Problem 46

Question

Sketch the graph of the equation. In each case determine whether the graph is that of a function. $$ |x|-|2 y|=1 $$

Step-by-Step Solution

Verified

Answer

The graph is not a function; some vertical lines intersect multiple line segments.

1Step 1: Understand the Equation

We are given the equation $|x| - |2y| = 1$. This equation involves absolute values, which suggests that the graph could be comprised of linear segments.

2Step 2: Analyze Simplified Components

Consider two separate cases for the absolute values: 1. $x \, \geq \, 0$ and $y \, \geq \, 0$2. $x \, \geq \, 0$ and $y \, < \, 0$3. $x \, < \, 0$ and $y \, \geq \, 0$4. $x \, < \, 0$ and $y \, < \, 0$

3Step 3: Solve for Each Case

For the cases, we get:1. $|x| = x$ and $|2y| = 2y$ leads to $x - 2y = 1$.2. $|x| = x$ and $|2y| = -2y$ leads to $x + 2y = 1$.3. $|x| = -x$ and $|2y| = 2y$ leads to $-x - 2y = 1$.4. $|x| = -x$ and $|2y| = -2y$ leads to $-x + 2y = 1$.

4Step 4: Graph Each Line

For each derived equation:1. $x - 2y = 1$: Rearrange to get $y = \frac{x - 1}{2}$.2. $x + 2y = 1$: Rearrange to get $y = \frac{1 - x}{2}$.3. $-x - 2y = 1$: Rearrange to get $y = -\frac{1 + x}{2}$.4. $-x + 2y = 1$: Rearrange to get $y = \frac{x + 1}{2}$.Plot these lines, each representing a part of the graph.

5Step 5: Determine Functionality

A graph represents a function if each vertical line intersects the graph at most once. Due to the symmetry and intersection of these lines, observe if any vertical line crosses more than one line segment of the graph.

Key Concepts

Absolute Value EquationsLinear EquationsFunction DeterminationGraph Symmetry

Absolute Value Equations

Absolute value equations often seem daunting at first, but their core concept is simple—they measure the distance from zero. In the equation $|x| - |2y| = 1$, the absolute value symbols $ |x|$ and $ |2y|$ indicate that both $ x $ and $ y $ could have either positive or negative values.

The key to handling absolute value equations is considering different cases based on the values of the variables. This is because the equation behaves differently when $ x $ and $ y $ are positive versus when they are negative.

For $x \geq 0$ and $y \geq 0$, the equation simplifies to $x - 2y = 1$.
For $x \geq 0$ and $y < 0$, it becomes $x + 2y = 1$.
For $x < 0$ and $y \geq 0$, it changes to $-x - 2y = 1$.
For $x < 0$ and $y < 0$, the equation is reformed to $-x + 2y = 1$.

By dissecting these cases separately, we unravel the behavior of the equation for different regions of the graph. This approach simplifies the graph into sections that are easier to sketch and analyze.

Linear Equations

Linear equations, like the ones we derived from the absolute value cases, are foundational in graphing. Each simplification of the absolute value equation leads to a straightforward linear equation, such as $ y = \frac{x-1}{2} $.

These equations portray lines in the coordinate plane, determined by the familiar form $ y = mx + b $, where $ m $ is the slope and $ b $ is the y-intercept.

The equation $x - 2y = 1$ translates to $y = \frac{x-1}{2}$, showing a slope of 0.5 and a y-intercept at $-0.5$.
For $x + 2y = 1$, the linear form $y = \frac{1-x}{2}$ reveals a slope of -0.5 and a y-intercept of 0.5.
\

Function Determination

Determining if a graph represents a function is crucial. By definition, a graph can be deemed a function if no vertical line intersects it at more than one point, known as the $"vertical line test"$.

Examining our absolute value equation's graph, there are multiple linear segments. For instance, when you draw a vertical line, it might intersect more than one line segment of the graph.

This occurrence directly implies that the graph is not a function, as there are multiple $ y $ values for at least one $ x $ value on the graph.

Passing this test is essential for confirming the functionality of the graph. In some cases, you might discover that despite having several equations mapped out, the overall graph might not qualify as a function if this test isn't satisfied.

Graph Symmetry

Graph symmetry can help us understand the overall shape and characteristics of a graph. Symmetry in graphs often simplifies the process of sketching and analyzing.

Consider the original equation $|x| - |2y| = 1$. The absolute values suggest that the graph has symmetry, particularly:

Y-axis Symmetry: If the equation remains unchanged when $ x $ is replaced with $-x$, the graph is symmetric with respect to the y-axis. Our equation exhibits this as both positive and negative $ x $ values are equally considered.

Additionally, the lines derived from the equation show symmetrical traits around the origin point due to their equal and opposite slopes:

The lines $ y = \frac{x-1}{2} $ and $ y = \frac{x+1}{2} $ display reflectional symmetry, providing further visual cues to the symmetry inherent in the graph.
Thus, symmetry aids in predicting the continuity and directionality of the graph's segments.

Understanding symmetry not only helps predict graph behaviors but can also simplify complex graphing tasks.

Problem 46

Other exercises in this chapter

Problem 46

Find an equation of the line that is perpendicular to the given line $l$ and passes through the given point $P$. $l: 3 x-y=0 ; P=(1,3)$

View solution

Problem 46

Let $f(x)=\sqrt{x^{2}+1}-1$ and $g(x)=\frac{x^{2}}{1+\sqrt{x^{2}+1}}$ a. Find the domains of $f$ and $g$. b. Show that $f=g$.

View solution

Problem 46

For which functions $f$ is there a function $g$ such that $f=1 / g$ ?

View solution

Problem 47

Suppose a ball of mass $m$ is attached to a string of length $L$ and is rotated in a vertical plane with enough velocity $v$ so that the string remains ta

View solution