3.3.1: Slopes Don't Have to be Positive

Lesson

Let's find out what a negative slope means.

Exercise \(\PageIndex{1}\): Which One Doesn't Belong: Odd Line Out

Which line doesn't belong?

Figure \(\PageIndex{1}\): 4 lines plotted on a grid, lines labeled s, t, u, v. line s, posiitve slope, intersects line v. line v, negative slope, intersects lines s, t, u. line t, positive slope, intersects lines v and u. line u, positive slope, intersects lines t and v.

Exercise \(\PageIndex{2}\): Stand Clear of the Closing Doors, Please

Noah put $40 on his fare card. Every time he rides public transportation, $2.50 is subtracted from the amount available on his card.

1. How much money, in dollars, is available on his card after he takes
   1. 0 rides?
   2. 1 ride?
   3. 2 rides?
   4. rides?
2. Graph the relationship between amount of money on the card and number of rides.
3. How many rides can Noah take before the card runs out of money? Where do you see this number of rides on your graph?

Exercise \(\PageIndex{3}\): Travel Habits in July

Here is a graph that shows the amount on Han's dare card for every day of last july.

1. Describe what happened with the amount on Han’s fare card in July.
2. Plot and label 3 different points on the line.
3. Write an equation that represents the amount on the card in July, \(y\), after \(x\) days.
4. What value makes sense for the slope of the line that represents the amounts on Han’s fare card in July?

Are you ready for more?

Let’s say you have taken out a loan and are paying it back. Which of the following graphs have positive slope and which have negative slope?

1. Amount paid on the vertical axis and time since payments started on the horizontal axis.
   Amount owed on the vertical axis and time remaining until the loan is paid off on the horizontal axis.
2. Amount paid on the vertical axis and time remaining until the loan is paid off on the horizontal axis.

Exercise \(\PageIndex{4}\): Payback Plan

Elena borrowed some money from her brother. She pays him back by giving him the same amount every week. The graph shows how much she owes after each week.

Figure \(\PageIndex{2}\): Graph of a line on grid, origin O. horizontal axis, time in weeks, scale 0 to 7, by 1's. vertical axis, scale 0 to 20, by 5's. line crosses y axis at 0 comma 18 and crosses the x axis at 6 comma 0.

Answer and explain your reasoning for each question.

1. What is the slope of the line?
2. Explain how you know whether the slope is positive or negative.
3. What does the slope represent in this situation?
4. How much did Elena borrow?
5. How much time will it take for Elena to pay back all the money she borrowed?

Summary

At the end of winter in Maine, the snow on the ground was 30 inches deep. Then there was a particularly warm day and the snow melted at the rate of 1 inch per hour. The graph shows the relationship between the time since the snow started to melt and the depth of the snow.

Figure \(\PageIndex{3}\): Graph on grid, origin O. horizontal axis, time since snow started to melt in hours, scale 0 to 11, by 1's. vertical axis, depth of snow in inches, scale 0 to 30, by 5's. line with negative slope drawn that goes through points 2 comma 28, 3 comma 27. triange with down 1 right 1 drawn below these 2 points. line also goes through 5 comma 25 and 10 comma 20. triangle with down 5 right 5 drawn below these 2 points.

The slope of the graph is -1 since the rate of change is -1 inch per hour. That is, the depth goes down 1 inch per hour. The vertical intercept is 30 since the snow was 30 inches deep when the warmth started to melt the snow. The two slope triangles show how the rate of change is constant. It just also happens to be negative in this case since after each hour that passes, there is 1 inch less snow.

Graphs with negative slope often describe situations where some quantity is decreasing over time, like the depth of snow on warm days or the amount of money on a fare card being used to take rides on buses.

Slopes can be positive, negative, or even zero! A slope of 0 means there is no change in the \(y\)-value even though the \(x\)-value may be changing. For example, Elena won a contest where the prize was a special pass that gives her free bus rides for a year. Her fare card had $5 on it when she won the prize. Here is a graph of the amount of money on her fare card after winning the prize:

Figure \(\PageIndex{4}\): Graph on a grid, origin O. horizontal axis, time since winning contest in days, scale 0 to 11, by 1's. vertical axis, balance on fare card in dollars, 0 to 6, by 1's. horizontal line through 2 days comma 5 dollars and 5 days comma 5 dollars.

The vertical intercept is 5, since the graph starts when she has $5 on her fare card. The slope of the graph is \(0\) since she doesn’t use her fare card for the next year, meaning the amount on her fare card doesn’t change for a year. In fact, all graphs of linear relationships with slopes equal to 0 are horizontal—a rate of change of 0 means that, from one point to the next, the \(y\)-values remain the same.

Practice

Exercise \(\PageIndex{5}\)

Suppose that during its flight, the elevation \(e\) (in feet) of a certain airplane and its time \(t\), in minutes since takeoff, are related by a linear equation. Consider the graph of this equation, with time represented on the horizontal axis and elevation on the vertical axis. For each situation, decide if the slope is positive, zero, or negative.

1. The plane is cruising at an altitude of 37,000 feet above sea level.
2. The plane is descending at rate of 1000 feet per minute.
3. The plane is ascending at a rate of 2000 feet per minute.

Exercise \(\PageIndex{6}\)

A group of hikers park their car at a trail head and walk into the forest to a campsite. The next morning, they head out on a hike from their campsite walking at a steady rate. The graph shows their distance in miles, \(d\), from the car after \(h\) hours of hiking.

1. How far is the campsite from their car? Explain how you know.
2. Write an equation that describes the relationship between \(d\) and \(h\).
3. After how many hours of hiking will they be 16 miles from their car? Explain or show your reasoning.

Figure \(\PageIndex{5}\)

(From Unit 3.2.3)

Exercise \(\PageIndex{7}\)

Elena’s aunt pays her $1 for each call she makes to let people know about her aunt’s new business.

The table shows how much money Diego receives for washing windows for his neighbors.

Table \(\PageIndex{1}\)

number of windows	number of dollars
\(27\)	\(30\)
\(45\)	\(50\)
\(81\)	\(90\)

Select all the statements about the situation that are true.

1. Elena makes more money for making 10 calls than Diego makes for washing 10 windows.
2. Diego makes more money for washing each window than Elena makes for making each call.
3. Elena makes the same amount of money for 20 calls as Diego makes for 18 windows.
4. Diego needs to wash 35 windows to make as much money as Elena makes for 40 calls.
5. The equation \(y=\frac{9}{10}x\), where \(y\) is number of dollars and \(x\) is number of windows, represents Diego’s situation.
6. The equation \(y=x\), where \(y\) is the number of dollars and \(x\) is the number of calls, represents Elena’s situation.

(From Unit 3.1.4)

Exercise \(\PageIndex{8}\)

Each square on a grid represents 1 unit on each side. Match the graphs with the slopes of the lines.

• \(-\frac{1}{4}\)
• \(\frac{1}{4}\)
• \(-4\)

Figure \(\PageIndex{6}\)