Topic 1.1 · CED: Change in Tandem

What a Function Is, and How Inputs and Outputs Vary

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01
A relation is a function when every input has exactly one output.
02
As $x$ moves right, $f(x)$ moves up.
03
As $x$ moves right, $f(x)$ moves down.

Practice 23 problems Next: Chapter 1.2 Rates of Change

What Is a Function?

A function is a rule that takes an input and produces exactly one output.

KEY RULE

A relation is a FUNCTION when every input has EXACTLY ONE output.

CONCEPT

Key Terms

Domain → all allowed input values ( $x$ )

Range → all resulting output values ( $y$ or $f(x)$ )

Independent variable → the input, usually $x$

Dependent variable → the output, usually $y$ or $f(x)$

The flow is always: input → rule → output

REAL-LIFE EXAMPLE

A Vending Machine

Think of a vending machine that gives soda for a $1 coin. Every time you put in a $1 coin (input), the machine follows its programmed rule and dispenses one specific soda (output). Same input, same output — every single time.

A vending machine models a function: coin → machine → drink.

If the same $1 coin sometimes gave soda and sometimes gave water, this machine would NOT be a function — because the same input would give two different outputs.

Quick check

Does the vertical line $x = 2$ represent $y$ as a function of $x$ ?

Four Representations of a Function

The same function can be shown four different ways. Exam questions often give you one form and ask about another. To see they all describe the same thing, we'll use $f(x) = 3x - 4$ for every representation below.

① Analytical (equation)

f(x) = 3x - 4

The rule is written as an algebraic formula — plug in any $x$ , calculate $f(x)$ .

② Numerical (a table of values)

Plug several x-values into $f(x) = 3x - 4$ :

x	f(x) = 3x − 4
0	3(0) − 4 = −4
1	3(1) − 4 = −1
2	3(2) − 4 = 2
3	3(3) − 4 = 5

③ Graphical (a picture)

Plot those same points — they form a straight line.

Graph of $f(x) = 3x - 4$ with the same points from the table.

④ Verbal (a description)

"Start at −4 when $x$ is 0. Every time $x$ goes up by 1, $f(x)$ goes up by 3."

CONCEPT

The Big Idea

All four representations above describe the exact same function.

Switching between them is one of the most important skills in this course.

Increasing and Decreasing Functions

These describe what happens to the output as the input increases — is the graph going up, or going down?

Increasing

KEY RULE

As $x$ moves right, $f(x)$ moves up.

Formally:

\textbf{If } a < b, \textbf{ then } f(a) < f(b)

The arrows show the direction of the curve as $x$ moves right.

Decreasing

KEY RULE

As $x$ moves right, $f(x)$ moves down.

Formally:

\textbf{If } a < b, \textbf{ then } f(a) > f(b)

The arrows show the curve moving downward as $x$ moves right.

The Same Function Can Be Both — on Different Intervals

Most functions aren't increasing or decreasing everywhere. They switch. That's why you must always state the interval.

This function increases, then decreases, then increases again.

CONCEPT

How to Describe the Function Above

Increasing on $(-\infty, 0)$ — rises up to $x = 0$ .

Decreasing on $(0, 2)$ — falls between $x = 0$ and $x = 2$ .

Increasing on $(2, \infty)$ — rises again after $x = 2$ .

Never say "it's increasing" alone — always say WHERE.

How to Identify It Quickly

From a graph → rising = increasing, falling = decreasing
From a table → compare consecutive rows
From an equation → plug in two x-values and compare outputs

Zeros and x-Intercepts

KEY RULE

A ZERO is an x-value where $f(x) = 0$ — where the graph crosses the x-axis.

$f(x) = x^{2} + x - 2$ has zeros at $x = -2$ and $x = 1$ .

CONCEPT

An Important Distinction

The x-value itself → "the zero" (e.g., $x = -2$ )

The point $(x, 0)$ → "the x-intercept" (e.g., $(-2, 0)$ )

Solving algebraically → you are solving $f(x) = 0$

Reading a graph → you are finding where $y = 0$

Worked Example

Worked example

Find the zeros of $f(x) = x^{2} + x - 2$ .

01
Set $f(x) = 0$ : $x^{2} + x - 2 = 0$
02
Factor: $(x + 2)(x - 1) = 0$
03
Set each factor to zero and solve: $x = -2$ or $x = 1$
04
The zeros are $x = -2$ and $x = 1$ . The x-intercepts are $(-2, 0)$ and $(1, 0)$ .

Concavity and Rate of Change

Concavity is about how the SLOPE of the graph is changing — getting bigger, or getting smaller? This is a completely different question from "is the function going up or down?"

