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Chapter 4: Problem 328

Determine the most convenient method to graph each line. \(y=\frac{2}{3} x-1\)

Short Answer

Expert verified
Plot the y-intercept (0, -1) and use the slope \( \frac{2}{3} \) to find another point (3, 1), then draw the line.

Step by step solution

01

- Identify the Equation

The given equation is in slope-intercept form, which is written as \( y = mx + b \). In this form, \( m \) represents the slope and \( b \) represents the y-intercept.
02

- Identify the Slope and Y-intercept

From the equation \( y = \frac{2}{3} x - 1 \), identify the slope \( m = \frac{2}{3} \) and y-intercept \( b = -1 \).
03

- Plot the Y-intercept

Start by plotting the y-intercept on the graph, which is the point (0, -1).
04

- Use the Slope to Find Another Point

The slope \( \frac{2}{3} \) means rise over run, so from the y-intercept point, move up 2 units and to the right 3 units to find another point on the line. This will be the point (3, 1).
05

- Draw the Line

Draw a line through the two points, (0, -1) and (3, 1), to complete the graph of the line.

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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 way to write the equation of a straight line. This form is written as \( y = mx + b\). Here, \(m\) represents the slope of the line, and \(b\) represents the y-intercept. Using this form makes it easy to see both the slope and the y-intercept directly from the equation.

For example, in the equation \(y = \frac{2}{3} x - 1\), \(\frac{2}{3}\) is the slope and \(-1\) is the y-intercept.

This form is especially convenient for graphing, as it gives you all the information needed to easily plot the line on a graph.
plotting points
Plotting points is a key step in graphing a line. To start, find the y-intercept, which is the point where the line crosses the y-axis. Next, use the slope to find another point on the graph.

Let's take the y-intercept from the equation \(y = \frac{2}{3} x - 1\), which is -1. Plot \( (0, -1)\) on the graph.

After plotting the y-intercept, use the slope to find another point. From \( (0, -1)\), move up 2 units (rise) and 3 units to the right (run) to reach \( (3, 1)\). Now you have two points: \( (0, -1)\) and \( (3, 1)\).

Draw a line through these points to complete the graph.
slope
The slope of a line tells you how steep the line is and in which direction it鈥檚 going. The slope is usually denoted by \(m\). It is calculated as the rise (the vertical change) over the run (the horizontal change) between two points on the line.

In the equation \(y = \frac{2}{3} x - 1\), the slope is \(\frac{2}{3}\). This means that for every 3 units you move to the right (run), you move 2 units up (rise).

Positive slopes go upwards from left to right, while negative slopes go downwards. Zero slopes mean the line is horizontal and infinite slopes mean the line is vertical.
y-intercept
The y-intercept is where the line crosses the y-axis. This is the value of \(y\) when \(x\) is 0. The y-intercept is denoted by \(b\) in the slope-intercept form \(y = mx + b\).

In our example, \(y = \frac{2}{3} x - 1\), the y-intercept is \(-1\). So, the line crosses the y-axis at \( (0, -1)\).

The y-intercept is the starting point for graphing a line. By beginning at the y-intercept and using the slope, you can determine other points on the line.

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Most popular questions from this chapter

Find the equation of a line containing the given points. Write the equation in slope-intercept form. (-2,-3) and (-5,-6)

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Find an equation of a line parallel to the given line and contains the given point. Write the equation in slope-intercept form. line \(y+7=0,\) point(1,-1)

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