 2.8 \text{ miles} = 7 \times 2.8 = 19.6 \text{ minutes}
\end{equation*}
That means while her dad would take 30 minutes to run the loop and do his stretches, Muna can run it in just under 20 minutes.
The equation for Muna is
\begin{equation*}
\textbf{Muna:}\quad T = 7D
\end{equation*}
The slope is 7 minutes per mile. What’s the intercept for this equation? There’s no time for stretching in her equation, so it’s like \(T = 0 + 7D\text{.}\) The intercept is 0 minutes.
Compare the graphs. Each intercept shows where that line meets the vertical axis. Kaleb’s crosses at 6 minutes, but Muna’s crosses at 0 minutes, at the origin (where the two axes cross).
By the way, Muna’s equation \(T = 7D\) is a direct proportionality because the only thing happening is that the independent variable is being scaled by a proportionality constant, \(k=7\text{.}\) Any direct proportionality fits this template.
Direct Proportionality Template
\begin{equation*}
\text{dep} = k \ast \text{indep}
\end{equation*}
To understand what “proportionality” means, recall that Muna can run 2.8 miles in 19.6 minutes. What happens if she goes for a run twice as long? Then she would be running \(2 \times 2.8 = 5.6\) miles. Her time would be
\begin{equation*}
T = 7 \ast 5.6 = 7 \times \underline{5.6} = 39.2 \text{ minutes}
\end{equation*}
Notice that \(2 \times 19.6 = 39.2\text{.}\) So, it would take her twice the time to run twice the distance. This general idea - that you get twice the value of the dependent variable if you have twice the value of the independent variable - characterizes direct proportions. We sometimes say that Muna’s time is proportional to how far she runs. Nothing special about twice here, as it would take her three times the time to run three times the distance, etc.
Not so for Kaleb. Remember it takes him 29.8 minutes to run that 2.8 miles. If he runs twice the distance, which is 5.6 miles, it takes
\begin{equation*}
T = 6+8.5 \ast 5.6 = 6+ 8.5 \times \underline{5.6} = 53.6 \text{ minutes}
\end{equation*}
which is not quite twice the time, since \(2 \times 29.8 = 59.6 \text{ minutes}\text{.}\) The key is that Kaleb does not stretch twice, only once, for the longer run so double the distance does not count the 6 minutes again. Kaleb’s equation is not a direct proportionality. Another way to say that is that Kaleb’s time is not proportional to how far he runs. It is a function of how far he runs, yes, but not proportionally so.
Do you know …
- What the intercept of a linear function means in the story and what it tells us about the graph?
- How to calculate the intercept given the slope and an example (another point on the graph)?
- Why an intercept might not make sense, for example if it’s outside the domain of the function?
- When a linear function is a direct proportion?
- Why you cannot reason proportionally if the linear function is not a direct proportion?
- What the graph of a direct proportion looks like?
If you’re not sure, work the rest of exercises and then return to these questions. Or, ask your instructor or a classmate for help.
Exercises Exercises
Exercises 1-4 are available in a separate workbook format.
1.
In each of the following stories, the temperature changes over time. It might be confusing to call either variable \(T\text{,}\) so use \(H\) for the time in hours and \(D\) for the temperature in degrees (°F). In each case, time should be measured from the start of the story.
(a)
It was really cold at 8:30 this morning when Raina arrived at the office. Luckily the heating system warms things up very quickly, 4°F per hour. By 11:00 a.m. it was a very comfortable 72°F.
(i)
Figure out what the temperature was at 8:30 a.m.
(ii)
Write an equation illustrating the function.
(b)
While 72°F is a perfectly good temperature for an office, not so for ballroom dancing. When Raina arrived for her practice at 5:30 that evening, she began to sweat before she even took the floor. Turns out the air conditioner had been running since 4:00 p.m. but it only cools down the room 3°F per hour.
(i)
Figure out what the temperature was at 4:00 p.m.
(ii)
Write an equation illustrating the function.
2.
Maryn is very happy. Her interior design business is finally showing a profit. She has logged a total of 471 billable hours at $35 per hour since she started her business. Accounting for start up costs, her net profit now totals $2,194.
(a)
How much were Maryn’s start up costs?
(b)
Identify the slope and intercept (including their units and sign) and explain what each means in terms of the story.
(c)
Calculate what Maryn’s profits will be once she has logged a total of 1,000 hours.
(d)
Name the variables and write an equation relating them.
(e)
Graph the function.
3.
(a)
Jerome has gained weight since he took his power training to the next level ten weeks ago, at the rate of around 1 pound a week. He now weighs 198 pounds.
(b)
Vanessa’s doctor put her on a sensible diet and exercise plan to get her back to a healthy weight. She will need to lose an average of 1.25 pounds a week to reach her goal weight of 148 pounds in a year. Use 1 year = 52 weeks.
