Table of Contents

Introduction
Functions
- Formal Definition
- Key Features of Functions
Increasing and Decreasing Functions
Frequently Asked Questions

1.1 Change in Tandem in AP Precalculus

Utkarsh
December 24, 2024

Table of Contents

Introduction
Functions
- Formal Definition
- Key Features of Functions
Increasing and Decreasing Functions
Frequently Asked Questions

Introduction

In mathematics, understanding how two quantities change together forms the bedrock of advanced problem-solving. This idea—referred to as 1.1 change in tandem—is central to everything around AP Precalculus. Unit 1.1 emphasizes interpreting relationships between variables, constructing graphs, identifying intervals of growth, concavity, and zeros of a function.

Functions

A function is a fundamental concept in mathematics that describes a special relationship between two sets of values: the input (domain) and the output (range). In simple terms, a function assigns exactly one output for every input.

Formal Definition

A function $f$ is a rule that takes an input $x$ from the domain and produces a unique output $y$ , often written as:

$f(x)=y$

Here, $x$ is the input variable (independent variable), and $y$ is the output variable (dependent variable).

For example:

$f(x) = 2x + 3$ : When $x = 1$ , the output is $f(1) = 2(1) + 3 = 5$ .
$f(x) = x^2$ : This function squares the input value $x$ . For $x = -2$ , $f(-2) = (-2)^2 = 4$ .

Key Features of Functions

Domain: The set of all possible input values $x$ .

Example: For $f(x) = \sqrt{x}$ , the domain is $x \geq 0$ because square roots of negative numbers are not real.

Range: The set of all possible output values $y$ .

Example: For $f(x)=x^2$ , the range is $y \geq 0$ because squares of numbers are always non-negative.

Graph of a Function:

The graph of a function visually represents how the output $y$ changes with the input $x$ . A function passes the vertical line test: any vertical line drawn on the graph will intersect it at most one point.

Example:

- A linear function like $f(x) = 2x + 3$ produces a straight line.

- A quadratic function like $f(x) = x^2$ produces a parabolic curve.

Zeros of a Function:

The zeros (or roots) of a function are the points where the graph intersects the x-axis. Mathematically, this means $f(x) = 0$ .

Example:

For $f(x) = x^2 - 4$ , the zeros occur at $x=2$ and $x=-2$ $. Hence it has two zeroes (or roots)$ .

Increasing and Decreasing Functions

One of the key concepts related to change in tandem is analyzing increasing and decreasing functions. These terms describe the behavior of a function’s output as the input increases. By understanding where a function is increasing or decreasing, we can analyze its overall trend and solve real-world problems.

Definition

Increasing Function:

A function $f(x)$ is said to be increasing on an interval if, as the input $x$ increases, the output $f(x)$ also increases.

Mathematically:

$f(x_1) < f(x_2) \quad \text{whenever} \quad x_1 < x_2$

In simpler terms, as you move to the right along the x-axis, the graph of the function moves upward.

- Example: $f(x) = x$ (a linear function).

For $x = 1$ and $x = 2$ , we have $f(1) = 1$ and $f(2) = 2$ , which shows the output increases as the input increases.

- Graphically: The slope of the graph is positive.

Decreasing Function:

A function $f(x)$ is said to be decreasing on an interval if, as the input $x$ increases, the output $(x)$ decreases.

Mathematically:

$f(x_1) > f(x_2) \quad \text{whenever} \quad x_1 < x_2$

In simpler terms, as you move to the right along the x-axis, the graph of the function moves downward.

- Example: $f(x) = -x$ (a linear function).

For $x = 1$ and $x = 2$ , we have $f(1)=-1$ and $f(2)=-2$ , which shows the output decreases as the input increases.

- Graphically: The slope of the graph is negative.

Key Visual Clues in Graphs

On a graph, if you follow the curve from left to right:

If it goes upward, the function is increasing.
If it goes downward, the function is decreasing.

