1.1 Change in Tandem — AP Precalculus | The Complete Equation

AP PRECALCULUS — UNIT 1A · POLYNOMIAL & RATIONAL FUNCTIONS

1.1 — Change in Tandem

Notes — How Two Quantities Vary Together

💡 Learning Objectives (1.1.A and 1.1.B)

By the end of this lesson you will be able to:

Describe how the input and output of a function change together by comparing function values
Identify intervals on which a function is increasing or decreasing
Construct a graph that represents two real-world quantities that change in tandem
Identify intervals where a graph is concave up or concave down
Locate the zeros of a function from its graph

1. Functions Recap

A function is a rule that pairs every value in a set called the domain with exactly one value in another set called the range. The variable representing the input is the independent variable, and the variable representing the output is the dependent variable. We often write y = f(x) to indicate that the output y depends on the input x.

This pairing — input matched to output — is what we mean by ‘two quantities varying in tandem.’ When the input changes, the output responds. Studying how the output responds to the input is the central activity of precalculus.

2. Four Ways to Show a Function's Behavior

The same input–output relationship can be displayed in four equivalent ways, and a strong precalculus student moves freely between them:

Verbally: ‘a car's distance from home increases as time passes.’
Numerically: a table of (time, distance) pairs.
Analytically: a formula such as d = 5t².
Graphically: a curve in the t-d plane.

A numerical view of distance varying in tandem with time. Notice that the distance values are not evenly spaced — the pairs ‘change together’ but not at a constant rate.

3. Increasing and Decreasing Behavior

💡 Definitions

On an interval I in the domain of f:

f is increasing on I if for any two inputs a, b in I with a < b, we have f(a) < f(b). Outputs grow as inputs grow.
f is decreasing on I if for any two inputs a, b in I with a < b, we have f(a) > f(b). Outputs shrink as inputs grow.

Reading increasing and decreasing intervals from a graph. Walk along the curve from left to right; if you go uphill the function is increasing, if you go downhill it is decreasing.

⚠️ Common mistake

Increasing and decreasing intervals are stated as intervals of the input (x-values), not output. ‘f is increasing on (−1.5, 2.5)’ refers to x-values, not the y-values that result.

4. Concavity — How the Rate of Change Itself Changes

A graph can be increasing while still curving upward or downward. Concavity describes that curvature.

Concave up: the graph bends upward like a smile. Equivalently, the rate of change of f is itself increasing on that interval.
Concave down: the graph bends downward like a frown. The rate of change of f is decreasing on that interval.
Inflection point: a point where the curve switches between concave up and concave down.

A function can be increasing the whole way and still change concavity. The point where the bending switches direction is an inflection point.

📘 Example — Identifying concavity from a verbal description

‘At first the cost rises slowly, then it shoots up sharply’ — increasing AND concave up.

‘The temperature climbs rapidly, then levels off’ — increasing AND concave down.

‘The water level drops slowly at first, then plummets’ — decreasing AND concave down.

5. Zeros of a Function

The zeros (or roots) of a function f are the input values x for which f(x) = 0. On a graph, they are the x-coordinates where the curve crosses or touches the x-axis.

Zeros are important because they often signal something meaningful in context: when a population goes extinct, when a business breaks even, when an object hits the ground.

6. Building a Graph from a Verbal Description

📘 Example — Sketching from words

Description: ‘A bathtub starts empty. The faucet is opened, and water fills it at a constant rate for 4 minutes. The drain is then opened and the water empties at a constant (but slower) rate, becoming empty again after another 8 minutes.’

• On 0 ≤ t ≤ 4: water level rises linearly — increasing, slope positive.

• On 4 ≤ t ≤ 12: water level falls linearly — decreasing, slope negative (and shallower than the rising slope).

• Maximum value at t = 4; zeros at t = 0 and t = 12.

7. Summary

A function pairs each input with exactly one output; we can show this verbally, numerically, analytically, or graphically
Increasing means outputs grow with inputs; decreasing means they shrink
Concave up means the curve smiles; concave down means it frowns
An inflection point is where concavity changes
Zeros are the inputs where f(x) = 0; they appear as x-intercepts on a graph
Translating verbal descriptions into accurate sketches is a core AP skill