/*! 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 61 What is the graph of \(x=0\) ? W... [FREE SOLUTION] | 91Ó°ÊÓ

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What is the graph of \(x=0\) ? What is the graph of \(y=0\) ? Explain your answers.

Short Answer

Expert verified
The graph of \(x=0\) is a vertical line on the y-axis. The graph of \(y=0\) is a horizontal line on the x-axis.

Step by step solution

01

Understanding the Equation

Let's consider the equation \(x = 0\). This equation implies that no matter what value \(y\) takes, \(x\) will always be 0. Therefore, this is a vertical line on the Cartesian plane that crosses the x-axis at 0.
02

Visualizing the Graph for x = 0

Since \(x = 0\) is a vertical line, it runs parallel to the y-axis, crossing the x-axis at 0. Hence, the graph of \(x = 0\) is a vertical line passing through the y-axis, and this line continues infinitely in both the positive and negative directions of the y-axis.
03

Understanding the Equation for y

Now, let's consider the equation \(y = 0\). This equation implies that \(y\) is always zero, regardless of the value of \(x\). Thus, this is a horizontal line that crosses the y-axis at 0.
04

Visualizing the Graph for y = 0

Since \(y = 0\) is a horizontal line, it runs parallel to the x-axis, crossing the y-axis at 0. Therefore, the graph of \(y = 0\) is a horizontal line along the x-axis, extending infinitely in both the positive and negative directions of the x-axis.

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Key Concepts

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

Vertical Line
A vertical line on the Cartesian plane is a line that goes straight up and down. Let's take a moment to break this down:
  • In a vertical line, all the points have the same x-coordinate, while the y-coordinate can be any value. This characteristic makes vertical lines unique as they don't deviate left or right.
  • The equation of a vertical line can be written in the form of \(x = c\), where \(c\) represents a constant. For instance, \(x = 0\) is a perfect example of a vertical line where \(x\) remains at 0, no matter the \(y\) value.
  • Visualizing this is easier if you draw a straight line from top to bottom on your graph, intersecting the x-axis only once at the point \((0, y)\).
Vertical lines are handy in graphing as they clearly demarcate areas on the plane where x is constant.
Horizontal Line
A horizontal line is quite the opposite of a vertical line. It runs from left to right and showcases some fascinating properties:
  • In a horizontal line, every point on the line has the same y-coordinate, while the x-coordinate can vary freely.
  • The equation of a horizontal line is typically presented as \(y = c\), with \(c\) being a constant. Take \(y = 0\) for instance, representing a horizontal line right along the x-axis, precisely at the zero y-coordinate.
  • When you imagine this line, picture a straight, unending line running side to side, never moving up or down from the point \((x, 0)\).
Knowing how to identify and graph horizontal lines helps deepen one's understanding of coordinate systems.
Graphing Equations
Graphing equations is a fundamental skill in mathematics. It bridges abstract equations with visual representation:
  • The process involves plotting points on the Cartesian plane, based on the equation you have.
  • For vertical and horizontal lines, the task is straightforward. After identifying the form of the equation (either \(x = c\) or \(y = c\)), you then draw the corresponding vertical or horizontal line, always ensuring it extends infinitely in its respective direction.
  • This graphical interpretation allows you to not only see the relationship between variables but also solve intersection and optimization problems visually.
Mastering how to graph equations efficiently enables students to translate theoretical concepts into practical interpretations.

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Most popular questions from this chapter

Contains the point \((1,3)\) and is parallel to the line \(x+5 y=9\)

Contains the point \((2,-4)\) and is parallel to the \(y\) axis

$$ \begin{array}{ll|llll} 2 x-y=6 & \mathbf{x} & -2 & 0 & 2 & 4 \\ \hline \mathbf{y} & & & & \end{array} $$

Graph \(|x-y|>2\).

Now let's use a graphing calculator to get a graph of \(\mathrm{C}=\frac{5}{9}(\mathrm{~F}-32)\). By letting \(\mathrm{F}=x\) and \(\mathrm{C}=y\), we obtain Figure 7.15. Pay special attention to the boundaries on \(x\). These values were chosen so that the fraction \(\frac{\text { (Maximum value of } x \text { ) minus (Minimum value of } x \text { ) }}{95}\) would be equal to 1 . The viewing window of the graphing calculator used to produce Figure \(7.15\) is 95 pixels (dots) wide. Therefore, we use 95 as the denominator of the fraction. We chose the boundaries for \(y\) to make sure that the cursor would be visible on the screen when we looked for certain values. \(7.2\) = Linear Inequalities in Two Variables 337 Now let's use the TRACE feature of the graphing calculator to complete the following table. Note that the cursor moves in increments of 1 as we trace along the graph. \begin{tabular}{l|lllllllll} \(\mathbf{F}\) & \(-5\) & 5 & 9 & 11 & 12 & 20 & 30 & 45 & 60 \\ \hline \(\mathbf{C}\) & & & & & & & & & \end{tabular} (This was accomplished by setting the aforementioned fraction equal to 1.) By moving the cursor to each of the F values, we can complete the table as follows. \begin{tabular}{r|rrrrrrrrr} \(\mathbf{F}\) & \(-5\) & 5 & 9 & 11 & 12 & 20 & 30 & 45 & 60 \\ \hline \(\mathbf{C}\) & \(-21\) & \(-15\) & \(-13\) & \(-12\) & \(-11\) & \(-7\) & \(-1\) & 7 & 16 \end{tabular} The \(C\) values are expressed to the nearest degree. Use your calculator and check the values in the table by using the equation \(\mathrm{C}=\frac{5}{9}(\mathrm{~F}-32)\).

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