Problem 63 - Introductory Algebra for College Students

Question

Graph each equation in a rectangular coordinate system. $$y=\frac{1}{3} x-2(\text { Section } 3.4, \text { Example } 3)$$

Step-by-Step Solution

Answer

The graph of the equation, \(y=\frac{1}{3}x -2\) has a positive slope and intersects the y-axis at -2. From the y-intercept, the line rises 1 unit and runs 3 units to the right.

Step 1: Identify the slope and y-intercept

In the equation \(y=\frac{1}{3}x - 2\), the slope (m) is \(\frac{1}{3}\) and the y-intercept (b) is -2.

Step 2: Plot the y-intercept

The y-intercept is the point where the line crosses the y-axis. Since the y-intercept is -2, plot this point on the y-axis at (-2).

Step 3: Use the slope to find a second point

The slope of a line is the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. Since the slope is positive \(\frac{1}{3}\), from the y-intercept point, move 1 unit upward (rise) and 3 units to the right (run). Plot this second point.

Step 4: Draw the line

After plotting the second point, draw a straight line passing through these two points.This line represents the graph of the equation \(y=\frac{1}{3}x-2\).

Key Concepts

Slope-Intercept Form

When dealing with linear equations, one of the most common forms you'll encounter is the slope-intercept form. This form of the equation is expressed as \(y = mx + b\). But what do these values mean? Let's break it down:
\(m\) represents the slope of the line, which tells us how steep the line is. It is a measure of how much \(y\) increases or decreases as \(x\) increases. The slope is calculated as "rise over run," or the change in \(y\) divided by the change in \(x\).
\(b\) is the y-intercept. This is the point where the line crosses the y-axis, meaning \(x=0\). The value of \(b\) shows us where the line will land when graphed.
Understanding this form is crucial, as it allows us to quickly gather information about the behavior of the line just by looking at the equation.

Graphing

Graphing a linear equation is a crucial skill that visually represents how an equation behaves. To graph the equation \(y = \frac{1}{3}x - 2\), you follow a series of straightforward steps:

• First, identify the slope and y-intercept from the equation (here the slope \(m\) is \(\frac{1}{3}\), and the y-intercept \(b\) is -2).
• Next, plot the y-intercept on the y-axis. For \(b = -2\), place a point where x is zero, and y is -2.
• With the slope, find a second point. Starting at the y-intercept, use the slope \(\frac{1}{3}\) to determine your next point: move 1 unit up (rise) and 3 units right (run).
• Connect these points with a straight line, extending it across the graph to fully represent the equation.

Graphing helps you see not just where a line is, but how it moves across the plane. Remember, the slope gives it direction, while the y-intercept anchors it.

Rectangular Coordinate System

The rectangular coordinate system is like a map for plotting equations such as \(y = \frac{1}{3}x - 2\). It consists of two perpendicular lines called axes:

• The **x-axis** runs horizontally, serving as a base for measuring x-values.
• The **y-axis** is vertical, measuring y-values.

Where these axes intersect is the origin, marked as \((0, 0)\). Any point on the graph can be represented by a pair of coordinates \((x, y)\), which tell you how far along each axis the point is located. To plot our line's y-intercept \((-2)\), you'd identify \(x = 0\) on the x-axis and then move down to \(y = -2\) on the y-axis. Using the coordinate system, you can map out not just simple points, but explore how equations form lines and shapes across the grid.