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Chapter 2: Problem 50

Graph each equation in a rectangular coordinate system. $$y=4$$

Short Answer

Expert verified
The graph of the equation \(y=4\) is a horizontal line that cuts across the y-axis at \(y=4\).

Step by step solution

01

Understand the equation

The equation is \(y=4\). It does not involve any x-components hence it is a straight horizontal line which crosses the y-axis at \(y=4\).
02

Draw a Cartesian Plane

Draw a two-dimensional Cartesian Plane. Mark the values on the y-axis, make sure to include the number 4.
03

Plot the Line

On the plane, draw a horizontal line on the point where \(y=4\). The line extends indefinitely in either directions.
04

Final Verification

Make sure the line represents the equation \(y=4\). Any point on this line will have the y-coordinate 4.

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Key Concepts

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

Rectangular Coordinate System
The rectangular coordinate system, also known as the Cartesian coordinate system, is a mathematical approach to visually represent algebraic equations. It comprises two number lines: the x-axis (horizontal) and the y-axis (vertical). These axes intersect at a point known as the origin, denoted by the coordinates (0,0).

Using this system:
  • Points are expressed as ordered pairs \(x, y\).
  • The system allows us to graphically depict relationships between variables.
  • The axes divide the plane into four quadrants.
When graphing, each point on the plane corresponds to one solution of the equation. The rectangular coordinate system provides a straightforward way to visualize how different equations translate to lines, curves, and more.
Horizontal Lines
Horizontal lines in the Cartesian plane are unique because they have a constant y-value across all x-values. For instance, the equation \(y=4\) defines a horizontal line that crosses the y-axis at \(y=4\) without any slope.

Characteristics of horizontal lines:
  • They are parallel to the x-axis.
  • Every point along the line shares the same y-coordinate.
  • The slope is zero, indicating no vertical change.
This makes understanding and plotting these lines intuitive. Simply locate the y-coordinate line where the equation equalizes and draw the line parallel to the x-axis.
Cartesian Plane
The Cartesian plane is a two-dimensional surface where mathematical equations are plotted. It consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). The point where these axes intersect is called the origin.
  • Quadrants are formed by the intersections, labeled I to IV in a counter-clockwise fashion.
  • Positive x-values lie to the right of the origin, while negative x-values are to the left.
  • Positive y-values are above the origin, while negative are below.
This setup allows for a precise graphical representation of equations. By plotting points and lines on the Cartesian plane, we gain insights into the behavior of functions and variable interactions.

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Most popular questions from this chapter

In Exercises \(105-108,\) you will be developing functions that model given conditions. A company that manufactures bicycles has a fixed cost of \(\$ 100,000 .\) It costs \(\$ 100\) to produce each bicycle. The total cost for the company is the sum of its fixed cost and variable costs. Write the total cost, \(C,\) as a function of the number of bicycles produced, \(x .\) Then find and interpret \(C(90)\)

What is a circle? Without using variables, describe how the definition of a circle can be used to obtain a form of its equation.

give the center and radius of the circle described by the equation and graph each equation. Use the graph to identify the relation's domain and range. $$ (x-3)^{2}+(y-1)^{2}=36 $$

a. Graph the functions \(f(x)=x^{n}\) for \(n=2,4,\) and 6 in a \([-2,2,1]\) by \([-1,3,1]\) viewing rectangle. b. Graph the functions \(f(x)=x^{n}\) for \(n=1,3,\) and 5 in a \([-2,2,1]\) by \([-2,2,1]\) viewing rectangle. c. If \(n\) is positive and even, where is the graph of \(f(x)=x^{n}\) increasing and where is it decreasing? d. If \(n\) is positive and odd, what can you conclude about the graph of \(f(x)=x^{n}\) in terms of increasing or decreasing behavior? e. Graph all six functions in a \([-1,3,1]\) by \([-1,3,1]\) viewing rectangle. What do you observe about the graphs in terms of how flat or how steep they are?

give the center and radius of the circle described by the equation and graph each equation. Use the graph to identify the relation's domain and range. $$ x^{2}+y^{2}=49 $$
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