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Chapter 2: Problem 38

Graph the equations by plotting points. $$ |x|+y=3 $$

Short Answer

Expert verified
The graph forms a V-shape with vertices at (0,3) extending downward.

Step by step solution

01

Understand the Absolute Value Equation

The given equation is \(|x| + y = 3\). Absolute value equations involve two cases for \(||x||\): one for when \(x\) is positive or zero and another for when \(x\) is negative.
02

Consider the First Case (x 鈮� 0)

For values of \(x \ge 0\), the absolute value function is \(|x| = x\). Substitute \(x\) in the equation: \(x + y = 3\). Solve for \(y\): \[y = 3 - x\].
03

Consider the Second Case (x < 0)

For values of \(x < 0\), the absolute value function is \(|x| = -x\). Substitute \(-x\) in the equation: \(-x + y = 3\). Solve for \(y\): \[y = 3 + x\].
04

Plot Points for x 鈮� 0

Using the equation \[y = 3 - x\], choose initial values for \(x\) such as 0, 1, 2, and 3. \( (0, 3)\), \( (1, 2)\), \( (2, 1)\), \( (3, 0)\).
05

Plot Points for x < 0

Using the equation \[y = 3 + x\], choose values for \(x\) such as -1, -2, -3, and -4. \( (-1, 2)\), \( (-2, 1)\), \( (-3, 0)\), \( (-4, -1)\).
06

Draw the Graph

Sketch the points found in Steps 4 and 5 on the graph. Connect the dots, noting that the graph is going to form a V-shape due to the behaviour of the absolute value function.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Plotting Points
When graphing any equation, plotting points is a fundamental step. It involves selecting specific x-values, calculating the corresponding y-values, and then marking these coordinates on the graph. For the equation \(|x| + y = 3\), we start by choosing values for x, both positive and negative, and then solving for y.
Absolute Value Functions
Understanding absolute value functions is crucial for solving the equation \(|x| + y = 3\). Absolute value means the distance from zero, so \(|x| = x\) for \(x \ge 0\) and \(|x| = -x\) for \(x < 0\). This piecewise nature impacts how we set up our two cases for graphing.
Graph Interpretation
Interpreting the graph helps us visually understand the equation \(|x| + y = 3\). By plotting points and sketching the graph, we see that it forms a V-shape. This shape happens because of the absolute value function, which creates two linear segments joined at the vertex (when x = 0). The V-shape illustrates how y changes based on different x-values, providing a clear picture of the equation's behaviour.

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Most popular questions from this chapter

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