Problem 44 - Calculus with Concepts in Calculus

Question

Sketch the graph of the equation. In each case determine whether the graph is that of a function. $$ |x|=|y| $$

Step-by-Step Solution

Answer

The graph is two intersecting lines, not a function.

Step 1: Understanding the Equation

The equation given is \(|x| = |y|\). This equation involves absolute values, meaning it describes situations where two numbers have the same magnitude regardless of their sign.

Step 2: Breakdown of Absolute Value

Since the absolute value of a number can be either positive or negative, \(|x| = |y|\) implies two scenarios: either \(x = y\) or \(x = -y\).

Step 3: Graphing the First Scenario

Let's consider \(x = y\). This equation represents a straight line passing through the origin with a slope of 1, creating a diagonal line from the bottom left to the top right of the graph.

Step 4: Graphing the Second Scenario

Now, consider \(x = -y\). This equation represents another line through the origin but with a slope of -1, creating a diagonal line from the top left to the bottom right of the graph.

Step 5: Combining the Lines

By combining both scenarios, the graph is two intersecting lines forming an "X" shape through the origin, with angles of 45 degrees between them.

Step 6: Determining Function Status

To determine if the graph is a function, use the vertical line test. Draw vertical lines across the graph: if any vertical line touches two or more points from the graph, it is not a function. In this graph, vertical lines will intersect the graph at two points, failing the test.

Key Concepts

Graphing Equations

Graphing absolute value equations like \( |x| = |y| \) involves understanding how the absolute value affects the variables. \( |x| = |y| \) simplifies into two possible lines: \( x = y \) and \( x = -y \). Each of these lines represents a possible scenario where the magnitude of \( x \) and \( y \) are the same, either by being equal or being opposites.

To graph these equations, we begin by considering each line individually. Let's take \( x = y \). This line, having a slope of 1, stretches diagonally across the graph from the bottom left corner to the top right corner. It passes directly through the origin, which is where \( x \) and \( y \) equal zero.

On the other hand, \( x = -y \) has a slope of -1. This creates a different diagonal path—from the top left to the bottom right of the graph. Again, it crosses through the origin where both variables are zero. By plotting these two lines, we get a crossing shape often referred to as an 'X', showing the complete solution to the equation. These lines demonstrate all the values fulfilling \( |x| = |y| \).

Functions vs Non-Functions

The concept of functions is fundamental, but not all equations qualify as such. An equation represents a function if every input, or \( x \) value, corresponds to exactly one output value, or \( y \).

In the graph of \( |x| = |y| \), you have two different outcomes for certain \( x \) values. For example, when \( x = 1 \), \( y \) can be either \-1 or 1. This is because the absolute value simply indicates equal magnitude, not necessarily the same direction.

Due to having different possible outputs for a single \( x \) input, \( |x| = |y| \) does not abide by the definition of a function. This distinction is crucial when identifying and working with graphs that may appear straightforward but hide nuances of multiple outputs. Recognizing such characteristics helps in handling more complex graphing scenarios effectively.

Vertical Line Test

The vertical line test is a simple yet powerful tool for determining if a graph represents a function. Imagine drawing a vertical line across your graph. For the graph to depict a function, each vertical line should intersect the graph at no more than one point.

In the case of the graph of \( |x| = |y| \), when you draw vertical lines at certain points, such as \( x = 1 \), these lines intersect the 'X' shape at two different points, reflecting \( y = 1 \) and \( y = -1 \). A graph failing this test, like our example, means it can't be a function because it shows that a single \( x \) input can lead to multiple outputs.

Using the vertical line test helps quickly verify whether a given graph meets the strict definition of a function. It is a very practical method for students working to understand different types of relations between \( x \) and \( y \) on a graph.