AP® Precalculus review sheet from Aim for Five (aimforfive.com/precalc/units/1/1-1)
Unit 1 · Topic 1.1
1.1 Change in Tandem
A function links two quantities so that each input gets exactly one output. This topic is about reading how the output changes as the input grows: where a graph rises or falls, how it bends, and where it hits zero. Every later topic in the course uses this vocabulary.
Key terms
- function
- domain and range
- increasing / decreasing
- concave up / concave down
- zero of a function
Functions, inputs and outputs
A function is a rule that gives exactly one output for each input. If f(3) = 7, the input 3 always produces the output 7, never 7 one time and 9 another.
The domain is the set of inputs the function accepts. The range is the set of outputs it actually produces. On a graph, inputs run along the horizontal axis and outputs along the vertical axis, so each point (x, f(x)) is one input-output pair.
The input variable is called the independent variable, and the output variable is the dependent variable, because the output depends on the input. The one output an input produces is its image. Going the other way, the preimage of an output is the set of all inputs that produce it: for f(x) = x², the preimage of 9 is {−3, 3}.
“Change in tandem” just means the two quantities change together. When you read a graph from left to right, you are watching what the output does as the input increases.
Increasing and decreasing
A function is increasing on an interval if bigger inputs always give bigger outputs there: whenever a < b in the interval, f(a) < f(b). The graph rises as you move right.
A function is decreasing on an interval if bigger inputs give smaller outputs: whenever a < b, f(a) > f(b). The graph falls as you move right.
You won't be tested on whether an endpoint belongs in the interval. Saying a function increases on (1, 4) or on [1, 4] is treated the same way on the AP exam, so don't spend time on that.
Concavity: how the rate of change behaves
Concavity describes how the rate of change is changing. A graph is concave up on an interval where its rate of change is increasing; it bends like a cup that holds water. A graph is concave down where its rate of change is decreasing; it bends like an upside-down cup.
Concavity and increasing/decreasing are separate questions, so there are four combinations:
| Behavior | What it looks like | Everyday example |
|---|---|---|
| Increasing, concave up | Rising faster and faster | A video going viral |
| Increasing, concave down | Rising, but more and more slowly | Learning a skill that levels off |
| Decreasing, concave up | Falling, but more and more slowly | Hot coffee cooling toward room temperature |
| Decreasing, concave down | Falling faster and faster | A ball dropped from a roof |
Zeros of a function
A zero of f is an input where the output is 0, so f(a) = 0. On the graph, zeros are the x-intercepts: the points (a, 0) where the graph meets the x-axis.
Zeros matter because they are often where the output switches sign, from positive to negative or the other way. In a context, a zero might be the moment a ball hits the ground or the price where profit is exactly zero.
Building a graph from words
Questions often describe a situation in words and ask which graph matches. Translate each phrase into a feature: “grows steadily” means a straight rising segment, “grows faster and faster” means increasing and concave up, “levels off” means increasing and concave down, and “returns to where it started” means the output comes back to its first value.
Then check the order of events from left to right. The graph has to show each change at the right time, not just the right shapes.
Worked examples
Try each one yourself first, then open the solution.
- Example 1
Graph from a description
Water is poured at a steady rate into a vase that is wide at the bottom and narrow at the top. Describe the graph of the water's height h as a function of time t.
Show the solutionHide the solution
- Step 1: The vase is always filling, so the height keeps going up: h is increasing for the whole time.
- Step 2: Near the bottom the vase is wide, so each cup of water raises the level only a little. Near the top it is narrow, so the same amount of water raises the level a lot.
- Step 3: That means the rate at which h increases gets bigger over time. A rate of change that is increasing means the graph is concave up.
Answer: The graph of h is increasing and concave up for the whole pour: it rises slowly at first, then more and more steeply.
- Example 2
Reading behavior from a table
The table gives values of a function f at equally spaced inputs: f(0) = 10, f(1) = 16, f(2) = 20, f(3) = 22, f(4) = 22.5, f(5) = 21, f(6) = 17. Describe where f appears to increase or decrease and its concavity.
Show the solutionHide the solution
- Step 1: Find the change in output over each interval of length 1: 16 − 10 = 6, 20 − 16 = 4, 22 − 20 = 2, 22.5 − 22 = 0.5, 21 − 22.5 = −1.5, 17 − 21 = −4.
- Step 2: The changes are positive from x = 0 to x = 4, then negative from x = 4 to x = 6. So f increases from 0 to about 4, then decreases.
- Step 3: The changes themselves keep getting smaller: 6, 4, 2, 0.5, −1.5, −4. The rate of change is decreasing the whole time, which means concave down.
- Step 4: A table only shows a few points, so these are conclusions about what the data suggest, not a guarantee about every input in between.
Answer: f appears to increase on 0 < x < 4 and decrease on 4 < x < 6, and it appears to be concave down on the whole interval.
- Example 3
Trap: falling but concave up
Over four consecutive intervals of equal length, a function's average rates of change are −8, −5, −3 and −1. Is the graph concave up or concave down on this stretch?
Show the solutionHide the solution
- Step 1: Every rate is negative, so the function is decreasing. Many students stop here and say “concave down.” That is the trap.
- Step 2: Concavity depends on whether the rate of change is increasing or decreasing, not on whether the function is.
- Step 3: The rates go −8, −5, −3, −1. Each one is bigger than the one before, so the rate of change is increasing.
Answer: Concave up. The function is decreasing, but more and more slowly.
Common mistakes
- Confusing decreasing with concave down. A graph can be falling and concave up (falling more slowly) or rising and concave down (rising more slowly). Check the rate of change separately.
- Reading a graph right to left. Increasing and decreasing are always judged as the input increases, moving left to right.
- Giving the zero as a point when a value is asked for, or the reverse. The zero is the input a; the x-intercept is the point (a, 0).
On the exam
- Multiple-choice questions often describe a situation in words and ask which graph or statement fits. Match each phrase to increasing/decreasing and concavity before looking at the choices.
- When a free-response question asks you to describe behavior on an interval, name both direction and concavity, and give a reason based on the rate of change.
Connected topics
Videos
Check yourself
4 questions on 1.1 Change in Tandem. Pick an answer to see if you got it, and why.
A cup of hot cocoa is left on a counter. Its temperature drops quickly at first and then more and more slowly as time passes. Which describes the graph of the cocoa's temperature as a function of time?
Water is poured at a constant rate into an empty vase that is narrow at the bottom and gets steadily wider toward the top. Which of the following describes the graph of the height of the water as a function of time?
The function f is decreasing on the interval 2 < x < 7. Which of the following must be true?
Maya walks away from her house at a steady pace for 10 minutes. She stops and talks with a neighbor for 5 minutes, and then she walks straight home at a faster steady pace than before. Which of the following describes the graph of Maya's distance from her house as a function of time?
0 of 4 answered