Question

Use the definition of absolute value to graph the equation \(|x|-|y|=1 .\) Use a graphing utility to check your work.

Short answer

The graph of the equation \(|x| - |y| = 1\) is a square or a rhombus shape formed by the four lines y = x - 1, y = 1 - x, y = -x - 1, and y = x + 1. This shape is created by considering different cases for the values of x and y based on the absolute value function.

Step-by-step solution

Understand the absolute value function

The absolute value function is defined as follows: \(|a| = a\), if \(a \ge 0\) \(|a| = -a\), if \(a < 0\) So, we can rewrite the equation as: $$ |x| = \begin{cases} x, & \text{if } x>0 \\ -x, & \text{if } x<0 \\ \end{cases} $$ $$ |y| = \begin{cases} y, & \text{if } y>0 \\ -y, & \text{if } y<0 \\ \end{cases} $$ We will be using these definitions to evaluate the equation based on different cases for the values of x and y.

Solve for y based on the cases for x and y

Since we are considering 4 cases: 1. \(x \ge 0\) and \(y \ge 0\) 2. \(x \ge 0\) and \(y < 0\) 3. \(x < 0\) and \(y \ge 0\) 4. \(x < 0\) and \(y < 0\) For Case 1: $$ x - y = 1 \\ y = x - 1 $$ For Case 2: $$ x + y = 1 \\ y = 1 - x $$ For Case 3: $$ -x - y = 1 \\ y = -x - 1 $$ For Case 4: $$ -x + y = 1 \\ y = x + 1 $$

List the points for each case

Now that we have four equations, we need to find several points (x, y) that satisfy each equation and plot them on the coordinate plane: For Case 1: \(y = x - 1\) Points: \((1, 0), (2, 1), (3, 2)\) For Case 2: \(y = 1 - x\) Points: \((0, 1), (1, 0), (2, -1)\) For Case 3: \(y = -x - 1\) Points: \((-1, 0), (-2, 1), (-3, 2)\) For Case 4: \(y = x + 1\) Points: \((0, -1), (-1, 0), (-2, -1)\)

Plot the points and draw lines

Now, plot these points on the coordinate plane and connect them with lines. You should see a square or a rhombus shape on the graph.

Check the graph with a graphing utility

Finally, use a graphing utility (like Desmos, Geogebra, or a graphing calculator) and input the equation \(|x|-|y|=1\). The graph should match the one you've drawn in Step 4. If it does, then your work is correct.

Key concepts

Graphing Equations

Graphing equations involves representing algebraic equations visually on a plane. In this exercise, we are dealing with the equation \(|x|-|y|=1\). Usually, equations involving absolute values can create interesting shapes on a graph because of the piece-wise nature of the absolute value function.
When graphing such an equation, it is important to look at all possible cases for the variables, as absolute value can produce different equations depending on whether the variable inside the absolute value is positive or negative.

• Break down the equation into simpler forms depending on the sign of variables. This means creating sub-equations that represent each scenario: both variables are positive, one is positive and the other is negative, and so forth.
• For \(|x|\) and \(|y|\), consider cases for each of x and y being greater or less than zero.
• Use these separate equations to find points that lie on these lines by simply plugging in suitable values for x.

When you plot these points and connect them for each case, you start seeing the shape that represents the solution to the original equation.

Coordinate Plane

The coordinate plane is a two-dimensional surface on which we can plot points, lines, and curves. It consists of two axes, the x-axis, which is horizontal, and the y-axis, which is vertical. These axes create four quadrants in which we can place points using coordinates \((x, y)\).
For this task, the coordinate plane allows you to visually express the relationships between the variables in an equation.

• When plotting points from the equations derived from \(|x|-|y|=1\), each point represented by an \((x, y)\) pair will fit into one of these quadrants based on its sign.
• The intersections of these points, when connected by lines from each case, will outline a specific shape or pattern indicative of the equation worked on.

Overall, understanding how to use the coordinate plane is essential for taking abstract algebraic concepts and turning them into tangible visual forms, helping you understand the properties and solutions of equations.

Graphing Utility

A graphing utility is a tool, such as a computer application or calculator, that assists you in plotting mathematical equations efficiently and accurately. Popular tools include Desmos and GeoGebra.
Using a graphing utility is especially helpful for complex equations, providing visual verification to ensure the hand-drawn graphs are correct.

• Enter your original equation, \(|x|-|y|=1\), into the graphing utility.
• The tool will automatically plot the graph according to the equation you provided, showing how x and y interact to form the line or shape described by your equation, which in this case is likely a rhombus pattern given the step-by-step solution already defined certain points and lines we expect to cross each other.
• Seeing a correctly shaped graph appear can reassure you that your manual plotting matches computational results, highlighting any areas needing adjustment if there are discrepancies.

In practice, a graphing utility solidifies understanding by providing an exact visual reference that reinforces the manual calculations and plots.