8.5 Average Rate of Change, Qualitative Graphs, & Comparing Representations
Key Takeaways
- The average rate of change of a function between two inputs is the change in output divided by the change in input — the slope of the line joining the two points.
- For a linear function the average rate of change is the same on every interval; for a nonlinear function it varies from interval to interval.
- A qualitative graph is read by describing where the function increases, decreases, or stays constant, and where it turns, without computing exact values.
- Comparing two functions given in different forms — one as a table, one as an equation, one as a graph — requires converting them to a common feature such as rate of change or initial value.
Average Rate of Change, Qualitative Graphs, & Comparing Representations
Three Level A and Level D standards sit together here: calculate and interpret the average rate of change of a function over a specified interval; describe qualitatively the functional relationship between two quantities by analyzing a graph; and compare properties of two functions each represented in a different way.
Average Rate of Change
This is the slope of the straight line joining $(a, f(a))$ and $(b, f(b))$ — the secant line. It answers "on average, how fast did the output change over this stretch?"
A patient's weight was 214 lb in January and 190 lb in July. Average rate of change over those six months:
Always attach units in the form "output units per input unit." "Negative 4" is incomplete; "a loss of 4 pounds per month" is the answer.
Linear vs. nonlinear
For a linear function, the average rate of change is identical on every interval — that is what "constant rate of change" means.
For a nonlinear function, it differs. Take $f(x) = x^2$:
| Interval | Computation | Average rate of change |
|---|---|---|
| $[0, 2]$ | $(4 - 0) \div 2$ | 2 |
| $[2, 4]$ | $(16 - 4) \div 2$ | 6 |
| $[4, 6]$ | $(36 - 16) \div 2$ | 10 |
Increasing average rates confirm the graph is curving upward. This is exactly how TABE asks you to distinguish linear from quadratic growth from a table.
Estimating the rate from a graph
When only a graph is given, read the two endpoint coordinates off the axes and divide. Precision to the nearest gridline is enough; TABE answer choices are separated widely enough to absorb reading error.
Qualitative Graph Analysis
A qualitative question asks you to describe shape, not compute values. The vocabulary is small and it is worth memorizing.
| Feature | What the graph does | What it means in context |
|---|---|---|
| Increasing | rises left to right | the quantity is growing |
| Decreasing | falls left to right | the quantity is shrinking |
| Constant | flat horizontal segment | no change; a pause, a plateau, a hold |
| Maximum | a peak, then turns down | the largest value reached |
| Minimum | a valley, then turns up | the smallest value reached |
| Intercept on the vertical axis | value at input 0 | the starting amount |
| Intercept on the horizontal axis | output reaches 0 | when the tank empties, the debt clears, the object lands |
| Linear | straight | constant rate |
| Nonlinear | curved | changing rate |
Reading a story from a graph
A graph of distance from home versus time rises steeply, then flattens, then falls gradually to zero.
- Rises steeply: traveling away from home quickly.
- Flattens: stopped — distance is not changing, though time is. A horizontal segment on a distance-time graph means at rest, not moving slowly.
- Falls gradually to zero: returning home at a slower speed, arriving when distance is 0.
Sketching from a description
"A kettle heats from room temperature to boiling in 4 minutes, holds at boiling for 3 minutes, then cools slowly."
Sketch: a rising curve from 0 to 4 minutes, a horizontal segment from 4 to 7 minutes, then a curve decreasing with a flattening tail. TABE presents four candidate sketches and asks which matches — the horizontal middle segment is usually the discriminator.
Comparing Functions in Different Representations
The tested version of this standard gives you two functions in two different forms and asks which has the greater rate of change, greater initial value, or greater value at some input. The method is always the same: convert both to the same feature.
Problem. Function $A$ is $y = 7x + 15$. Function $B$ is given by this table:
$x$ 0 2 4 6 $y$ 20 38 56 74 Which has the greater rate of change, and which has the greater initial value?
| Feature | Function $A$ | Function $B$ | Winner |
|---|---|---|---|
| Rate of change | 7 | $(38 - 20) \div 2 = 9$ | B |
| Initial value | 15 | 20 (the $y$ at $x = 0$) | B |
| Value at $x = 10$ | 85 | $9(10) + 20 = 110$ | B |
Function $B$ wins on every count here — but TABE more often splits them, giving one function the higher rate and the other the higher start, then asking where they cross.
Where do they cross?
Set the two rules equal. With $A: y = 7x + 15$ and $B: y = 9x + 20$:
Since a negative input is usually outside the realistic domain, these two never cross in context — B is above A everywhere that matters.
Key Features Checklist
When a TABE item shows a graph and asks about "key features," it wants some subset of:
- Vertical intercept (starting value)
- Horizontal intercept(s) (zeros)
- Intervals of increase and decrease
- Maximum or minimum, if any
- Whether the shape is linear or nonlinear
- End behavior — what happens as the input grows large
Run that six-item checklist and you will have answered nearly any qualitative question the test can pose.
A company's monthly revenue was $48,000 in month 2 and $79,200 in month 8. What is the average rate of change in revenue over that interval?
A graph shows the water level in a bathtub over time. The line rises steadily, then runs horizontally for several minutes, then falls steeply to zero. What does the horizontal segment represent?
Function P is defined by y = 6x + 30. Function Q is given by the table below. x: 0, 3, 6, 9 y: 18, 42, 66, 90 Which statement correctly compares the two functions?