Chapter 2: Problem 38

Graph the equations by plotting points. \(|x|+y=3\)

### Short Answer

Expert verified

Plot the points (0, 3), (1, 2), (2, 1), (3, 0), (-1, 2), (-2, 1), (-3, 0) and connect them with straight lines to form a 'V' shape.

## Step by step solution

01

## Identify the Equation

The given equation is \(|x| + y = 3\)

02

## Substitute Positive x Values

Substitute some positive values for x into the equation and solve for y. For example, if \( x = 0 \), \( |0| + y = 3 \) gives \( y = 3 \). Another example, if \( x = 1 \), \( |1| + y = 3 \) gives \( 1 + y = 3 \) which simplifies to \( y = 2 \). Continue for \( x = 2 \) and \( x = 3 \).

03

## Substitute Negative x Values

Substitute some negative values for x into the equation and solve for y. For example, if \( x = -1 \), \( |-1| + y = 3 \) gives \( 1 + y = 3 \) which simplifies to \( y = 2 \). Repeat for \( x = -2 \) and \( x = -3 \).

04

## Create the Plot Points

List all the plot points from the previous steps: \( (0, 3) \), \( (1, 2) \), \( (2, 1) \), \( (3, 0) \), \( (-1, 2) \), \( (-2, 1) \), and \( (-3, 0) \).

05

## Plot the Points on the Graph

Plot each of the points on a Cartesian coordinate system. Ensure accuracy in the placement of each point: \( (0, 3) \), \( (1, 2) \), \( (2, 1) \), \( (3, 0) \), \( (-1, 2) \), \( (-2, 1) \), and \( (-3, 0) \).

06

## Draw the Graph

Connect the points with straight lines to visualize the graph of the equation \( |x| + y = 3 \). The graph will form a 'V' shape.

## Key Concepts

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

###### Plotting Points

Plotting points is a fundamental skill in graphing equations. It involves placing specific coordinates on a graph based on given equations. Let's break this down. To start, you'll need:

- The equation to graph
- A list of points derived from substituting values into the equation
- A graph paper or a graphing tool

For example, in the equation \(|x| + y = 3\), you substitute positive and negative \(\text{values of } x\text { to } find y:\). This gives you points you can plot: \((0, 3), (1, 2), (2, 1), (3, 0), (-1, 2), (-2, 1), (-3, 0)\). After plotting these points, connect them to form the shape of the graph. Make sure each point is accurately placed using the coordinates, and draw straight lines to connect them. Plotting points is essential for understanding how the equation translates visually.

###### Cartesian Coordinate System

The Cartesian coordinate system is a two-dimensional plane divided by a horizontal line \(x-axis\) and a vertical line \(y-axis\). This system helps us to visually represent mathematical equations. Every point on the plane is defined by an ordered pair \((x, y)\). Here’s how it works:

- Each point is determined by how far along the x-axis and y-axis it is.
- The origin \( (0, 0)\) is the point where the x-axis and y-axis intersect.
- Positive values of x are to the right of the origin, and negative values are to the left.
- Positive values of y are above the origin, and negative values are below.

To plot the equation \( |x| + y = 3\), you’ll use the Cartesian plane. Place each calculated point \((0, 3), (1, 2), (2, 1), (3, 0), (-1, 2), (-2, 1), (-3, 0)\)while keeping track of their exact locations in relation to the origin. Understanding this system is crucial for accurate graphing.

###### Absolute Value Graph

An absolute value graph represents equations that include absolute value expressions, like \(|x|\text{ in } |x| + y = 3\). The absolute value \(|x|\) makes sure that only non-negative values of x are represented. Here’s the step-by-step:

- When x is positive or zero, \(|x| = x\).
- When x is negative, \(|x| = -x\).
- In essence, \(|x|\) reflects all negative x values as positive, creating symmetry.

For this equation \(|x| + y = 3\), this symmetry manifests in the plotted points: \((1, 2)\) mirrors \((-1, 2)\), \((2, 1)\) mirrors \((-2, 1)\). Plotting these on the coordinate system forms a 'V' shape. Absolute value graphs are unique due to this V-shaped symmetry and are extremely helpful in visualizing how equations behave differently with positive and negative inputs.

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