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Constant rate of change

To calculate a constant rate of change m, divide the change in the dependent variable by the change in the independent variable.

$m=\frac{change\;in\;the\;dependent\;variable}{change\;in\;the\;independent\;variable}$

If given ordered pairs (x1,y1) and (x2,y2),

$m=\frac{y_2-y_1}{x_2-x_1}$

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.

$Slope=\frac{Vertical\;change}{Horizontal\;change}=\frac{Rise}{Run}$

 The graph represents the distance travelled while driving on a highway. Use the graph to find the constant rate of change.

To find the rate of change, pick any two points on the line, such as (1, 60) and (2, 120).

$m=\frac{change\;in\;miles}{change\;in\;hours}=\frac{120-60}{2-1}=\frac{60}{1}=60$

The distance increases by 60 miles in 1 hour. So, the constant rate of traveling on a highway is 60 miles per hour.

 Todd had 5 gallons of gasoline in his motorbike. After driving 100 miles, he had 3 gallons left. The graph at the right shows Todd's situation. a.  Find the slope of the line. b.  What does this slope tell us? c.  What is Todd's mpg?

a.  Find the slope of the line.

$Slope=\frac{Vertical\;change}{Horizontal\;change}=\frac{Rise}{Run}=\frac{-2}{100}=\frac{-1}{50}$

b.  What does this slope tell us?

Since $\frac{-1}{50}=-0.02$, we know that Todd's bike is burning 0.02 gallons of gasoline for every mile that he travels.   The negative value of the slope tells us that the amount of gasoline in the tank is decreasing.

c.  What is Todd's mpg?

The  $\frac{-1}{50}=\frac{change\;in\;gallons}{change\;in\;miles}$  tells us that Todd can drive 50 miles on one gallon of gasoline (an mpg of 50 miles per gallon).