Direction of Covariation: Does the Output Increase or Decrease?
Textbook Reference
| Primary source | OpenStax Calculus Volume 1, Section 1.1: “Review of Functions” |
| Direct link | https://openstax.org/books/calculus-volume-1/pages/1-1-review-of-functions |
| Supplementary | OpenStax Precalculus 2e, Section 1.2: “Domain and Range” |
| Supplementary link | https://openstax.org/books/precalculus-2e/pages/1-2-domain-and-range |
Try This First
Below are four tables. For each one, as the input increases from row to row, decide: does the output go up, go down, or do both at different parts?
Table A:
| Input $x$ | Output $y$ |
|---|---|
| 1 | 5 |
| 2 | 8 |
| 3 | 11 |
| 4 | 14 |
Table B:
| Input $x$ | Output $y$ |
|---|---|
| 0 | 20 |
| 2 | 14 |
| 4 | 8 |
| 6 | 2 |
Table C:
| Input $x$ | Output $y$ |
|---|---|
| 0 | 1 |
| 1 | 4 |
| 2 | 9 |
| 3 | 4 |
Table D:
| Time (s) | Height (m) |
|---|---|
| 0 | 0 |
| 1 | 15 |
| 2 | 20 |
| 3 | 15 |
| 4 | 0 |
Write your answers before reading further: which tables show output always increasing, always decreasing, or first one then the other?
Key Idea
When two quantities covary, the most basic question is: as the input grows, does the output grow with it, or move in the opposite direction?
A function is increasing on an interval when larger inputs produce larger outputs throughout that interval. It is decreasing when larger inputs produce smaller outputs. Some functions do both, depending on where you look.
Noticing direction of covariation is a skill you can exercise on a table or graph without computing anything. It trains your eye for what calculus will later measure precisely.
Two Connected Representations
From a table: scan down the output column as you move down the input column. If the outputs grow, the function is increasing. If the outputs shrink, it is decreasing. If they grow then shrink (or shrink then grow), the direction changes somewhere.
From a graph: moving left to right along the horizontal axis represents increasing input. If the graph goes uphill, the function is increasing. If it goes downhill, decreasing.
Translation prompt: Pick any increasing section you identified in Table D. Draw what that section would look like on a graph. Then pick the decreasing section and add that. Where do the two sections meet on the graph?
Worked Example: Tables A through D (check your predictions)
Table A: Output goes 5, 8, 11, 14 -- always increasing. As time passes, the value climbs steadily.
Table B: Output goes 20, 14, 8, 2 -- always decreasing. As the input grows, the output falls.
Table C: Output goes 1, 4, 9, 4 -- increases from $x=0$ to $x=2$, then decreases from $x=2$ to $x=3$. Direction changes. Note: $x = 2$ is where the output peaks at 9.
Table D (height of a ball): Height goes 0, 15, 20, 15, 0 -- increases from $t=0$ to $t=2$, then decreases from $t=2$ to $t=4$. The ball rises, reaches a peak, then falls.
How close were your predictions? If you correctly identified the change of direction in C and D, that is the key insight.
When Direction Changes: Noticing the Turning Point
In Table C and D, there is a moment where the output stops increasing and starts decreasing. That moment -- an input value where direction reverses -- is important in calculus. For now, the goal is simply to see it in the data: the output column stops going up and starts going down at some row.
Finding exactly where this happens, and why, is what calculus is built to answer.
a graph that goes up then down “looks like” a hill, so it represents an actual hill. This is the iconic-graph misconception. Table D describes a ball thrown upward and falling back down, and its graph does rise then fall. But the graph’s shape is not a picture of the ball’s path in space -- the ball falls straight down, not along the arch shown on the graph. The horizontal axis of the graph is time, not horizontal position. The graph shows how height varies with time. Treating it as a bird’s-eye view of the trajectory is a common and significant error.
Predict-Then-Check Practice
Problem 1 (Low floor): Without computing anything, is the function $f(x) = 100 - 3x$ increasing or decreasing on $[0, 20]$? Predict, then check by computing $f(0)$ and $f(20)$.
Check your answer
Predict: subtracting $3x$ from 100 means as $x$ grows, the value shrinks. So decreasing.
Check: $f(0) = 100$, $f(20) = 100 - 60 = 40$. The output went from 100 to 40 as the input went from 0 to 20. Decreasing. Prediction confirmed.
Problem 2: A table shows the following data.
| $x$ | $g(x)$ |
|---|---|
| 0 | 3 |
| 1 | 1 |
| 2 | 1 |
| 3 | 4 |
Describe the direction of covariation across the whole table. Is there more than one possible interpretation?
Check your answer
From $x=0$ to $x=1$: output drops from 3 to 1 (decreasing). From $x=1$ to $x=2$: output stays at 1 (neither increasing nor decreasing -- constant). From $x=2$ to $x=3$: output rises from 1 to 4 (increasing).
So the direction changes twice. One valid description: the function decreases, then is constant, then increases. Another approach: say it is non-monotone (not consistently in one direction). Both descriptions are correct.
Problem 3 (High ceiling): Sketch any graph of a function that is decreasing on $(-\infty, 0)$, has a minimum at $x = 0$, and is increasing on $(0, \infty)$. Now describe the output’s covariation direction in both intervals. Is there one function with this behavior, or many? What does this tell you about how much direction alone specifies?
Check your answer
Many functions fit: $x^2$, $|x|$, $x^4 + 1$, and others all decrease left of 0 and increase right of 0. Knowing the direction of covariation tells you the shape category (decreasing then increasing) but not the specific curve. Direction is one piece of information; later in calculus you will also describe how fast the output changes (rate of change) and how the rate itself changes (concavity), building a full picture.
Ask Why
Why might it matter to distinguish “increasing” from “increasing quickly” at this stage? What information does direction give you that pace does not, and what does pace give you that direction does not?
Consider: two investments both grow for ten years. One doubles slowly each year; the other triples each year. Both are increasing -- but are they equally good? What additional information do you need?