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Chapter 7: Problem 24

Graph each equation using the slope and \(y\) -intercept. $$y=x-2$$

Short Answer

Expert verified
The line crosses the y-axis at (0, -2) and follows a slope of 1 to another point (1, -1).

Step by step solution

01

Identify the Slope and Y-Intercept

The given equation is in the slope-intercept form, which is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. From the equation \( y = x - 2 \), we identify the slope \( m = 1 \) and the y-intercept \( b = -2 \).
02

Plot the Y-Intercept

Begin by plotting the y-intercept on the coordinate plane. The y-intercept \( b = -2 \) means the point (0, -2) is where the line crosses the y-axis. Plot this point on the graph.
03

Use the Slope to Find Another Point

The slope \( m = 1 \) can be interpreted as the rise over run, which means for every 1 unit you move right (positive x-direction), you move 1 unit up (positive y-direction). From the y-intercept point (0, -2), move right 1 unit to (1, -2), then up 1 unit to reach the point (1, -1). Plot this new point.
04

Draw the Line

With the points (0, -2) and (1, -1) plotted, draw a straight line through them to represent the equation \( y = x - 2 \). Extend the line across the graph to show it in both directions.

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Key Concepts

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

Slope-Intercept Form
Linear equations can often be seen in the slope-intercept form, which is written as \( y = mx + b \). This format is incredibly helpful when graphing because it gives you two pieces of critical information right away: the slope and the y-intercept.
The slope \( m \) tells you how steep the line is, and the y-intercept \( b \) is where the line crosses the y-axis. The equation \( y = x - 2 \) is already in this form. Here, the slope \( m \) is 1, and the y-intercept \( b \) is -2. Knowing these values allows us to draw the initial point and establish the line's direction easily.
Coordinate Plane
A coordinate plane is like a map where each location is identified by a pair of numbers, known as coordinates. These numbers are written as \((x, y)\). The horizontal line is called the x-axis, and the vertical line is the y-axis.
This grid helps us graph equations visually. When you're asked to plot an equation like \( y = x - 2 \), you'll specifically look for the y-intercept and use the slope to plot additional points. The first step is to locate the y-intercept on the y-axis.
  • The x-coordinate indicates how far to the left or right a point is.
  • The y-coordinate indicates how far up or down a point is.
This understanding allows us to accurately position each point based on our mathematical findings.
Slope
The slope of a line is a measure of its steepness and direction. It's often described as "rise over run," meaning how much you go up or down (rise) for a certain move to the right (run). In equations like \( y = mx + b \), the slope \( m \) guides us on this move.
For the equation \( y = x - 2 \), the slope \( m \) is 1. This means for every unit you move to the right on the x-axis, you also move 1 unit up on the y-axis. The result is a line that moves diagonally upward across the coordinate plane.
  • A positive slope moves upwards as you go from left to right.
  • A negative slope descends downwards as you move right.
Mastering slope benefits your graphing skills greatly, enabling you to trace lines accurately.
Y-Intercept
The y-intercept of a line is the specific point where the line crosses the y-axis, meaning the x-coordinate is 0 at this point. In the formula \( y = mx + b \), the y-intercept \( b \) allows you to start graphing the line.
For the equation \( y = x - 2 \), the y-intercept \( b \) is -2. This is your starting point for the graph. It corresponds to the coordinates \((0, -2)\). This point is crucial because it anchors your line on the graph.
From the y-intercept, you use the slope to find other points on the line. This combination of y-intercept and slope quickly gives you a full picture of the line's placement and direction on the graph.

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