TL;DR:
Slope is rise over run: the change in y for each one-unit change in x.
Its units are "y-units per x-unit", such as dollars per gigabyte.
The y-intercept is the value when x is zero, such as a fixed fee.
A phone plan charges a fixed $15 a month plus $4 for every gigabyte of data. How much does 5 GB cost? How much does each extra gigabyte add? Those two questions are slope and intercept, in a setting you can picture.
Slope in one sentence
Slope is how much the output changes when the input goes up by one. On a graph, it is the rise (vertical change) divided by the run (horizontal change). A steeper line has a bigger slope because y climbs more for each step across.
Worked example: a plan with a fixed fee and a price per gigabyte
Let g be the gigabytes used and C the monthly cost in dollars. The plan is C = 4g + 15. Build a small table first.
Gigabytes (g) | Cost (C) |
|---|---|
0 | 15 |
2 | 23 |
5 | 35 |
9 | 51 |
Now pick two points and find the slope. With (0, 15) and (5, 35): rise = 35 − 15 = 20, run = 5 − 0 = 5, slope = 20/5 = 4. With a different pair, (2, 23) and (5, 35): rise 12, run 3, slope 4. A straight line gives the same slope from any two points, and that is the test for linearity.
Reading the units
The slope is not just 4. It is 4 dollars per gigabyte. Dividing rise (dollars) by run (gigabytes) gives units of dollars/gigabyte. Whenever a problem asks "how fast" or "how much per", the answer is a slope, and the units tell you which quantity is on top.
The 15 is the y-intercept, the value of C when g = 0. In this story it is the fixed fee you pay even with no data. Always ask what the intercept means in the situation. If it has no sensible meaning, the model may not extend to zero.
How to check your answer
Same slope from two pairs. We got 4 both times.
Sign. The cost goes up as data goes up, so the slope should be positive. It is.
Predict and test. For a $51 budget, solve
4g + 15 = 51, so4g = 36andg = 9. The table says 9 GB costs $51. It does.Units. Dollars divided by gigabytes gives dollars per gigabyte.
A negative slope
A tank holds 40 litres and drains at 3 litres each minute. Then V = 40 − 3t. The slope is −3 litres per minute: the volume falls as time rises. The intercept 40 is the starting volume. The line hits zero when 40 − 3t = 0, so t = 40/3 ≈ 13.3 minutes.
Slope between two measured points
Suppose a hiker is at an elevation of 1,200 m at the start and at 1,500 m after 2 hours of steady climbing. The slope is (1500 − 1200)/(2 − 0) = 300/2 = 150 metres per hour. The line is E = 150t + 1200, and after 3 hours the model predicts 150 × 3 + 1200 = 1650 m.
This is the slope-intercept form y = mx + b, where m is the slope and b is the intercept. In the phone plan, m = 4 and b = 15. In the hike, m = 150 and b = 1200. The form is a recipe for building a line from a rate and a starting value, so most word problems with a steady rate fit it.
Slope and the shape of the table
You can spot a linear relationship from a table alone. Look at how y changes while x moves in equal steps. In the phone-plan table, the gigabytes jump by 2, then 3, then 4, so the steps are unequal, which means you must divide each y-change by its own x-change. Doing that gives 8/2 = 4, 12/3 = 4 and 16/4 = 4. Every ratio is 4, so the data lie on one straight line.
If the ratios had been 4, 5 and 7, the relationship would not be linear, and a single slope would not describe it. Then the best you can say is that the average rate between two chosen points is a certain number, which is an idea that returns in calculus.
Common mistakes
Subtracting in a different order on top and bottom. If you use
y₂ − y₁on top, usex₂ − x₁on the bottom.Writing run over rise.
Dropping the units, so the answer loses its meaning.
Treating a table with changing steps as a straight line without checking that the slope stays constant.
Mixing up slope 0 (a horizontal line) and an undefined slope (a vertical line).
Where you will use it next
Slope appears everywhere a graph is read. Your next step is to turn an equation into a picture in graphing a line from its equation, and slope is also the first step toward the derivative, which asks how steep a curve is at a single point. Two lines with a slope each also form a system, solved in solving systems of equations by elimination.
Frequently asked questions
Can slope be a fraction?
Yes. A slope of 1/2 means y rises 1 for every 2 across. Fractions are common when the run is bigger than the rise.
Why is the slope of a vertical line undefined?
The run is zero, and dividing by zero has no value.
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