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Chapter 1: Problem 10

Sketch a graph of the line. $$g(x)=\frac{1}{3} x-1$$

Short Answer

Expert verified
The line \(g(x)=\frac{1}{3} x-1\) crosses the y-axis at point (0, -1) and rises by \(\frac{1}{3}\) units for each unit move to the right. It passes through the points (0,-1) and (1, -\frac{2}{3}).

Step by step solution

01

Identify the Slope and Y-Intercept

The equation is in the form \(y=mx+b\), where \(m\) is the slope and \(b\) is the y-intercept. Here, the slope \(m=\frac{1}{3}\), and the y-intercept \(b=-1\).
02

Plot the Y-Intercept on the Graph

Start by plotting the y-intercept, which is at the point (0, -1). This gives the point where the line crosses the y-axis.
03

Use the Slope to Find Another Point on the Line

The slope of the line, \(\frac{1}{3}\), tells you that for every step right by 1 unit, you go up by \(\frac{1}{3}\) unit. So, if you start at the y-intercept (0, -1) and move right one unit to x=1, the y-value will be \(-1 + \frac{1}{3} = -\frac{2}{3}\). Therefore, another point on the line is (1, -\frac{2}{3}).
04

Draw the Line

Finally, draw a straight line that passes through the points identified at step 2 and step 3. This line represents the equation \(g(x)=\frac{1}{3} x-1\).

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