Comparing Which Interval Has The Greater Rate

Why Compare Rates, Not Just Changes?

Suppose a function describes an output that varies with an input. On one interval, the output might increase by 6; on another, it might increase by 10. It is tempting to conclude that the second interval has the greater rate. But if the first change happens over 2 input units and the second over 10, the first rate is greater: it has a larger change per input unit.

Comparing rates on separate intervals therefore requires two pieces of information for each interval: the change in output and the change in input. The comparison also depends on where the intervals lie. A function can rise quickly in one region and slowly in another, or rise on one interval and fall on another.

The Average Rate on an Interval

For a function \(f\) and distinct inputs \(a\) and \(b\), the average rate of change from \(a\) to \(b\) is the change in function value divided by the change in input:

This quotient is the slope of the secant line through the points \((a,f(a))\) and \((b,f(b))\). Its sign indicates whether the function’s output is, on average, increasing or decreasing as the input increases across the interval. A positive rate means the function value at the larger input is greater; a negative rate means it is smaller. A zero rate means the endpoint values are equal.

The magnitude also matters. A rate of \(5\) is greater than a rate of \(2\), while a rate of \(-1\) is greater than a rate of \(-4\). The latter comparison can feel counterintuitive: \(-1\) is greater because it is closer to zero. When comparing signed rates, use their ordinary numerical order rather than comparing absolute values.

A Reliable Comparison Method

For two intervals, first compute each rate separately. Keep the endpoint order consistent within each quotient, and include the input width in the denominator. Then compare the resulting signed numbers. If the intervals have positive widths and you want to compare without first dividing, cross multiplication is also possible—but only after accounting for the positive denominators.

Why Dividing by the Input Change Is Essential

The average rate is a secant slope: vertical change divided by horizontal change. This explains why two intervals with different widths cannot be compared by their output changes alone. A larger vertical change may be spread across a much larger horizontal distance.

For intervals with positive widths, write their rates as \(r_1=\frac{\Delta y_1}{\Delta x_1}\) and \(r_2=\frac{\Delta y_2}{\Delta x_2}\), where \(\Delta x_1>0\) and \(\Delta x_2>0\). Multiplying both sides of a comparison by the positive product \(\Delta x_1\Delta x_2\) preserves the inequality:

This is a valid way to compare rates without evaluating the quotients, provided both input widths are positive. If an interval is written with its endpoints in reverse order, its input change is negative; do not apply this positive-denominator comparison without first handling the signs.

Worked Comparisons

Interval Placement and Direction

Intervals at different input locations can have different average rates even if they have the same width. The quadratic example shows this: both intervals have width 2, but the later interval has the larger secant slope. More generally, the values at the interval endpoints determine its average rate. A function need not change at a uniform rate across its entire domain.

Interval direction requires careful interpretation. If a rate is calculated from \(a\) to \(b\), swapping both endpoints changes the signs of both the output change and the input change. Their quotient remains the same:

Thus, reversing the order in which the endpoints are written does not reverse the secant slope. A negative rate instead means that the function value at the larger input is less than the value at the smaller input. Do not confuse the direction used to list endpoints with whether the function increases or decreases as the input increases.

Common Pitfall: Comparing Output Changes Alone

Consider an interval where the output rises by 6 over 2 input units, and another where it rises by 8 over 4 input units. A mistaken comparison might say, “The second interval has the greater rate because \(8>6\).” That compares total changes, not rates.

The correct rates are \(6/2=3\) on the first interval and \(8/4=2\) on the second. Therefore, the first interval has the greater average rate, even though its total output change is smaller. Always divide each output change by its own input change before comparing.

Why These Comparisons Matter

Comparing average rates helps describe how a function behaves across different parts of its input range. It can show that an output is changing more quickly on one interval than another, or that its average direction of change differs between intervals. The conclusion is about the endpoints and the average change between them; it does not by itself claim that the function changes at a constant rate at every point inside either interval.

When working from a formula or a table, preserve exact arithmetic when possible. When estimating from a graph, keep the estimates clearly labeled and avoid claiming more precision than the graph supports. In either case, a sound comparison rests on the same essentials: correct endpoint values, signed changes, the appropriate interval widths, and a final comparison of the rates.