1. Rate of Change and Slope

2. The table below shows the total distance that a car traveled over various time intervals.

3. Time (hours)
Distance (in km) 1 72
2.5
168 3
205
4.5
316 5
456

4. To find how fast the car was traveling, we use the rate of change.
Rate of change is a ratio that compares how much a quantity changes, on average, relative to the change in other quantity.
If x in the independent variable and y is the dependent variable, then
change in y
Rate of change =
change in x

5. change in distance
Rate of change =
change in time =
5 – 1
384 4 =

6. The rate of change of 96 means that the distance was increasing by 96 km every hour.

7. The word slope (gradient, incline, pitch) is used to describe the measurement of the steepness of a straight line.  The higher the slope, the steeper the line.  The slope of a line is a rate of change.                      

8. Slope is a ratio and can be expressed as:
change in y
slope =
change in x =

9. Consider the line y = 2x + 1, shown at the right
Consider the line  y = 2x + 1, shown at the right. By how much has the value of y changed between the two points (-4,-7) and (-3,-5)?  This will be a vertical change.       Answer:  2 units
By how much has the value of x changed between the two points (-4,-7) and (-3,-5)?   This will be a horizontal change.   Answer:  1 unit
The slope = 2  
Notice that this slope will be the same if the points (1,3) and (2, 5) are used for the calculations.  For straight lines, the rate of change (slope) is constant (always the same). 

10. The higher is the slope, the steeper is the line.

11. Negative slope
Positive slope
Slope undefined
Slope zero