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Chapter 2: Problem 14

Exer. \(13-14\) : Sketch the graph of \(y=m x\) for the given values of \(m\) $$m=5,-3, \frac{1}{2},-\frac{1}{3}$$

Short Answer

Expert verified
Graph four lines with different slopes for each \( m \), highlighting their direction and steepness.

Step by step solution

01

Understanding the Linear Equation

The given equation is of the form \( y = mx \), where \( m \) is the slope of the line. The slope \( m \) determines the steepness and the direction of the line on the graph. A positive \( m \) means the line rises from left to right, while a negative \( m \) means the line falls from left to right.
02

Sketching the Graph for \( m = 5 \)

For \( m = 5 \), the equation becomes \( y = 5x \). This means that for every 1 unit increase in \( x \), \( y \) increases by 5 units. To sketch, plot points like (0,0), (1,5), (-1,-5), and draw the line through these points.
03

Sketching the Graph for \( m = -3 \)

For \( m = -3 \), the equation becomes \( y = -3x \). This means that for every 1 unit increase in \( x \), \( y \) decreases by 3 units. Plot points like (0,0), (1,-3), (-1,3), and draw the line through these points, showing a downward slope.
04

Sketching the Graph for \( m = \frac{1}{2} \)

For \( m = \frac{1}{2} \), the equation becomes \( y = \frac{1}{2}x \). This means that for every 1 unit increase in \( x \), \( y \) increases by 0.5 units. Plot points like (0,0), (2,1), (-2,-1), and draw the line through these points, indicating a gentle upward slope.
05

Sketching the Graph for \( m = -\frac{1}{3} \)

For \( m = -\frac{1}{3} \), the equation becomes \( y = -\frac{1}{3}x \). This means that for every 1 unit increase in \( x \), \( y \) decreases by about 0.33 units. Plot points like (0,0), (3,-1), (-3,1), and draw the line, showing a gentle downward slope.

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Key Concepts

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

Slope
The slope of a line is a number that tells us how steep the line is and in which direction it tilts. In the equation of a line, often written as \( y = mx + b \), the \( m \) represents the slope. In the cases we review here, \( b = 0 \), making the equation \( y = mx \). Understanding slope is crucial in mathematics because it gives us a detailed look at the behavior of a line.
  • When \( m \) is positive, the line ascends from left to right.
  • When \( m \) is negative, the line descends from left to right.
  • A larger absolute value of \( m \) means a steeper line. For instance, a slope of 5 is steeper than a slope of 1.
  • A slope of zero would create a horizontal line.
The slope is essentially the amount that \( y \) changes for a one-unit change in \( x \). To interpret this, just imagine skiing: a positive slope feels like going uphill, and a negative slope feels like going downhill.
Graph Sketching
Graph sketching is a visual way to represent equations on a coordinate plane. To sketch a graph of a linear equation, we need to identify key points that the line will pass through.With linear equations such as \( y = mx \), plot the y-intercept, which is the point where the line crosses the y-axis. Because in \( y = mx \), the y-intercept is 0, our graph passes through the origin (0,0). After identifying the y-intercept, consider the slope to determine other points. Here鈥檚 a simple approach to sketching:
  • Start at the origin (0,0).
  • Use the slope, \( m \), to determine the rise-over-run to the next point.
  • Plot a few points by applying the slope to consistent intervals.
  • Draw a straight line through the points, extending it in both directions.
This method allows a basic graph sketch, showcasing the line's general direction and intersection points.
Positive and Negative Slope
The slope of a line indicates its direction, and here is how it influences the graph:A **positive slope** implies that the line rises as it moves from left to right. Think of a climbing trail: starting low and going higher.
  • Examples include equations such as \( y = 5x \) or \( y = \frac{1}{2}x \).
  • The point (1,5) for \( y = 5x \) shows a steep rise as compared to (2,1) for \( y = \frac{1}{2}x \).
A **negative slope** suggests that the line falls as you go from left to right. Imagine a slide: starting high and ending low.
  • Examples include \( y = -3x \) or \( y = -\frac{1}{3}x \).
  • For instance, the fall from (1,-3) in \( y = -3x \) is sharper than from (3,-1) in \( y = -\frac{1}{3}x \).
Negative and positive slopes are essential in understanding how a line behaves in relation to the x-axis.
Coordinate Plane
The coordinate plane is a two-dimensional space formed by two perpendicular lines called axes. The horizontal line is the x-axis, and the vertical one is the y-axis.Understanding the coordinate plane is vital to effectively sketch graphs:
  • Each point in this space is represented as an ordered pair \((x, y)\).
  • The intersection of the axes is the origin, noted as \((0, 0)\).
  • Points to the right of the origin have a positive x-coordinate, while those to the left have a negative x-coordinate.
  • Above the origin, points have a positive y-coordinate, and below, they have a negative y-coordinate.
The coordinate plane forms the backdrop upon which we plot linear equations and understand the spatial relationship between points and lines. It provides a clear format for determining movement vertically or horizontally when applying the slope of a line.

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