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Chapter 0: Problem 2

Graph. (Unless directed otherwise, assume that "Graph" means "Graph by hand.") $$y=x-1$$

Short Answer

Expert verified
Plot the points (0, -1) and (1, 0), then draw a line through them.

Step by step solution

01

- Identify the type of equation

Identify that the equation is in slope-intercept form, which is generally given by \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
02

- Determine the slope and y-intercept

From the equation \(y = x - 1\), we can see that the slope \(m\) is 1, and the y-intercept \(b\) is -1.
03

- Plot the y-intercept

Start by plotting the y-intercept on the graph. The y-intercept is the point where the line crosses the y-axis. For the equation \(y = x - 1\), this point is (0, -1).
04

- Apply the slope to find another point

The slope is 1, meaning for every unit increase in x, y increases by 1. From the y-intercept (0, -1), move right 1 unit and up 1 unit to find another point on the line, which is (1, 0).
05

- Draw the line

Using a ruler, draw a straight line through the points (0, -1) and (1, 0). This line represents the graph of the equation \(y = x - 1\).

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Key Concepts

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

slope-intercept form
The slope-intercept form is a powerful tool for graphing linear equations. It is written as:
\( y = mx + b \)
In this form,
  • \( m \) represents the slope of the line.
  • \( b \) represents the y-intercept, where the line crosses the y-axis.
Recognizing and using this form allows us to easily graph the equation by identifying the slope and y-intercept.
slope
The slope \( m \) of a line indicates its steepness and direction. In the context of the equation \( y = x - 1 \), the slope is 1.
The slope tells us how much the y-value changes for a corresponding change in the x-value. Specifically,
  • A slope of 1 means that for every 1 unit increase in x, y increases by 1 unit.
To find this on a graph, you can start from any point on the line and use the slope to determine another point. This is crucial for correctly drawing the line.
y-intercept
The y-intercept \( b \) is the point where the line crosses the y-axis. For the equation \( y = x - 1 \), the y-intercept is -1.
This means that the line will pass through the point (0, -1) on the graph.
  • To plot the y-intercept, find the value of y when x is 0 (which is simply \( b \)).
In this scenario, when x is 0, y is -1, making (0, -1) the y-intercept.
graphing steps
To graph the equation \( y = x - 1 \), follow these steps:
1. **Start by plotting the y-intercept**: Since the y-intercept is -1, put a point at (0, -1) on the graph.
2. **Use the slope to find another point**: With a slope of 1, from (0, -1), move right 1 unit and up 1 unit to reach (1, 0).
3. **Draw the line**: Connect these two points with a straight line, extending it in both directions.
These steps ensure an accurate representation of the equation on the graph.

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