The slope of the graph (rate of change) helps determine this behavior:

Positive slope → Increasing function.
Negative slope → Decreasing function.

Intervals of Increase and Decrease

A function may not be entirely increasing or decreasing. Instead, it might increase on certain intervals and decrease on others. These intervals are crucial for understanding the function’s behavior.

Example: $f(x) = -x^2 + 4$ :

- This quadratic function is concave down, and its graph is a downward-facing parabola
- Increasing on $(-\infty, 0]$ : The graph moves upward as $x$ approaches 0.
- Decreasing on $[0, \infty)$ : The graph moves downward as $x$ increases past 0.

What is Concavity?

Concavity describes how the slope (or rate of change) of a function behaves:

Concave Up:

- A function is said to be concave up when the slope of the function is increasing.
- Graphically, the curve opens upward, resembling the shape of a “U”.
- If you think of pouring water into the curve, it would “hold” the water.
- Mathematically, the second derivative (rate of change of the slope) is positive: $f''(x) > 0$

Example: The function $f(x) = x^2$ is concave up.

As you move from left to right, the slope goes from negative to zero (at the vertex) and then positive.

Key Points:

- - The graph of $f(x) = x^2$ is a parabola that opens upward.
  - The slope increases as you move along the curve.

Concave Down:

- A function is said to be concave down when the slope of the function is decreasing.
- Graphically, the curve opens downward, resembling the shape of an inverted “U”.
- If you poured water into the curve, it would “spill” out.
- Mathematically, the second derivative (rate of change of the slope) is negative: $f''(x) < 0$

Example: The function $f(x) = -x^2$ is concave down.

As you move from left to right, the slope goes from positive to zero (at the vertex) and then negative.

Key Points:

- - The graph of $f(x) = -x^2$ is a parabola that opens downward.
  - The slope decreases as you move along the curve.

Points of Inflection

The point of inflection is where a graph changes concavity (from concave up to concave down, or vice versa). At this point:

The second derivative $f''(x)=0$ or does not exist.

Example: For , $f(x)=x^3$ the graph changes from concave down to concave up at $x = 0$ .

Download AP Precalculus worksheets along with video solutions

Frequently Asked Questions

Q. 1. What is Change in Tandem?

Change in tandem refers to the relationship between two quantities where a change in one causes a corresponding change in the other. This concept is fundamental in AP Precalculus and is often represented through functions and their graphs.

For example, if we consider time and distance, as time increases, the distance traveled can increase or decrease depending on the situation. Graphically, such relationships are shown as curves or lines on a coordinate plane, where the x-axis represents one variable (like time) and the y-axis represents the other (like distance).

Q. 2. How Long is the AP Precalculus Exam?

The AP Precalculus exam is 3 hours long. It consists of two main sections:

Section I: Multiple Choice

Number of questions: 40

Part A: 28 questions; 80 minutes; 43.75% of exam score (calculator not permitted)
Part B: 12 questions; 40 minutes; 18.75% of exam score (graphing calculator required)

Time : 2 hours

This section tests your ability to interpret and solve problems efficiently.

2. Section II: Free Response

Number of questions : 4

Part A: 2 questions; 30 minutes; 18.75% of exam score (graphing calculator required)
- Free-Response Question 1: Function Concepts
- Free-Response Question 2: Modeling a Non-Periodic Context
Part B: 2 questions; 30 minutes; 18.75% of exam score (calculator not permitted)
- Free-Response Question 3: Modeling a Periodic Context
- Free-Response Question 4: Symbolic Manipulations

Time : 1 hour

In this section, you are required to provide detailed solutions and justify your answers.

To keep yourself updated regarding the Exam information, you’re advised to have a frequent look on Official College Board website and if you’re struck while choosing the relevant resources for your AP Precalculus Preparation, Check this out

Need personalized help? Book a private online AP Precalculus tutor now!

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