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Chapter 3: Problem 110

In the following exercises, graph the line of each equation using its slope and \(y\) -intercept. $$ y=2 x-3 $$

Short Answer

Expert verified
Plot y-intercept at (0, -3) and use slope \(2\) to find another point (1, -1). Draw the line through these points.

Step by step solution

01

Identify the Slope-Intercept Form

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

Determine the Slope and Y-Intercept

From the equation \(y = 2x - 3\), you can see that the slope \(m\) is 2, and the y-intercept \(b\) is -3.
03

Plot the Y-Intercept

Start by plotting the y-intercept. This is the point where the line intersects the y-axis. For this equation, the y-intercept is -3, so plot a point at (0, -3) on the graph.
04

Use the Slope to Find Another Point

The slope is 2, which can be written as a fraction \(\frac{2}{1}\). This means for every 1 unit you move to the right on the x-axis, you move 2 units up on the y-axis. Starting from the y-intercept (0, -3), move 1 unit to the right to x = 1, then move 2 units up to y = -1 and plot this point at (1, -1).
05

Draw the Line

Finally, draw a straight line through the two points: (0, -3) and (1, -1). Extend the line in both directions and add arrowheads to indicate that it continues infinitely.

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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 of a linear equation is one of the simplest ways to understand how to graph a straight line.
It is written as: \(y = mx + b\)
In this form:
  • The coefficient \(m\) represents the slope of the line.
  • The constant \(b\) represents the y-intercept, or the point where the line crosses the y-axis.

Using this form makes graphing straightforward because it tells you exactly how the line behaves and where it is positioned on the graph.
For example, in the equation \(y = 2x - 3\), the slope \(m\) is 2 and the y-intercept \(b\) is -3. This provides all the information you need to start graphing.
Slope
The slope of a line indicates how steep the line is. It shows the rate of change between the y-values and x-values.
It is calculated as: \[m = \frac{\Delta y}{\Delta x}\] which means the 'change in y' over the 'change in x'.
  • Positive slope means the line rises from left to right.
  • Negative slope means the line falls from left to right.

In the equation \(y = 2x - 3\), the slope \(m\) is 2. This can be written as a fraction \(\frac{2}{1}\), meaning for every 1 unit you move horizontally (to the right on the x-axis), you move 2 units vertically (up on the y-axis). This consistent rise ensures the line maintains a specific angle.
Y-Intercept
The y-intercept is where the line crosses the y-axis, which means the x-value at this point is always 0.
In the slope-intercept form \(y = mx + b\), the y-intercept is represented by the constant \(b\).
  • This point is easy to plot because you only need the value of \(b\).

From the equation, \(y = 2x - 3\), we see the y-intercept is -3. Therefore, you can immediately plot this point on the graph at (0, -3). This first plotted point serves as a starting point for drawing the line.
Plotting Points
To graph a line, you need at least two points.
The y-intercept provides the first point, and the slope helps determine additional points.
Starting from the y-intercept (-3), you use the slope to find other points.
From \(y = 2x - 3\), we know the slope is \(\frac{2}{1}\). Starting at the y-intercept (0, -3):
  • Move 1 unit right to get the x-coordinate (1).
  • Move 2 units up to get the y-coordinate (-1).

This gives you another point at (1, -1). Plot this second point. Lastly, draw a straight line through these two points and extend the line in both directions with arrowheads, indicating the line goes on forever.

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Most popular questions from this chapter

In the following exercises, use the set of ordered pairs to ( ) determine whether the relation is a function, (b) find the domain of the relation, and \(\odot\) find the range of the relation. \(\\{(-3,-27),(-2,-8),(-1,-1),\) (0,0),(1,1),(2,8),(3,27)\\}

Graph the linear inequality: \(x-y \leq 3\)

In the following exercises, for each relation ( ) find the domain of the relation (B) find the range of the relation. $$ \\{(1,7),(5,3),(7,9),(-2,-3),(-2,8)\\} $$

In the following exercises, evaluate the function: (@) \(f(2)\) (b) \(f(-1)\) (c) \(f(a) .\) $$ f(x)=3 x^{2}+x-2 $$

Graph the linear inequality: \(y \geq-\frac{1}{3} x-2\)
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