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Chapter 13: Problem 2

On the same axes, graph \(y=m x+2\) for \(m=3,-3, \frac{1}{3},\) and \(-\frac{1}{3}\).

Short Answer

Expert verified
The graph contains four lines with slopes of 3, -3, 1/3, and -1/3. All lines intersect the y-axis at y=2. The lines with positive slope rise from left to right, while the lines with negative slope fall from left to right. Lines with a larger absolute value of slope are steeper than those with a smaller absolute value.

Step by step solution

01

Understanding the equation

The equation given is of the form \(y = mx + 2\). This is a line equation where \(m\) is the slope and '2' is the y-intercept.
02

Graph for \(m=3\)

Plot the line with slope 3 and y-intercept 2. It means for every unit increase in x, y increases by 3 units.
03

Graph for \(m=-3\)

Similarly, plot the line with slope -3 and y-intercept 2. This means for every unit increase in x, y decreases by 3 units.
04

Graph for \(m=1/3\)

Now, plot the line with slope 1/3 and y-intercept 2. It indicates for each unit increase in x, y increases by 1/3rd unit.
05

Graph for \(m=-1/3\)

Lastly, plot the line with slope -1/3 and y-intercept 2. It shows that for each unit increase in x, y decreases by 1/3rd unit.
06

Finalize the Graph

Now, all the required lines for the given slopes are plotted on the same graph with y-intercept 2. Observe how the slope affects the steepness and direction of the line.

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Key Concepts

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

Graphing Lines
When you hear about graphing lines, it means you're plotting equations on a graph to visually understand their behavior. Each line represents an equation, and on a graph, it tells you how the values of 'y' change as the values of 'x' change. This is crucial because visualizing it can make complex concepts become a lot simpler. In this exercise, you'll graph multiple lines on the same axes, allowing you to compare how different slopes affect the line's appearance.

You'll need an XY grid to begin plotting these lines.
  • Each point on the graph represents an (x, y) pair that satisfies the equation of the line.
  • Lines on the graph can go up, down, or be horizontal or vertical, depending on the slope.
Creating a graph of a line involves knowing the slope and the y-intercept, which will guide where and how steep your line will be.
Slope-Intercept Form
The slope-intercept form is a common way of writing the equation of a line. It's expressed as \( y = mx + c \), where 'm' is the slope and 'c' is the y-intercept. This form is particularly useful because you can immediately see the slope and the point where the line crosses the y-axis.

In the equation you're working with, \( y = mx + 2 \), the number '2' represents the y-intercept. This means that no matter the slope, all lines will cross the y-axis at (0, 2). This is why the y-intercept is crucial. It gives you a starting point for drawing your line.
  • The slope tells how fast and in what direction you climb or descend along the line.
  • The y-intercept ensures you begin your graph in the correct place.
Using slope-intercept form helps in quickly sketching lines and understanding their characteristics.
Slope of a Line
The slope of a line is a measure of how steep the line is. It is denoted as 'm' in the slope-intercept form of a line, \( y = mx + c \). The slope indicates the ratio of the 'rise' over 'run'. This means for every change in 'x' units, 'y' will change by 'm' units.

Here's how to understand what different slopes mean:
  • If the slope is positive, the line goes upwards as it moves from left to right.
  • If the slope is negative, the line goes downwards across the graph.
  • A larger positive slope means a steeper upward line.
  • A larger negative slope means a steeper downward line.
  • If the slope is zero, the line is horizontal, indicating no change in 'y'.
  • An undefined slope indicates a vertical line.
In the exercise, you'll notice that as the slope changes, the steepness and direction of each line on the graph also change. This helps you visually comprehend the powerful concept of slope.

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