(c)
After the past 6 weeks of terrible migrane headaches, Carlos is down to 158 pounds. He has lost 4 pounds a week.
(d)
Since she has been pregnant, Zoe has gained the recommended \(\sfrac{1}{2}\) pound per week. Now 30 weeks pregnant and 168 pounds, she wonders if she will ever see her feet again.
4.
Each story describes a situation that we are assuming is linear. Decide whether it is proportional, meaning the intercept equals zero. If it is proportional, explain why the intercept would be zero. If it is not proportional, explain what the intercept would mean in the story.
(a)
The price of kiwis (a kind of fruit) depends on how many kiwis you buy.
(b)
The price of a bag of tortillas depends on how many tortillas are in the bag.
(c)
The time it takes to vacuum a rug depends on the area of the rug.
(d)
The time it takes to wash dishes depends on how many dirty dishes there are.
(e)
The amount of laundry detergent I have left depends on how many loads of laundry I did.
5.
Different runners run at different paces. And take a different amount of extra time to warm-up and/or cool down. The table lists six runners, their training time to run a 5K (rounded to the nearest minute), and their pace (in minutes per mile).
| Name | Yannick | Olga | Aziz | Hitomi | Galen | Fiona |
|---|---|---|---|---|---|---|
| Pace | 8.2 | 8.6 | 9.5 | 10 | 10 | 11.2 |
| 5K time | 32 | 35 | 33 | 36 | 31 | 44 |
| extra time | ? | ? | ? | ? | ? | ? |
(a)
We are interested in each runner’s extra time, but first convert 5K, which is short for 5 kilometers, to miles using \(1 \text{ mile} \approx 1.609 \text{ kilometers}\text{.}\)
(b)
Now, determine the extra (warm up/cool down) time for each runner and list your answer in the table. Report your answer to the nearest minute.
(c)
List the runners in order from least to most warm up/cool down time.
6.
At 10:00 a.m. we’ve got snowy skies and 4 inches of new snow on the ground. It’s coming down fast out there at a rate of an inch per hour.
(a)
Name the variables, measuring time in hours since 10:00 a.m.
(b)
Write an equation illustrating the dependence.
(c)
When did the snowstorm start?
(d)
Name a new variable for the time measured in hours since the snowstorm started.
(e)
Write an equation illustrating the dependence using this new variable instead.
(f)
Check that this equation confirms 4 inches of new snow at 10:00 a.m.
(g)
Explain why the two equations have different intercepts.
7.
The public beach near Paloma’s house has lost about 3′9″ feet a year of beach depth (measured from the dunes to the high water mark) due to erosion since they started keeping records 60 years ago. Currently it’s 210 feet deep.
(a)
The county is considering filling in sand to offset the erosion, back to the historical mark (60 years ago). How deep was it then? Notice that you need to convert 3′9″ to (decimal) feet first.
(b)
Name the variables and write an equation relating them, assuming the county does not fill in the beach now. Measure time from 60 years ago.
(c)
The country agrees to start filling in sand when the depth drops below 180 feet. How many (more) years will that take to happen? First estimate the answer using successive approximation. Then set up and solve an inequality to find the answer.
(d)
Draw a graph showing the sand erosion over the past 60 years and including the next 20 years, assuming the county does not do any filling.
(e)
Identify the slope and intercept and explain their meaning in the story.
8.
Clyde is loading bricks weighing 4.5 pounds each onto his wheelbarrow. The wheelbarrow weighs 89 pounds when it has 16 bricks in it. (That weight includes both the bricks and the wheelbarrow itself.)
(a)
How much would Clyde’s wheelbarrow weigh if it were empty?
(b)
Name the variables and write an equation relating them.
(c)
How much (total) will the wheelbarrow weigh if he loads a total of 30 bricks?
(d)
Clyde continues loading bricks until the wheelbarrow full of bricks weighs 206 pounds. How many bricks are in it?
(e)
Graph and check.
9.
The city offers bus “convenience” passes - 20 rides for $12.95 or 80 rides for $51.80.
(a)
Calculate the rate of change.
(b)
Is there a convenience charge?
(c)
What is the name for this type of function?
10.
To make cookies it takes a few minutes to prepare the dough. After that it takes 12 minutes per batch to bake in the oven. Last time I made 3 batches of cookies and it took a total of 54 minutes.
(a)
How long does it take me to prepare the dough?
(b)
How long would it take me to make 10 batches of cookies for the cookie swap? Assume the time to prepare the dough remains the same and only one batch bakes in the oven at a time.
(c)
Name the variables and write an equation describing the function.
(d)
Identify the slope and intercept and explain their meaning in the story.
