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Chapter 1: Problem 50

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

Short Answer

Expert verified
The graph of the equation \(y = 4\) is a straight line parallel to the x-axis, passing through the point (0,4).

Step by step solution

01

Identify the equation

Given equation is \(y = 4\), which clearly indicates that y-coordinate is 4 for every value of x. In other words, regardless of x's position, y will always be the same, at 4.
02

Plot the line

On the coordinate plane, mark a point anywhere on y-coordinate 4. Because every point on this line has a y-coordinate of 4, we can choose an arbitrary 'x' value. Now, draw a straight line that passes through this point parallel to the x-axis. The line represents the solution set for the equation \(y = 4\).

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Key Concepts

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

Graph Equation
Graphing an equation on a rectangular coordinate system involves placing points on a plane to visually represent a mathematical relationship. In our case, the equation is simple: \( y = 4 \). This states that no matter the value of \( x \), \( y \) remains fixed at 4. This special type of equation is called a horizontal line equation. You're essentially illustrating that every point on this line shares the same \( y \)-coordinate of 4.
  • Understand that the entire equation is already solved because it doesn't depend on \( x \).
  • Visual representation: You're turning an algebraic equation into accessible visual information.
This makes it a straightforward equation to graph since it indicates a constant relationship.
y-coordinate
The \( y \)-coordinate determines a point's vertical position on a graph. In the equation \( y = 4 \), this value is consistent across the board. That's because the \( y \)-coordinate tells us how high or low we are on the plane. In every point that satisfies \( y = 4 \), the height or distance from the x-axis is the same.
  • For all points on this line, the \( y \)-coordinate remains unaffected by changes in \( x \).
  • This simplicity defines the line's horizontal nature on the graph.
Such constancy helps in quickly understanding where the line will lie when visualizing.
x-axis
The \( x \)-axis is the horizontal line on a graph where the \( y \)-coordinate is zero. However, in our example equation \( y = 4 \), the \( x \)-axis plays a different role. It serves purely as a reference point since our line will run parallel to it.
  • The line is not dependent on \( x \) and doesn鈥檛 intersect with the \( x \)-axis, showing its parallel relationship.
  • Any \( x \)-value is acceptable for our graph, illustrating that \( x \) is unrestricted.
This means no adjustments to the \( x \)-axis are needed because the line's placement is wholly based on \( y = 4 \).

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Most popular questions from this chapter

Complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. $$x^{2}+y^{2}+12 x-6 y-4=0$$

If you are given a function's graph, how do you determine if the function is even, odd, or neither?

Suppose that \(h(x)=\frac{f(x)}{g(x)} .\) The function \(f\) can be even,odd, or neither. The same is true for the function \(g .\) a. Under what conditions is \(h\) definitely an even function? b. Under what conditions is \(h \quad\) definitely an odd function?

In your own words, describe how to find the midpoint of a line segment if its endpoints are known.

A company that sells radios has yearly fixed costs of \(\$ 600,000 .\) It costs the company \(\$ 45\) to produce each radio. Each radio will sell for \(\$ 65 .\) The company's costs and revenue are modeled by the following functions, where \(x\) represents the number of radios produced and sold: \(C(x)=600,000+45 x\) This function models the company's costs. \(R(x)=65 x\) This function models the company's revenue. Find and interpret \((R-C)(20,000),(R-C)(30,000),\) and \((R-C)(40,000)\)
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