/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 44 Find the slope of the line satis... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 44

Find the slope of the line satisfying the given conditions. vertical, through \((-8,5)\)

Short Answer

Expert verified
The slope is undefined.

Step by step solution

01

Understand the Problem

Identify that a vertical line goes up and down and has the same x-coordinate for any point on the line.
02

Identify the Given Point

We are given that the line passes through the point (-8,5). For a vertical line, all points will have the x-coordinate -8.
03

Understand the Slope of a Vertical Line

A vertical line has an undefined slope because its run (change in x) is 0, and division by zero is undefined in mathematics.

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Key Concepts

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

vertical line
A vertical line is a unique type of line in the coordinate plane. It extends up and down, parallel to the y-axis. One of the most important characteristics of a vertical line is that it has the same x-coordinate for any point along the line. For example, if we have a vertical line passing through \(-8, 5\), any other points on this line would also have an x-coordinate of \(-8\), regardless of their y-coordinates. Therefore, all points on this vertical line look like: \((-8, y)\). This property makes vertical lines quite distinct from other types of lines, which usually have different x-coordinates for different points.
undefined slope
The concept of slope is essential in understanding lines in a coordinate plane. Slope measures the steepness of a line and is calculated by the ratio of the change in y-coordinates (rise) to the change in x-coordinates (run). For vertical lines, however, this calculation poses a problem. Since all points on a vertical line share the same x-coordinate, the change in x is zero. Mathematically, when we try to calculate the slope, it corresponds to division by zero, which is undefined. Thus, vertical lines are said to have an undefined slope. This means it's impossible to assign a numerical value to their slope.
coordinates
The coordinate system is a standardized way to describe positions on a plane using pairs of numbers, called coordinates. The first number is the x-coordinate, which specifies the horizontal position, and the second number is the y-coordinate, indicating the vertical position. For instance, in the coordinates \((-8, 5)\), \(-8\) is the x-coordinate and \`5\ is the y-coordinate. Understanding how coordinates work is crucial when working with vertical lines or any other geometrical concepts since it helps pinpoint exact locations in the plane. It's important to remember that changing either coordinate will place a point in a different location on the plane.

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Most popular questions from this chapter

Solve each problem. Emission of Pollutants When a thermal inversion layer is over a city (as happens in Los Angeles), pollutants cannot rise vertically but are trapped below the layer and must disperse horizontally. Assume that a factory smokestack begins emitting a pollutant at 8 A.M. Assume that the pollutant disperses horizontally over a circular area. If \(t\) represents the time, in hours, since the factory began emitting pollutants \((t=0 \text { represents } 8 \text { A.M.), assume that the radius of the circle of pollutants at time } t\) is \(r(t)=2 t\) miles. Let \(\mathscr{A}(r)=\pi r^{2}\) represent the area of a circle of radius \(r\) (a) Find \((\mathscr{A} \circ r)(t)\) (b) Interpret \((\mathscr{A} \circ r)(t)\) (c) What is the area of the circular region covered by the layer at noon?

Given functions \(f\) and \(g,\) find ( \(a\) ) \((f \circ g)(x)\) and its domain, and ( \(b\) ) \((g \circ f)(x)\) and its domain. See Examples 6 and 7 . $$f(x)=x^{3}, \quad g(x)=x^{2}+3 x-1$$

For the pair of functions defined, find \((f+g)(x),(f-g)(x),(f g)(x),\) and \(\left(\frac{f}{g}\right)(x)\) Give the domain of each. $$f(x)=3 x+4, g(x)=2 x-5$$

Decide whether each relation defines \(y\) as a function of \(x\). Give the domain and range. $$y=\sqrt{4 x+1}$$

The tables give some selected ordered pairs for functions \(f\) and \(g\). $$\begin{array}{|c|c|c|c|}\hline x & 3 & 4 & 6 \\\\\hline f(x) & 1 & 3 & 9 \\\\\hline\end{array}$$ $$\begin{array}{|c|c|c|c|c|}\hline x & 2 & 7 & 1 & 9 \\\\\hline g(x) & 3 & 6 & 9 & 12 \\\\\hline\end{array}$$ Find each of the following. $$(g \circ g)(1)$$
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