3.2 Rate of Change and Comparing Function Features
Key Takeaways
- Average rate of change over an interval [a, b] is (f(b) − f(a)) / (b − a); it equals the slope of the secant line connecting the two points on the graph (MA.912.F.1.3).
- For a linear function, average rate of change is constant; for quadratics and exponentials it varies by interval, so the interval must be specified.
- Key features to compare include x- and y-intercepts, maximum or minimum values, domain, range, and end behavior (MA.912.F.1.6).
- When two functions are given in different representations (one algebraically, one graphically), convert or extract the same feature from each before comparing.
- End behavior describes how the function values change as x approaches positive or negative infinity; for exponentials, one end approaches the asymptote.
Quick Answer: Average rate of change measures how much a function's output changes per unit of input across an interval. The EOC also asks you to compare two functions given in different forms by extracting the same feature from each.
Calculating and Interpreting Average Rate of Change
Average rate of change (MA.912.F.1.3) is defined as the change in output divided by the change in input over a closed interval [a, b]:
Average rate of change = (f(b) − f(a)) / (b − a)
Geometrically, this is the slope of the secant line connecting the points (a, f(a)) and (b, f(b)) on the graph. For a linear function, the average rate of change is the same constant slope on every interval. For a quadratic or exponential function, the value depends on which interval you choose, so the EOC always specifies the interval.
Worked example: f(x) = x² + 2x. Find the average rate of change from x = 1 to x = 4. Compute f(1) = 1 + 2 = 3 and f(4) = 16 + 8 = 24. Then (24 − 3) / (4 − 1) = 21 / 3 = 7. The output rises by an average of 7 units per 1-unit increase in x across that interval. A common trap is computing only f(b) − f(a) and forgetting to divide by b − a, which gives the total change, not the rate.
Interpreting the value in context matters. If h(t) gives height in feet of a projectile after t seconds, then an average rate of change of −12 ft/s over [1, 4] means the projectile's height decreased by an average of 12 feet per second between 1 and 4 seconds. A negative rate indicates decrease; a positive rate indicates increase. A rate of zero means the net change over the interval was zero (the function started and ended at the same height, even if it moved in between).
Comparing Key Features of Two Functions
Comparing key features of two functions (MA.912.F.1.6) requires extracting the same feature from each function even when they are presented differently. The features the EOC tests are:
| Feature | How to find from equation | How to find from graph |
|---|---|---|
| y-intercept | Substitute x = 0 | Read the point where the curve crosses the y-axis |
| x-intercept(s) | Set f(x) = 0 and solve | Read the point(s) where the curve crosses the x-axis |
| Maximum or minimum | Vertex of parabola; none for linear | Highest or lowest point on the graph |
| Domain | Input values that keep the expression defined | Horizontal extent of the graph |
| Range | Resulting output values | Vertical extent of the graph |
| End behavior | Substitute large positive and negative x | Trace the graph left and right |
Worked comparison: Function f is given by the graph, with a highest point at (3, 10) and x-intercepts at x = 1 and x = 5. Function g is given by g(x) = −x² + 4x + 5. Which function has the greater maximum? The graph of f has a maximum of 10. For g, the vertex x-coordinate is −b/(2a) = −4 / (2 · −1) = 2, and g(2) = −4 + 8 + 5 = 9. Because 10 > 9, the function shown on the graph has the greater maximum.
A frequent error is comparing an intercept from one function with a maximum from the other. Always confirm you are comparing the same feature. Another trap is ignoring the sign of end behavior. A quadratic with a positive leading coefficient rises on both ends; one with a negative leading coefficient falls on both ends. An exponential growth function rises on the right and approaches the asymptote (a horizontal line) on the left, so its left-end behavior is 'approaches a constant,' not 'decreases without bound.'
Domain, Range, and End Behavior in Context
Domain and range from context: When a function models a real-world situation, the domain may be restricted. A function modeling the height of a ball thrown upward is defined only for non-negative time until the ball lands, not for all real numbers. The EOC rewards students who state the contextual domain (for example, 0 ≤ t ≤ 3) rather than the mathematical domain (all real numbers).
Comparing rates qualitatively: Some items ask which of two functions is increasing faster over a given interval without requiring a numeric answer. Read the slopes from the graph, or compute the average rate of change from the table, and compare the signed values. The function with the larger positive average rate of change is increasing faster; the function with the more negative value is decreasing faster. If both functions are increasing but one curve is steeper on the interval, that steeper function has the greater rate of change, even if the other function reaches a higher overall value at the endpoint.
End behavior for the four families: Linear functions rise on one end and fall on the other (unless slope is zero, a constant function). Quadratic functions either rise on both ends (a > 0) or fall on both ends (a < 0). Exponential growth functions rise without bound on the right and approach the asymptote on the left; exponential decay functions fall toward the asymptote on the right and approach a finite value on the left. Absolute value functions rise or fall on both ends depending on the sign of the coefficient, forming a V that never approaches an asymptote.
If f(2) = 7 and f(6) = 23, what is the average rate of change of f from x = 2 to x = 6?
Function g(x) = −x² + 4x + 5 has its vertex at x = 2. What is the maximum value of g?
An exponential growth function y = 3(1.5)^x is compared with a linear function y = 2x + 1. Which statement about end behavior is correct as x increases without bound?