We explore the slope of lines.
Considering the definition for rate of change, this means that when and are two variables where the rate of change between any two points is always the same, then you can calculate slope, , by finding two distinct data points and , and calculating
A slope is a rate of change. So if there are units for the horizontal and vertical variables, then there will be units for the slope. The slope will be measured in
Here are some linear scenarios with different slopes. As you read each scenario, note how a slope is more meaningful with units.
- If a tree grows 2.5 feet every year, its rate of change in height is the same from year to year. So the height and time have a linear relationship where the slope is 2.5 ftyr.
- If a company loses 2 million dollars every year, its rate of change in reserve funds is the same from year to year. So the company’s reserve funds and time have a linear relationship where the slope is -2 million dollars per year.
- If Sakura is an adult who has stopped growing, her rate of change in height is the same from year to yearit’s zero. So the slope is 0 inyr. Sakura’s height is constant with respect to time.
Yara keeps her resolution, and her account balance increases steadily by $20 each week. That’s a constant rate of change, so her account balance and time have a linear relationship with slope of 20 .
We can model the balance, , in dollars, in Yara’s savings account weeks after she started making deposits with an equation. Since Yara started with $50 and adds $20 each week, then weeks after she started making deposits, where is a dollar amount. Notice that the slope, , serves as the multiplier for weeks.
We can also consider Yara’s savings using a table
In first few rows of the table, we see that when the number of weeks increases by 1, the balance increases by 20. The row-to-row rate of change is the slope. In any table for a linear relationship, whenever increases by 1 unit, will increase by the slope.
In further rows, notice that as row-to-row change in increases, row-to-row change in increases proportionally to preserve the constant rate of change. Looking at the change in the last two rows of the table, we see increases by 5 and increases by 100, which gives a rate of change of the value of the slope again.
We can see this constant rate of change on the graph by drawing in slope triangles between points on the graph, showing the change in as a horizontal distance and the change in as a vertical distance.
The Relationship Between Slope and Increase/Decrease
In a linear relationship, as the -value increases (in other words as you read its graph from left to right):
- if the -values increase (in other words, the line goes upward), its slope is
positive.
- if the -values decrease (in other words, the line goes downward), its slope is
negative.
- if the -values don’t change (in other words, the line is flat, or horizontal), its
slope is 0.
Finding the Slope by Two Given Points
Whenever you know two points on a line, you can find the slope of the line directly from the definition of slope.
We don’t actually need the picture, though, to find the slope. From the definition of slope, we have that
We know that after 3 yr, the height is 15 ft. As an ordered pair, that information gives us the point (3,15) which we can label as Similarly, the background information tells us to consider (6,27), which we label as . Here, and represent the first point’s -value and -value, and and represent the second point’s -value and -value.
Substiuting in our values for , , , and into our definition of slope, we have