Question

Sawtooth wave Graph the sawtooth wave defined by $$f(x)=\left\{$$\begin{array}{ll}⋮& \\ \text{x}+1& \text{ if }-1\le x<0\\ \text{x}& \text{ if }0\le x<1\\ \text{x}-1& \text{ if }1\le x<2\\ \text{x}-2& \text{ if }2\le x<3\\ \\ ⋮& ⋮\end{array}$$\right.$$

Short answer

Short Answer: To graph the given sawtooth function, follow these steps: 1) Understand the piecewise function and its intervals. 2) Graph each piece of the function on its specific interval, connecting them appropriately. 3) Extend the graph to infinity in both the positive and negative x-direction, repeating the same slope and translation observed in the previous segments.

Step-by-step solution

Understand the piecewise function

First, note that the given function f(x) is a piecewise function defined for different intervals of x (from -1 to 3). The function is described by the following equations: - x + 1, for -1 ≤ x < 0 - x, for 0 ≤ x < 1 - x - 1, for 1 ≤ x < 2 - x - 2, for 2 ≤ x < 3 The vertical dots indicate that the function repeats its behavior similarly as x increases or decreases further.

Graph each piece of the function

We will now plot the function over each defined interval: A. For -1 ≤ x < 0, the function is given by f(x) = x+1. It is a straight line with a slope of 1 and a y-intercept of 1. Plot this interval with these properties. B. For 0 ≤ x < 1, the function is given by f(x) = x. It is also a straight line with a slope of 1, but with a y-intercept of 0. Plot this interval right after the first one, connecting it to the point (0,1). C. For 1 ≤ x < 2, the function is given by f(x) = x - 1. It is a straight line with a slope of 1 and a y-intercept of -1. Plot this interval right after the second one, making sure it connects to the point (1,1). D. For 2 ≤ x < 3, the function is given by f(x) = x - 2. It is a straight line with a slope of 1 and a y-intercept of -2. Plot this interval after the third one, making sure it connects to the point (2,1).

Extend the graph to infinity

Since the vertical dots indicate that the function repeats its behavior for all x, we can extend the sawtooth graph to infinity in both the positive and negative x-direction. The function would continue with the same slope and translation to the previous segments. By following these steps, you should now have a graph of the sawtooth wave function as defined for f(x).

Key concepts

Piecewise Function

A piecewise function is like a chameleon of the mathematical world—it changes its formula depending on the value of the input, or in mathematical terms, the variable. In essence, a piecewise function is composed of multiple other functions, each with its domain where it 'comes to life' and defines the output.

For example, the sawtooth wave function is defined in pieces: for certain intervals of the independent variable 'x', the function behaves differently. It is designed to be a collection of linear equations that, together, form the distinctive jagged graph that resembles the teeth of a saw.

Understanding how to dissect and work with each piece is essential in grasping the overall behavior of the function. It's like understanding a story by examining its individual chapters—each interval is a chapter that tells part of the story of the function's behavior. When combined, you see the full narrative that is the function's graph.

Graphing Functions

Graphing functions is a visual way of interpreting equations. It’s where the abstract meets the concrete; numbers and equations become lines, curves, and interesting shapes on a graph. The graph of a function is a representation of all of the viable solutions to the equation of the function plotted on the Cartesian coordinate system.

When graphing a sawtooth wave, you are essentially plotting these different solutions along the number line, one interval at a time. The graph provides a visual narrative of the function's behavior: as 'x' increases or decreases, how does 'f(x)' respond? With the sawtooth wave, we see a repeating pattern of sharp increases—a direct result of the linear nature of each piece.

The goal when graphing is to ensure continuity where needed and to properly represent all the 'pieces' in their respective domains. This can be particularly challenging with piecewise functions, but when done correctly, the graph reveals the intricate dance between the various parts of the function.

Linear Equations

Linear equations are the simplest form of equations where the highest power of the variable is one—straightforward, no-nonsense kind of relationships where the graph is always a straight line. They come in the general form of 'y = mx + b', where 'm' is the slope and 'b' is the y-intercept, the point where the line meets the y-axis.

Within the sawtooth wave, each piece is a linear equation that continues until the next piece takes over. The graph of each equation is a straight line segment with a slope of 1, meaning that as 'x' increases by 1, 'f(x)' also increases by 1. These lines are consistently rising at the same rate but are displaced, or 'translated,' according to their y-intercept. This gives the sawtooth wave its unique shape as the graph restarts at lower values, creating the visual 'teeth'.

A reliable way to tackle these is to graph each equation over its specified interval, note the point where each segment ends, and ensure that the next segment begins right where the previous one left off. This creates the step-like structure necessary for the sawtooth pattern to emerge on the graph.