Graphs and Functions in Trigonometry

Introduction

Understanding graphs and functions is a foundational step in mastering trigonometry. This lesson covers the essential properties of functions, how to graph them, and how these concepts apply to trigonometric functions.

What is a Function?

A function is a relationship where each input (independent variable, typically denoted as $x$ ) has exactly one output (dependent variable, typically denoted as $y$ ). This relationship is often written as:

$y = f(x)$

For example, the equation $y = x^2$ represents a function, as each value of $x$ corresponds to only one value of $y$ .

Domain and Range

The domain of a function consists of all possible input values ( $x$ ), while the range consists of all possible output values ( $y$ ). For instance:

For $y = x^2$ , the domain is all real numbers ( $x \in \mathbb{R}$ ), and the range is $y \geq 0$ .
For the trigonometric function $y = \sin(x)$ , the domain is all real numbers ( $x \in \mathbb{R}$ ), and the range is $-1 \leq y \leq 1$ .

Graphing Functions

Graphing a function involves plotting points that satisfy the equation $y = f(x)$ on a coordinate plane. Follow these steps:

Choose a set of input values from the domain.
Calculate the corresponding output values using the function.
Plot the points $(x, y)$ on the graph.
Connect the points smoothly, keeping in mind the behavior of the function.

For example, graphing $y = \sin(x)$ over one period involves plotting values of $x$ from $0$ to $2\pi$ and corresponding $y$ values:

$\sin(0) = 0$
$\sin\left(\frac{\pi}{2}\right) = 1$
$\sin(\pi) = 0$
$\sin\left(\frac{3\pi}{2}\right) = -1$
$\sin(2\pi) = 0$

Key Properties of Trigonometric Functions

Trigonometric functions have unique properties, such as periodicity, symmetry, and amplitude. Let’s examine these using $y = \sin(x)$ :

Periodicity: The function repeats every $2\pi$ .
Amplitude: The maximum value of $|\sin(x)|$ is $1$ .
Symmetry: $\sin(x)$ is an odd function, meaning $\sin(-x) = -\sin(x)$ .

Practice Problems

Try these exercises to solidify your understanding:

Graph the function $y = \cos(x)$ over one period and identify its key properties.
Determine the domain and range of $y = \tan(x)$ .
Explain why $y = x^3$ is a function, but a vertical line $x = 3$ is not.

Graphs and Functions

Graphs and Functions in Trigonometry

Introduction

What is a Function?

Domain and Range

Graphing Functions

Key Properties of Trigonometric Functions

Practice Problems

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One thought on “Graphs and Functions”

Introduction

What is a Function?

Domain and Range

Graphing Functions

Key Properties of Trigonometric Functions

Practice Problems

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One thought on “Graphs and Functions”