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Chapter 11: Problem 48

In the following exercises, graph by plotting points. $$ y=x-3 $$

Short Answer

Expert verified
Calculate y for chosen x values, plot the points, and draw the line.

Step by step solution

01

Choose Values for x

Select a few values for x to substitute into the equation. Common values to use are -2, -1, 0, 1, and 2.
02

Calculate Corresponding y Values

Substitute each chosen value of x into the equation y = x - 3 and solve for y.
03

Create Ordered Pairs

Pair each x value with its corresponding y value to get ordered pairs (x, y). For example, if x = 0, y = 0 - 3 = -3, so the ordered pair is (0, -3).
04

Plot the Points

Plot each ordered pair on a coordinate plane. Place a dot at each point (x, y).
05

Draw the Line

Connect the plotted points with a straight line. This line represents the graph of the equation y = x - 3.

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Key Concepts

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

Coordinate Plane
The coordinate plane is a two-dimensional surface. It is made up of two number lines: a horizontal line called the x-axis and a vertical line called the y-axis.
The point where these two axes intersect is called the origin, labeled as \(0,0\).
Each point on the coordinate plane can be identified by an ordered pair (x, y), which tells you its position along the x and y axes.
Ordered Pairs
An ordered pair is a pair of numbers used to locate a point on the coordinate plane. It is written as (x, y).
The first number, x, represents the point鈥檚 horizontal position, while the second number, y, represents its vertical position.
For example, in the ordered pair (2, -3), 2 means move 2 units to the right (along the x-axis) and -3 means move 3 units down (along the y-axis).
Plotting Points
Plotting points means placing them on the coordinate plane according to their ordered pairs.
Here are the steps:
  • Start at the origin (0,0).
  • Move horizontally to the x-coordinate.
  • From there, move vertically to the y-coordinate and place a dot.
For example, to plot the point (1, -3):
  • Start at the origin.
  • Move 1 unit to the right on the x-axis.
  • Then, move 3 units down on the y-axis.
  • Place a dot at this position.
Linear Equations
A linear equation is an equation whose graph forms a straight line.
The general form of a linear equations in two variables is y = mx + b, where:
  • m represents the slope, the rate at which y changes with respect to x.
  • b represents the y-intercept, the point where the line crosses the y-axis.
In our exercise, the equation y = x - 3 represents a line that crosses the y-axis at -3 and has a slope of 1. This means for every increase in x by 1, y increases by 1.

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