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Magazine Chapter 3: Problem 13
Exer. 13-14: Sketch the graph of \(y=m x\) for the given values of \(m\). $$ m=3,-2, \frac{2}{3},-\frac{1}{4} $$
Short Answer
Expert verified
Sketch lines through the origin with slopes 3, -2, 2/3, -1/4.
Step by step solution
01
Understand the equation
The equation given is a linear equation: \( y = mx \), where \( m \) represents the slope of the line. Our objective is to sketch the graph for different values of \( m \).
02
Plot the graph for m=3
For \( m = 3 \), the line equation is \( y = 3x \). This means that for every unit increase in \( x \), \( y \) increases by 3 units. The line passes through the origin (0,0) and has a slope of 3. It will be a steep line rising to the right.
03
Plot the graph for m=-2
For \( m = -2 \), the line equation is \( y = -2x \). Here, for every unit increase in \( x \), \( y \) decreases by 2 units. The line passes through the origin and has a slope of -2. It will be a steep line falling to the right.
04
Plot the graph for m=2/3
For \( m = \frac{2}{3} \), the line equation is \( y = \frac{2}{3}x \). This means that for every increase of 3 units in \( x \), \( y \) increases by 2 units. The line passes through the origin and is less steep compared to previous cases.
05
Plot the graph for m=-1/4
For \( m = -\frac{1}{4} \), the line equation is \( y = -\frac{1}{4}x \). For every increase of 4 units in \( x \), \( y \) decreases by 1 unit. The line passes through the origin and has a mild slope, with a gentle decline to the right.
06
Compare and understand the slopes
Notice that positive values of \( m \) result in lines that rise as \( x \) increases, whereas negative values of \( m \) result in lines that fall. The absolute value of \( m \) determines the steepness of the line.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Slope of a Line
In the context of linear equations, the slope is a crucial concept. The slope, represented by the letter m in the equation \( y = mx \), indicates how steep a line is. Essentially, it tells us how much \( y \) changes for each unit change in \( x \).
When \( m \) is positive, the line rises as you move from left to right. Conversely, if \( m \) is negative, the line falls as you move in the same direction. The slope is expressed as a ratio, like \( \frac{rise}{run} \), which compares the vertical change (rise) to the horizontal change (run).
When \( m \) is positive, the line rises as you move from left to right. Conversely, if \( m \) is negative, the line falls as you move in the same direction. The slope is expressed as a ratio, like \( \frac{rise}{run} \), which compares the vertical change (rise) to the horizontal change (run).
- A steeper line means a larger absolute value of the slope.
- A slope of zero results in a horizontal line.
- An undefined slope occurs for vertical lines.
Graph Plotting
Graph plotting involves drawing a line on the graph based on a given equation. In this exercise, the equation is \( y = mx \), a linear equation that passes through the origin (0,0). To plot such a graph, you need to determine how the line behaves based on the slope \( m \).
Let's go through the steps:
Let's go through the steps:
- Find the origin (0,0) on the graph; this is the point where all lines from \( y = mx \) equations pass through.
- Use the slope \( m \) to find another point. For example, with \( m = 3 \), starting from the origin, move up 3 units and right 1 unit to find another point on the graph.
- Draw the line using these two points.
Linear Graph Characteristics
Understanding the characteristics of a linear graph is essential for analyzing and interpreting equations. These graphs have distinct features worth noting:
- Every line in the form of \( y = mx \) is straight and passes through the origin. This makes them easy to identify.
- The slope \( m \) dictates the tilt and direction of the line. High positive slopes make the line steep and rising, while high negative slopes make it steep and falling.
- A smaller absolute value of \( m \) results in a flatter line, such as the line with \( m = -\frac{1}{4} \), which declines gently.
