Key takeaway: The slope of a line is the ratio of its vertical change to its horizontal change, rise over run, and it measures how steeply the line climbs or falls.
Grade 88.EE.B.5Functions and Graphs
Definition of Slope
The slope of a line tells you how steep it is and which way it leans. Walk along the line from left to right: the rise is how far you go up, the run is how far you go across, and the ratio of the two is the number we are after.
A line that climbs as you move right has a positive value. A line that drops has a negative value. A flat, horizontal line scores zero, and a vertical line has no defined value at all because its run is zero. The ratio stays identical between any two points of a straight line, which is exactly what makes the line straight. In real life it turns into a rate such as feet climbed per mile, or dollars earned per hour.
Formula and notation
The letter \(m\) stands for this ratio:
\[m = \frac{\text{rise}}{\text{run}} = \frac{\text{change in } y}{\text{change in } x}\]
A value of \(\frac{3}{4}\) means: up 3 for every 4 steps to the right.
Key facts
- Lines that rise from left to right have a positive ratio; lines that fall have a negative one.
- Horizontal lines give 0, and vertical lines give an undefined result.
- A larger absolute value of the ratio means a steeper line.
- Every pair of points on one straight line produces the same number.
Slope example
On a hiking trail, the elevation is 450 feet at mile 2 and 750 feet at mile 7. Find the steepness of the trail’s elevation graph and say what it means.
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Rise (change in elevation): \(750 – 450 = 300\) feet.
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Run (change in distance): \(7 – 2 = 5\) miles.
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Ratio: \(m = \dfrac{300}{5} = 60\).
The value is 60: the trail climbs 60 feet for every mile walked.
Common mistake
Watch out
Rae reads the triangle in the graph and puts the run on top, getting \(\frac{4}{3}\).
Why it is wrong
The ratio is rise over run, the vertical change on top. For that line the correct slope is \(\frac{3}{4}\). Putting the run on top describes a line steeper than 45 degrees, which this one is not. Picture climbing stairs: the number of inches you go up is compared with the depth of the step.
Related terms and lessons
- Grade 8 · Proportional Relationships and SlopeLessonWorksheetQuiz
- Algebra 1 · Linear Functions: Slope-Intercept, Point-Slope and Standard FormLessonWorksheetQuiz
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