Problem 30 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-30, plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\)-intercepts. $$ |x|+|y|=4 $$

Step-by-Step Solution

Answer

The graph is a diamond centered at the origin with vertices at (4, 0), (-4, 0), (0, 4), and (0, -4).

Step 1: Identify the Equation Type

The given equation is \(|x| + |y| = 4\), which involves absolute values. This type of equation represents a boundary where the total distance from the origin, in terms of both x and y axes, equals 4.

Step 2: Check for Symmetries

The equation \(|x| + |y| = 4\) is symmetric about both the x-axis and the y-axis. This means that if you flip the graph over the x-axis or y-axis, it remains unchanged. This is confirmed because the equation is unaffected by replacing x with -x or y with -y.

Step 3: Finding x- and y-Intercepts

To find the x-intercept, set \(y = 0\), which gives \(|x| = 4\). Solving for x, we have two solutions: \(x = 4\) and \(x = -4\). Similarly, to find the y-intercept, set \(x = 0\), which gives \(|y| = 4\). Solving for y, we also have two solutions: \(y = 4\) and \(y = -4\).

Step 4: Identifying Lines and Region

The absolute value equation \(|x| + |y| = 4\) represents a diamond shape in the coordinate plane. We can rewrite it as a combination of lines: 1. For \(x, y \geq 0\), the line is \(x + y = 4\).2. For \(x \geq 0, y \leq 0\), the line is \(x - y = 4\).3. For \(x \leq 0, y \geq 0\), the line is \(-x + y = 4\).4. For \(x, y \leq 0\), the line is \(-x - y = 4\).

Step 5: Plot the Points and Connect

You can now plot the intercepts: (4, 0), (-4, 0), (0, 4), and (0, -4) on a graph. Then, draw lines connecting these points. This will form a diamond shape on the coordinate plane, with the center at the origin (0, 0).

Key Concepts

Symmetry in Graphs

The concept of symmetry in graphs is a fundamental aspect when analyzing absolute value equations, like \(\|x\| + \|y\| = 4\). Symmetry simplifies understanding and graphing the equation by revealing that certain transformations—such as reflections across axes—do not alter the graph's appearance. Here, the equation is symmetric with respect to both axes. This means:

• If you reflect the graph across the x-axis, the equation remains unchanged.
• The same is true for a reflection across the y-axis.

Because \(x\) and \(y\) are enclosed within absolute value symbols, replacing \(x\) with \(-x\) or \(y\) with \(-y\) makes no difference in the equation's outcome. This inherent symmetry aides in predicting the placement and pattern of the graph, making it easier to visualize the resulting shape, which in this case, resembles a diamond centered at the origin.

Intercepts of Equations

Finding the intercepts of an equation is crucial in determining where the graph will cross the axes. For the equation \(\|x\| + \|y\| = 4\), we analyze the x-intercepts and y-intercepts separately.

**Finding x-intercepts** involves setting \(y = 0\). Plugging this into the equation, we get \[\|x\| + \|0\| = 4 \] which simplifies to \[\|x\| = 4. \] Solving it, we find two solutions: \(x = 4\) and \(x = -4\). These tell us that the points (4, 0) and (-4, 0) lie on the x-axis.

**Finding y-intercepts** requires setting \(x = 0\). Substituting, we get \[\|0\| + \|y\| = 4, \] or \[\|y\| = 4. \] Solving this gives us \(y = 4\) and \(y = -4\), indicating the points (0, 4) and (0, -4) lie on the y-axis.

These intercepts are essential because they provide specific points that help shape the boundary of the diamond on the graph.

Plotting Points in Coordinate Plane

Plotting points in the coordinate plane involves placing calculated points correctly and connecting them to visualize the entire graph. Here, the equation \(\|x\| + \|y\| = 4\) becomes a diamond shape through appropriate plotting.

• After finding intercept points (4, 0), (-4, 0), (0, 4), and (0, -4), these points must be accurately placed on the coordinate plane.
• These points form the vertices of the diamond.

Next, you connect these points in straight lines, ensuring each point links to its neighbors to complete the diamond shape.

This task emphasizes a crucial step in visualizing and understanding absolute value graphs. The coordinate plane offers a way to not only represent numeric relationships but also see how symmetry and intercepts create geometric shapes, enhancing the comprehension of algebraic concepts.