which piecewise function is shown in the graph

Daniel Parker

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01-03-2025

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Determining the piecewise function represented by a graph involves carefully analyzing the different sections of the graph and identifying the function that describes each section. This guide will walk you through the process, providing a step-by-step approach and examples. Understanding piecewise functions is crucial in various fields, from computer science to engineering.

Understanding Piecewise Functions

A piecewise function is a function defined by multiple sub-functions, each applicable to a different interval of the domain. The graph of a piecewise function will often show distinct sections with different slopes or behaviors. Identifying these sections is key to determining the correct function.

Key Elements to Identify

Before we dive into examples, let's highlight the essential elements you need to identify when analyzing a graph of a piecewise function:

• The intervals: Where does each section of the graph begin and end? These define the domain intervals for each sub-function. Look for changes in slope or behavior.
• The function for each interval: What type of function describes the behavior within each interval? Is it linear (a straight line), quadratic (a parabola), or something else?
• Endpoints: Are the endpoints included or excluded in each interval? Closed circles (•) indicate inclusion, while open circles (◦) indicate exclusion.

Step-by-Step Guide: Deciphering Piecewise Functions from Graphs

Let's walk through an example. Imagine a graph showing a piecewise function with two distinct sections:

Step 1: Identify the Intervals

First, examine the graph and determine the intervals where the function changes its behavior. Let's say we have two intervals: x ≤ 2 and x > 2.

Step 2: Determine the Function for Each Interval

Next, focus on each interval individually. For x ≤ 2, the graph might show a straight line. Let's assume this line passes through points (0, 1) and (2, 5). Using the slope-intercept form (y = mx + b), we can determine the equation of this line.

The slope (m) is (5 - 1) / (2 - 0) = 2. The y-intercept (b) is 1. Therefore, the equation for this interval is y = 2x + 1.

For x > 2, the graph might show a constant function, a horizontal line at y = 5. The equation for this interval is simply y = 5.

Step 3: Write the Piecewise Function

Finally, combine the equations and their corresponding intervals into a piecewise function notation:

f(x) = 
   2x + 1,  if x ≤ 2
   5,        if x > 2

Example: Analyzing a More Complex Piecewise Function

Consider a graph exhibiting a more complex piecewise function, maybe involving a parabola and a linear function. The process remains the same:

1. Identify the intervals: Find where the function's behavior shifts. These might be at x = -1 and x = 3.
2. Determine the function for each interval: For each interval, determine the type of function—linear, quadratic, etc.—and find its equation using points on the graph. You might need to use the vertex form of a quadratic equation or other methods depending on the shape of the curve.
3. Write the piecewise function: Combine all the functions and their corresponding intervals using the piecewise function notation. Remember to include the correct inequality signs to denote whether the endpoints are included or excluded in each interval.

Tips and Tricks

• Use points on the graph: Choose clear points on the graph to help you determine the equations of the sub-functions.
• Check endpoints: Make sure you correctly identify whether the endpoints of each interval are included or excluded.
• Pay attention to slope and intercepts: The slope and y-intercept are crucial for determining the equation of a linear function.
• Consider the degree of the polynomial: For curved sections, consider quadratic (parabola) or cubic functions.

By systematically applying these steps and utilizing the tips provided, you can effectively determine the piecewise function depicted in any given graph. Remember to practice with different graphs to improve your understanding. The more you practice, the easier it becomes to identify the various components and write the correct piecewise function.