Learn to find rates of change and slopes.

Vocabulary rate of change slope

The rate of change of a function is a ratios that compares the difference between two output values to the difference between the corresponding input values.

Additional Example 1A: Using A Table to identify Rates of Change Tell whether the rates of change are constant or variable. +2 +3 +1 +2 Find the difference between consecutive data points. x 2 4 7 8 10 y 5 11 20 23 29 +6 +9 +3 +6 Find each ratio of the change in y to the change in x. The rate of change is constant.

Be careful to put the difference in y-values in the numerator and the differences in x-values in the denominator when you write a rate of change. Caution!

Additional Example 1B: Using A Table to identify Rates of Change Tell whether the rates of change are constant or variable. +1 +1 +1 +1 Find the difference between consecutive data points. x 1 2 3 4 y 5 8 10 +3 +1 +3 +2 Find each ratio of the change in y to the change in x. The rates of change are variable.

x 2 5 6 9 y 15 30 35 50 Check It Out: Example 1 Tell whether the rates of change are constant or variable. +2 +3 +1 +3 Find the difference between consecutive data points. x 2 5 6 9 y 15 30 35 50 +10 +15 +5 +15 Find each ratio of the change in y to the change in x. The rate of change is constant.

When the rate of change is constant, the segments form a straight line When the rate of change is constant, the segments form a straight line. The constant rate of change of a line is its slope.

Recall that a function whose graph is a straight line is a linear function. Reading Math

Additional Example 2: Driving Application The table shows the driving distances that Jesse recorded. A. Determine whether the rates of change are constant or variable. The rate of change is constant.

Additional Example 2: Driving Application B. Graph the data and connect the points with line segments. If the rate of change is constant, find and interpret the slope. The rate of change between any two points is . The slope of the line is . 35 35 The slope is . This means he drove 3 mi. every 5 min.

The table shows the driving distances that Barry recorded. Check It Out: Example 2 The table shows the driving distances that Barry recorded. Determine whether the rates of change are constant or variable. Time (min) 1 3 6 9 12 Distance (miles) 18 24 31 63 126 189 2412 = 3 = 2 = 2 = 2 = 2 The rates of change are variable.