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Chapter 2: Problem 64

Find the slope of each line, and sketch its graph. \(y=-4\)

Short Answer

Expert verified
The slope of the line is 0.

Step by step solution

01

Identify the Equation Type

The given equation is of the form \(y = c\) where \(c\) is a constant. In this case, the equation is \(y = -4\).
02

Understand the Characteristics

For an equation of the form \(y = c\), the line is horizontal. This means that the value of \(y\) does not change no matter what the value of \(x\) is.
03

Determine the Slope

Since the line is horizontal, the slope (m) is 0. In general, the slope of a horizontal line is always 0.
04

Sketch the Graph

To sketch the graph, draw a horizontal line that passes through \(y = -4\). This line should be parallel to the x-axis and intersect the y-axis at \(y = -4\).

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Key Concepts

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

Finding Slope
The slope of a line indicates its steepness and direction. It鈥檚 defined as the change in the vertical direction (rise) divided by the change in the horizontal direction (run).

The formula to calculate the slope, given two points \((x_1, y_1)\) and \((x_2, y_2)\), is:
\[m = \frac{{y_2 - y_1}}{{x_2 - x_1}}\]
Understanding this helps you determine how much a line goes up or down as it moves from left to right.
In our exercise, because the equation is \(y = -4\), it鈥檚 a horizontal line where the rise is zero. Hence, the slope \(m\) is 0.
Horizontal Line
A horizontal line is one where all points on the line have the same \(y\)-value but different \(x\)-values.

In the equation \(y = -4\), no matter what \(x\) value you choose, \(y\) always remains \(-4\).
This is because the line is completely flat without any tilt.
Such a line extends left and right infinitely. This makes it easy to determine that horizontal lines have a slope of 0,
since there is no rise (vertical change) as you move along the line.
Graphing Linear Equations
Graphing linear equations is about plotting points and drawing lines to represent solutions to the equation. Here鈥檚 how to do it for our given horizontal line \(y = -4\):
  • Identify points where \(y = -4\). This can be \((0, -4)\), \((2, -4)\), or any other \(x\)-value.
  • Plot these points on the graph.
  • Draw a straight, horizontal line through all these points.
    This line should be parallel to the x-axis, running left and right endlessly at \(y = -4\).
Practicing this helps you visually understand how different types of linear equations behave on a graph.

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Most popular questions from this chapter

Find the slope of each line in three ways by doing the following. (a) Give any two points that lie on the line, and use them to determine the slope. (b) Solve the equation for \(y\), and identify the slope from the equation. (c) For the form \(A x+B y=C,\) calculate \(-\frac{A}{B} .\) 3 x-y=-6

Determine whether each pair of lines is parallel, perpendicular, or neither. \(x+4 y=7\) and \(4 x-y=3\)

The upper deck at Guaranteed Rate Field in Chicago has produced, among other complaints, displeasure with its steepness. It is \(160 \mathrm{ft}\) from home plate to the front of the upper deck and \(250 \mathrm{ft}\) from home plate to the back. The top of the upper deck is \(63 \mathrm{ft}\) above the bottom. What is its slope?

Graph the union of each pair of inequalities. $$ x-2>y \text { or } x<1 $$

Graph the intersection of each pair of inequalities. $$ x+y \leq 1 \quad \text { and } \quad x \geq 1 $$
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