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Chapter 2: Problem 71

Graph each equation \(x=-\frac{5}{3}\)

Short Answer

Expert verified
The graph is a vertical line at \(x = -\frac{5}{3}\) on the Cartesian plane.

Step by step solution

01

Identify the Type of Equation

The equation given is \(x = -\frac{5}{3}\). This is a vertical line in a Cartesian coordinate system because it sets \(x\) to a constant value while \(y\) is not constrained by the equation. Such equations are straight vertical lines parallel to the \(y\)-axis.
02

Determine the Line's Location on the Graph

Since the equation is \(x = -\frac{5}{3}\), this means that every point on the line will have an \(x\)-coordinate of \(-\frac{5}{3}\). The line will cross the \(x\)-axis at the point \((-\frac{5}{3}, 0)\).
03

Draw the Vertical Line

To graph the line, draw a straight line parallel to the \(y\)-axis that passes through the point \((-\frac{5}{3}, 0)\). This line will extend indefinitely in the positive and negative \(y\) directions.
04

Verify with Additional Points

To ensure correctness, check a few more points along the line. All points will have an \(x\)-coordinate of \(-\frac{5}{3}\), such as \((-\frac{5}{3}, 1)\) and \((-\frac{5}{3}, -2)\), confirming the line's verticality.

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Key Concepts

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

Vertical Line
In mathematics, a vertical line is a type of straight line that runs parallel to the y-axis of a graph. For an equation given by \(x = c\), where \(c\) is a constant, the line is defined everywhere along the y-direction but stays fixed at \(x = c\). This line does not slope or tilt, hence it is termed "vertical."

Vertical lines have special characteristics:
  • They have an undefined slope. In contrast to horizontal lines which have a slope of 0, vertical lines don't have a clear rise-to-run ratio.
  • They cut through the x-axis at exactly one point, using the \(c\) value from the equation \(x = c\).
Understanding vertical lines helps visualize constraints, such as an unchanging x-coordinate for all points on the line. With this fundamental grasp of vertical lines, interpreting and plotting them on graphs becomes much easier.
Cartesian Coordinate System
The Cartesian coordinate system is a standardized system used in mathematics to locate points on a plane. It uses two perpendicular axes, the x-axis and the y-axis, which intersect at a point called the origin.

Key Features of this system include:
  • The **x-axis** runs horizontally while the **y-axis** runs vertically. Together, they create a plane divided into four quadrants.
  • Each point on the plane can be described using a pair of coordinates \((x, y)\), indicating its distance from the origin along the x and y directions.
  • This system allows for easy visualization of equations and geometric shapes, making it a foundation in graphing.
  • Different types of lines can be easily identified by their equations within this system, such as horizontal, vertical, and sloped lines.
When graphing equations, understanding the Cartesian plane is crucial. It guides how we interpret and draw lines, curves, and other figures accurately.
x-axis
The x-axis is the horizontal component of the Cartesian coordinate system. It plays a critical role in locating points and graphing equations.

What to know about the x-axis:
  • Points on the x-axis have a y-coordinate of 0 (e.g., \((a, 0)\) where \(a\) is any real number).
  • It acts as the reference line for measuring vertical distances (up or down) in the plane.
  • Equations that talk about constant x-values describe vertical lines that are either fully parallel to the y-axis or composed entirely along the x-axis for horizontal lines.
  • It intersects the y-axis at the origin \((0, 0)\).
By understanding the x-axis, you can effectively navigate graphs, assess the position of lines, and determine how changes in x-values affect other points in the coordinate plane.

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Most popular questions from this chapter

Let \(f(x)=-2 x+5 .\) For what value of \(x\) does function \(f\) have the given value? \(f(x)=5\)

Linear relationships between two quantities can be described by an equation or a graph. Which do you think is the more informative? Why?

Find \(h(5)\) and \(h(-2)\). \(h(x)=|x|+2\)

Find \(h(5)\) and \(h(-2)\). \(h(x)=\frac{x^{2}+x-2}{x^{2}-5 x}\)

What does it mean to translate a graph vertically? What does it mean to translate a graph horizontally?
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