Slope and Rate of Change Chapter

What you will learn

Slope measures how much a vertical quantity changes for each unit of horizontal change. For points (x1,y1) and (x2,y2), slope is (y2-y1)/(x2-x1) when x2 is not equal to x1. Using the same point order in numerator and denominator is essential. In an application, the numerical result should be followed by a sentence that names both quantities and their units.

Core ideas

• Positive slope rises from left to right; negative slope falls; zero slope is horizontal.
• A vertical line has undefined slope because its horizontal change is zero.
• Steeper appearance depends on graph scale, so calculate before comparing.
• Average rate of change between two points uses the slope formula even when the full relationship is not linear.

A practical way to use this resource

1. Label the coordinates and subtract y-values in a chosen order.
2. Subtract x-values in that same order and simplify the ratio.
3. Check the sign against the visual direction of the graph.
4. Interpret the result as units of y per one unit of x.

Check your understanding

The temperature changes from 68°F at 8 a.m. to 77°F at 11 a.m. The average rate is (77-68)/(11-8)=3°F per hour. Explain why this average does not require the temperature to rise exactly three degrees during every individual hour.

Accuracy and next steps

Use this page as a learning guide, compare your work with course requirements, and ask an instructor when a local rule or assignment direction differs.