Rate of Change and Slope

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

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

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

change in distance Rate of change = change in time 456 - 72 = 5 – 1 384 4 = 96

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

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.                      

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

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). 

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

Negative slope Positive slope Slope undefined Slope zero