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Chapter 3: Problem 40

Graph each linear equation. See Examples 4 through \(7 .\) \(y=2 \frac{1}{2}\)

Short Answer

Expert verified
Graph a horizontal line at \(y = 2.5\).

Step by step solution

01

Identify the Equation Type

The given equation is a linear equation in the form of a horizontal line since it is only composed of the constant term, which is the y-coordinate.
02

Interpret the Equation

The equation is written as \(y = 2\frac{1}{2}\). This means for every point on the line, the y-value is \(2\frac{1}{2}\), or equivalently, \(y = 2.5\).
03

Determine Constant Y-value

Since the equation states \(y = 2.5\), every point on the line will have the y-coordinate of 2.5. Thus, the line is horizontal and parallel to the x-axis.
04

Graph the Line

To graph the equation, draw a horizontal line across the graph at the y-coordinate \(2.5\). This line will extend infinitely in both the left and right directions across the graph.

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Key Concepts

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

Horizontal Line
A horizontal line is a straight line that runs from left to right or right to left across the coordinate plane. Unlike other lines that might have a tilted slope, horizontal lines have a slope of zero. This means they do not rise or fall as they move across the graph, and thus remain completely level.
To identify a horizontal line when looking at an equation, you can check if the equation is in the form of \(y = c\), where \(c\) is a constant number. This defines a line where all y-values along the line are equal, which visually appears as a flat line across the graph.
Y-Coordinate
The y-coordinate is one part of an ordered pair \((x, y)\) that specifies the position of a point on the Cartesian plane. It tells you how far up or down a point is from the x-axis, which is the horizontal axis.
  • In the equation of a horizontal line, such as \(y = 2.5\), the y-coordinate is the constant value of 2.5 for every point on the line.
  • This means that no matter the x-value, the position vertically never changes; it remains at 2.5.
  • This is a key characteristic of horizontal lines, where the y-coordinate is consistently the same.
Constant Term
In the context of linear equations, the constant term in an equation like \(y = b\) dictates the horizontal line's position on the graph. This constant term is the y-value where the line consistently sits. It does not rely on the changing x-values as it remains fixed.
  • The constant term provides the specific y-coordinate that all points on the horizontal line share.
  • It's what makes the line horizontal, defining a flat, unvarying bin in y-values across all x.
  • Understanding this helps in quickly and easily drawing horizontal lines on graphs without confusion.
Parallel to X-Axis
When a line is described as parallel to the x-axis, it means the line runs alongside the axis, never crossing it. Horizontally aligned, these lines keep a consistent distance from the x-axis.
  • For example, with the line given by \(y = 2.5\), every point on the line maintains a fixed height of 2.5 units above the x-axis.
  • This ensures that the line stays flat and does not incline or decline in relation to the axis.
  • The concept of parallel lines is useful for understanding why horizontal lines do not vary in slope鈥攖hey remain equidistant from the x-axis at all times.

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Most popular questions from this chapter

Scott Sambracci and Sara Thygeson are planning their wedding. They have calculated that they want the cost of their wedding ceremony \(x\) plus the cost of their reception \(y\) to be no more than \(\$ 5000\). a. Write an inequality describing this relationship. b. Graph this inequality below. C. Why is the grid showing quadrant I only?

Answer each exercise with true or false. Point (-1,5) lies in quadrant IV.

Write an equation in standard form of the line that contains the point (-1,2) and is perpendicular to the line \(y=3 x-1\)

Solve. See the Concept Checks in this section. In your own words, describe how to plot or graph an ordered pair of numbers.

Find an equation of each line described. Write each equation in slope- intercept form when possible. With undefined slope, through \(\left(-\frac{3}{4}, 1\right)\)
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