/*! 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 47 Graph each equation. SEE EXAMPLE... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 47

Graph each equation. SEE EXAMPLE 6. (OBJECTIVE 5 ) $$y=-5$$

Short Answer

Expert verified
The graph of the equation \(y = -5\) is a horizontal line parallel to the x-axis passing through the point where y coordinate is -5.

Step by step solution

01

Understand the equation

The equation is of the form \(y = c\) where \(c\) is a constant. This means the line will be parallel to the x-axis.
02

Determine the y-coordinate

In this case, the equation is \(y = -5\). This means that the y-coordinate for every point on the line will always be -5, no matter what x is.
03

Plot the line on the graph

Plot a horizontal line that goes through the y-coordinate at -5 on a graph. This line represents all the solutions to the equation. Every point on this line has y-coordinate -5, which satisfies the equation.

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Key Concepts

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

Horizontal Lines
Horizontal lines are among the simplest types of lines on a graph. They are characterized by having the same y-coordinate for all points on the line. This makes them parallel to the x-axis.
To understand why this happens, let's look at an example equation: \(y = -5\). Here, regardless of the x-value, the y-value is constantly -5.
  • These lines do not rise or fall as you move along the graph.
  • They all have a slope of zero, meaning they run perfectly flat across the y-value specified.
Horizontal lines may seem straightforward, but they are crucial in understanding equations and functions that do not change in one of their dimensions.
Coordinate System
The coordinate system is the grid upon which we plot points, lines, and shapes. It consists of two axes: the horizontal x-axis and the vertical y-axis. Together, these axes divide the plane into four quadrants.
Each point on this grid is identified using an ordered pair \((x, y)\):
  • The first number, \(x\), tells us how far to move in the horizontal direction.
  • The second number, \(y\), tells us how far to move in the vertical direction.
Understanding the coordinate system is foundational to graphing equations, as it allows us to accurately represent algebraic equations visually. When graphing horizontal lines like \(y = -5\), every point in that line will have a 'y' value continually the same, showing the idea of it being a constant horizontal line.
Constant Functions
Constant functions are very particular functions where the output value (or y-value) is always the same, no matter what input value (or x-value) you use.
For instance, the function \(y = -5\) is constant because it merely states that the y-value is -5 for all x-values.
This type of function:
  • Is represented by a horizontal line on a graph.
  • Shows that no two points have a change in the y-value.
  • In practical terms, illustrates situations that are steady and unchanging in their measured aspect.
This concept is essential as it outlines a fundamental characteristic of certain equations. Understanding constant functions aids in comprehending how situations or phenomena where one variable does not affect the outcome are represented graphically.

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Most popular questions from this chapter

Express each joint variation as an equation. Then find the requested value. Assume that all variables represent positive numbers. SEE EXAMPLE 3. (OBJECTIVE 3) A varies jointly with \(x\) and \(y .\) If \(A=18\) when \(x=3\) and \(y=3\), find \(A\) when \(x=7\) and \(y=9\).

Set up a variation equation and solve for the requested value. The distance that a car can travel without refueling varies directly with the number of gallons of gasoline in the tank. If a car can go 360 miles on 12 gallons of gas, how far can it go on 7 gallons?

Express each sentence as a formula. The distance \(d\) traveled varies jointly with the speed \(s\) and the time \(t\)

Express each variation as an equation. Then find the requested value. Assume that all variables represent positive numbers. \(y\) varies jointly with \(x\) and \(w^{2} .\) If \(y=12\) when \(x=2\) and \(w=3,\) find \(y\) when \(x=3\) and \(w=2\).

Express each inverse variation as an equation. Then find the requested value. Assume that all variables represent positive numbers. SEE EXAMPLE 2. (OBJECTIVE 2) \(r\) varies inversely with \(s .\) If \(r=40\) when \(s=10,\) find \(r\) when \(s=15\).
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