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Chapter 2: Problem 54

Find the slope and y-intercept of the line, and draw its graph. $$ 4 y+8=0 $$

Short Answer

Expert verified
The slope is 0, and the y-intercept is -2.

Step by step solution

01

Rewrite the Equation

Let's start by rewriting the given equation: \[ 4y + 8 = 0 \] We want to solve for \( y \) to express the equation in slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
02

Isolate y

Subtract 8 from both sides to start isolating \( y \): \[ 4y = -8 \] Next, divide both sides by 4: \[ y = -2 \] This simplified form shows that this is a horizontal line.
03

Identify the Slope and Y-Intercept

In the equation \( y = -2 \), this corresponds to \( y = 0x - 2 \). Here, the slope \( m \) is 0, indicating no incline (horizontal line), and the y-intercept \( b \) is -2.
04

Graph the Line

To graph the line, draw a horizontal line across the y-axis at \( y = -2 \). This line will run parallel to the x-axis and never touch it.

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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 indicates its steepness and the direction it moves. It's represented by the letter \( m \) in the linear equation \( y = mx + b \). The slope is calculated by taking the change in \( y \) over the change in \( x \) (often referred to as "rise over run").
  • If the slope is positive, the line ascends as it moves from left to right.
  • If the slope is negative, the line descends as it moves from left to right.
  • If the slope is zero, the line is perfectly horizontal, indicating no rise or drop.
  • An undefined slope corresponds to a vertical line.
In our exercise, the equation \( y = -2 \) reveals that the slope \( m \) is 0, as the line does not incline or decline. This defines it as a horizontal line.
Y-Intercept
The y-intercept of a line is the point where it crosses the y-axis. It's the value of \( y \) when \( x = 0 \) and is represented by \( b \) in the equation \( y = mx + b \). To find it, simply look at the constant term in the equation once it's in slope-intercept form.
In the equation \( y = -2 \), the y-intercept is \( -2 \). This means the line crosses the y-axis at the point \( (0, -2) \). Understanding the y-intercept helps us determine where the line starts on the y-axis before it extends in its direction.
Graphing Lines
Graphing a line involves plotting points on a graph to visually represent the equation of that line. You often start by identifying the y-intercept and then using the slope to determine the direction and spacing of the line.
  • Begin at the y-intercept from the equation \( y = mx + b \).
  • Use the slope \( m \) to find other points. For example, with \( m = 2 \), you rise 2 units and run 1 unit.
  • Connect these points with a straight line extending across your graph.
For the exercise where \( y = -2 \), plotting this involves drawing a horizontal line through the point \( (0, -2) \). It doesn't slope up or down but stretches infinitely in both directions along the y-axis.
Horizontal Line
A horizontal line has a slope of 0, meaning it doesn't rise or fall. Such lines are represented by equations of the form \( y = c \), where \( c \) is a constant.
Horizontal lines are unique because they run parallel to the x-axis and maintain a constant y-value for all values of \( x \). No matter where you are on the line, \( y \) remains the same.
In our problem \( y = -2 \), the line is horizontal with the y-value being -2 everywhere along the axis. These lines emphasize stability, and their graph will appear as a flat line across the grid, making them straightforward to plot and understand.

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