Topic 1.1 Change in Tandem: describe how the output values of a function change as the input values change, using increasing or decreasing behavior, concavity, and the relationship shown in a graph, table or context.

What this topic is asking

The College Board (Topic 1.1) wants you to describe how the output values of a function change as the input values change. This is called covariation, or change in tandem. You must say whether a function is increasing or decreasing over an interval, describe its concavity (whether the rate of change itself is rising or falling), and read all of this from a graph, a table, or a worded context.

Increasing and decreasing

On a graph, an increasing function rises from left to right and a decreasing function falls. In a table, read across: if the outputs grow as you move down the input column, the function is increasing on that stretch; if they shrink, it is decreasing.

Concavity and the rate of change

Saying a function increases is not the whole story. It can increase quickly or slowly, and the rate itself can change. Concavity captures this second layer.

The four combinations of direction (increasing or decreasing) and concavity (up or down) describe almost every smooth curve you meet in Unit 1, so being fluent with them is essential.

Reading change in tandem from a table

When the input values are equally spaced, you can detect concavity from a table by looking at successive differences. Compute the differences between consecutive outputs; these are a proxy for the rate of change. If those first differences are themselves increasing, the function is concave up; if they are decreasing, it is concave down; if they are constant, the function is linear with no concavity.

For example, outputs $1, 4, 9, 16, 25$ at equally spaced inputs have first differences $3, 5, 7, 9$. The outputs increase (so the function is increasing) and the differences increase (so it is concave up).

Reading change in tandem from a context

Worded problems hide the direction and concavity in their language. "Grows faster and faster" is increasing and concave up. "Grows, but more and more slowly" is increasing and concave down. "Cools toward room temperature" is decreasing and concave up (the drop slows as it approaches the limit). Training yourself to translate these phrases directly into the direction-plus-concavity pair is exactly the skill the exam rewards, because the same vocabulary recurs across polynomial, rational, exponential and logarithmic models.

A subtle point worth stating once: concavity is about the rate of change, not the output. A function can be decreasing while concave up, as in the worked example, because what matters for concavity is whether the rate is rising or falling, regardless of whether the function itself is going up or down. Keeping direction and concavity as two independent questions, answered separately, prevents the most common confusion in this topic and sets up the rate-of-change ideas in Topics 1.2 and 1.3.

Try this

Q1. A graph rises from left to right and bends downward. Describe its direction and concavity. [1 point]

• Cue. Increasing (it rises) and concave down (it bends downward), so it increases at a decreasing rate.

Q2. Outputs at equally spaced inputs are $2, 4, 8, 16, 32$. Is the function concave up or down? [1 point]

• Cue. First differences are $2, 4, 8, 16$, which are increasing, so the function is concave up.