Problem 79 - Algebra for College Students

Question

Graph \(y=1 x, y=2 x, y=3 x,\) and \(y=4 x\) together in the standard viewing window. These equations are all of the form \(y=m x .\) What effect does increasing \(m\) have on the graph of the equation? What are the slopes of these four lines?

Step-by-Step Solution

Answer

Increasing m makes the line steeper. The slopes are 1, 2, 3, and 4.

Step 1: Identify the Form of the Equations

All given equations are in the form of a linear equation: y = mx where m is the slope and x is the variable.

Step 2: List the Given Equations

The equations provided are: y = 1x y = 2x y = 3x y = 4x

Step 3: Determine the Slopes

The slope (m) for each equation is: y = 1x : slope (m) = 1 y = 2x : slope (m) = 2 y = 3x : slope (m) = 3 y = 4x : slope (m) = 4

Step 4: Graph the Equations

Plot each of these lines on a standard graph: 1. For y = 1x, start at the origin (0,0) and use the slope 1 to plot points (such as (1,1), (2,2)).2. For y = 2x, start at the origin and use the slope 2 to plot points (such as (1,2), (2,4)).3. For y = 3x, start at the origin and use the slope 3 to plot points (such as (1,3), (2,6)).4. For y = 4x, start at the origin and use the slope 4 to plot points (such as (1,4), (2,8)).

Step 5: Describe the Effect of Increasing the Slope

Increasing the value of m (the slope) makes the line steeper. As m increases from 1 to 4, each subsequent line becomes steeper than the previous one.

Key Concepts

Slope

The slope of a line, denoted by the letter \(m\), is a measure of how steep the line is. It indicates the rate at which \(y\) changes with respect to \(x\). In the equation of the form \(y=mx\), the slope \(m\) can take various values:

• If \(meq0\), the line is inclined and crosses the axis.
• If \(m>0\), the line ascends from left to right.
• If \(m<0\), the line descends from left to right.
• If \(m=0\), the line is horizontal.

For example, in the exercises provided:
\(y=1x\) has a slope of 1, \(y=2x\) has a slope of 2, \(y=3x\) has a slope of 3, and \(y=4x\) has a slope of 4. The larger the value of \(m\) (the slope), the steeper the line.

Graphing Linear Equations

Graphing a linear equation means plotting the line represented by the equation on a coordinate plane. For equations of the form \(y=mx\), the slope \(m\) dictates how the line moves. The steps to graph a linear equation are simple:
1. Start at the origin point (0,0), since all lines in the form \(y=mx\) pass through this point.
2. Use the slope \(m\) to determine the rise (change in \(y\)) over the run (change in \(x\)). For instance, if \(m=2\), move up 2 units for every 1 unit you move to the right.
3. Plot another point based on this slope.
4. Draw a straight line through the points.
For example, in the exercise:

• For \(y=1x\), the points (1,1), (2,2) help graph the line.
• For \(y=2x\), the points (1,2), (2,4) help graph a steeper line.
• For \(y=3x\), use (1,3), (2,6).
• For \(y=4x\), use (1,4), (2,8).

As the slope increases, the line gets steeper.

Effects of Slope on Graph

The slope of a line directly affects its steepness. Here's a breakdown of how changing the slope influences the graph:

• **Shallower Line**: If the slope is low (e.g., \(y=1x\)), the line rises slowly as \(x\) increases. It looks more like a gentle incline.
• **Steeper Line**: As the slope increases (e.g., \(y=4x\)), the line rises more sharply. It appears more vertical.

Observation from Exercise:
The graphs of \(y=1x\), \(y=2x\), \(y=3x\), and \(y=4x\) show a clear pattern: as the slope \(m\) increases, the lines get progressively steeper. You can visualize this effect by graphing each equation on the same set of axes and seeing how the lines diverge, growing steeper and steeper. This emphasizes how the value of \(m\) (slope) impacts the angle of the line relative to the horizontal.