Concave Up

Look at what the tangent line slopes do as we move left to right:

Slopes change from −4 → −2 → 0 → 2 → 4. The slopes are INCREASING.

KEY RULE

Concave UP → slopes are INCREASING. Shape: cup.

Concave Down

Now the opposite — the tangent slopes drop:

Slopes change from +4 → +2 → 0 → −2 → −4. The slopes are DECREASING.

KEY RULE

Concave DOWN → slopes are DECREASING. Shape: frown.

The Tricky Case: Decreasing BUT Concave Up

This is the case students always get wrong. A function can be going DOWN, yet still be concave UP — because "concave up" is about the SLOPES getting bigger, not the function itself.

The function is going down, but slopes go from −4 → −0.08 (getting bigger).

CONCEPT

Why This Matters

Direction: a function can be increasing OR decreasing

Concavity: independently, it can be concave up OR concave down

Every function has BOTH a direction AND a concavity at each point. They can appear in any combination.

Concavity Can Change: the Inflection Point

Just like functions switch between increasing and decreasing, they can switch between concave up and concave down. The exact point where concavity switches is called an inflection point.

Concave down on $(-\infty, 0)$ , concave up on $(0, \infty)$ . Switch at $x = 0$ .

KEY RULE

INFLECTION POINT: where concavity changes (up → down, or down → up).

REAL-LIFE EXAMPLE

Real-Life Meaning of Inflection Points

A graph of total cases in a pandemic reaches its inflection point on the day new cases per day stop increasing and start decreasing. The total is still rising — but no longer as steeply.

A student's grade over a semester might hit an inflection point when study habits change — grades keep rising, but no longer as steeply.

In business, sales growth hitting an inflection point often signals a market shift worth watching.

AP Exam Focus: Concavity vs. Direction

Concavity questions appear on the AP exam constantly, and the single biggest trap is confusing concavity with direction. This graphic drives home the correct interpretation:

LEFT: concave up while decreasing. RIGHT: concave up whether increasing or decreasing.

AP EXAM FOCUS

"Concave up" means the RATE OF CHANGE is increasing.

It does NOT mean the function itself is increasing.

"Concave down" means the RATE OF CHANGE is decreasing.

It does NOT mean the function itself is decreasing.

Read the question carefully: is it asking about the function, or its rate of change?

Quick check

A cooling cup of coffee loses heat more and more slowly. Direction and concavity of temperature vs. time?

Constructing a Graph from a Context

Sometimes the problem describes a situation in words and you have to sketch the graph. Focus on SHAPE (direction and concavity), not exact numbers.

Speed vs. time for a car that accelerates, then cruises at constant speed.

Worked Example

Worked example

A car starts from rest and speeds up faster and faster for 5 seconds, then continues at a constant speed. Sketch speed vs. time.

01
Choose the axes: $x$ -axis = time (seconds), $y$ -axis = speed.
02
Mark the starting point: at $t = 0$ , speed $= 0$ .
03
From 0 to 5 seconds: speed increases, and the rate of increase itself grows → increasing AND concave up.
04
After 5 seconds: speed is constant → a horizontal line.
05
Exact numbers aren't required — what matters is showing the correct SHAPE for each interval.

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8 cards · Functions and change

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Recap card

6 lines to re-read the night before.

01
Function → every input has exactly one output.
02
Four representations: equation, table, graph, words — all describe the same relationship.
03
Increasing = $f$ moves up as $x$ moves right. Decreasing = $f$ moves down. Always name the interval.
04
Zero = an x-value where $f(x) = 0$ . The x-intercept is the point $(x, 0)$ .
05
Concavity is about slopes: concave up = slopes increasing, concave down = slopes decreasing.
06
Concavity and direction are independent — a function can be decreasing but concave up.

Practice 23 problems Next: Chapter 1.2 Rates of Change

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What a Function Is, and How Inputs and Outputs Vary

What Is a Function?

Four Representations of a Function

① Analytical (equation)

② Numerical (a table of values)

③ Graphical (a picture)

④ Verbal (a description)

Increasing and Decreasing Functions

Increasing

Decreasing

The Same Function Can Be Both — on Different Intervals

How to Identify It Quickly

Zeros and x-Intercepts

Worked Example

Concavity and Rate of Change

Concave Up

Concave Down

The Tricky Case: Decreasing BUT Concave Up

Concavity Can Change: the Inflection Point

AP Exam Focus: Concavity vs. Direction

Constructing a Graph from a Context

Worked Example

Recap card

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