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Chapter 8: Problem 29

Graph each equation. $$ y=-\frac{1}{4} x $$

Short Answer

Expert verified
Graph the line through (0,0) with slope \(-1/4\) using points (0,0) and (4,-1).

Step by step solution

01

Identify the Equation Type

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

Determine the Slope and Y-intercept

From the equation \( y = -\frac{1}{4}x \), we can see that the slope \( m = -\frac{1}{4} \) and the y-intercept \( b = 0 \). This means the line crosses the y-axis at the origin (0,0).
03

Plot the Y-intercept

Since the y-intercept is 0, plot the point (0,0) on the graph. This is where the line will start.
04

Use the Slope to Find Another Point

The slope \( -\frac{1}{4} \) means that for every 4 units you move to the right (positive x-direction), you move 1 unit down (negative y-direction). Starting from (0,0), move 4 units to the right to (4,0) and 1 unit down to (4,-1), and plot this point.
05

Draw the Line

Draw a straight line through the points (0,0) and (4,-1). This is the graph of the equation \( y = -\frac{1}{4}x \).

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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 expressed as \( y = mx + b \). It is a very common way to write linear equations because it allows us to quickly identify key features of the line.

  • \( m \) represents the slope of the line.
  • \( b \) stands for the y-intercept, where the line crosses the y-axis.
This form is extremely useful for graphing linear equations. By having the slope and the y-intercept, you can easily draw the line on a coordinate plane. The given equation \( y = -\frac{1}{4}x \) fits perfectly into this form with \( m = -\frac{1}{4} \) and \( b = 0 \).
Slope
The slope of a line is a measure of its steepness and direction. It is calculated as the change in the \( y \)-values divided by the change in the \( x \)-values, often described as "rise over run."
For example, in the equation \( y = -\frac{1}{4}x \), the slope is \( -\frac{1}{4} \). This means:
  • For every 4 units you move to the right along the x-axis, you move 1 unit down on the y-axis.
  • A negative slope indicates that the line is descending from left to right.
The slope tells us how the line moves across the coordinate plane, dictating its angle and direction.
Y-intercept
The y-intercept of a line is the point where it crosses the y-axis, often denoted by \( b \) in the slope-intercept equation. This is the starting point when graphing a line as it provides an anchor on the coordinate grid.
In our example, the equation \( y = -\frac{1}{4}x \) reveals that the y-intercept is \( 0 \). This means the line passes through the origin at the point \( (0,0) \). To graph this equation, you start by placing a point at \( (0,0) \), serving as a pivotal reference to draw the rest of the line.
Coordinate Plane
The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface on which points are plotted and lines are drawn.
It is defined by two perpendicular axes:
  • The horizontal x-axis.
  • The vertical y-axis.
In the context of graphing linear equations, understanding the coordinate plane is essential. Each point on this plane can be represented by a pair of numbers \( (x, y) \).

To graph the line given by \( y = -\frac{1}{4}x \), begin at the origin \( (0,0) \), use the slope to identify another point \( (4,-1) \), and draw a line through these points to complete the graph of the equation.

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