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Chapter 7: Problem 14

State the slope and the \(y\) -intercept of the graph of each equation. $$y=4$$

Short Answer

Expert verified
Slope = 0, Y-intercept = 4.

Step by step solution

01

Identify the Form of the Equation

The equation given is \( y = 4 \). Recognize that this is a horizontal line equation where \( y \) is constant for all values of \( x \). This means the equation is in the format \( y = b \), where \( b \) is the constant \( y \)-value of the line.
02

Determine the Slope

The slope of a horizontal line like \( y = 4 \) is 0. Horizontal lines have no vertical change as \( x \) changes, thus the rise over run (change in \( y \) over change in \( x \)) is 0.
03

Determine the Y-Intercept

For the equation \( y = 4 \), the line crosses the \( y \)-axis at the point where \( y = 4 \). Therefore, the \( y \)-intercept is the point (0, 4).

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Key Concepts

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

Horizontal Lines
In mathematics, horizontal lines are a fascinating aspect of linear equations. These lines are defined by equations where the variable \( y \) remains constant, such as \( y = 4 \). This implies that no matter what \( x \) value you pick, \( y \) will always equal the constant.
This consistent \( y \)-value means that the line you'll plot is completely flat, running parallel to the \( x \)-axis. There is no rise or decline, because the vertical movement is zero.
Horizontal lines are unique in that they express a single numerical value for \( y \) across every \( x \), simplifying the calculation of their slope and intercept.
Linear Equations
Linear equations form the backbone of algebra, and they appear frequently in various mathematical contexts. A typical linear equation takes the form \( y = mx + b \), where \( m \) is the slope, and \( b \) is the \( y \)-intercept.
These equations graph as straight lines on the coordinate plane. The slope \( m \) determines the steepness and direction of the line, while the \( y \)-intercept \( b \) tells you where the line crosses the \( y \)-axis.
  • **Horizontal Lines:** Special cases of linear equations where \( y \) equals a constant. Here, the slope \( m \) is zero since there's no vertical change.
  • **Vertical Lines:** These occur when the \( x \) values are constant and have an undefined slope.

The ability to understand and manipulate these equations is foundational for solving real-world problems and advanced mathematical concepts.
Y-Intercept
The \( y \)-intercept of a line is a critical concept in understanding linear equations. It represents where the line will tragically intersect the \( y \)-axis on a graph. This occurs when \( x = 0 \).
For the given horizontal line equation \( y = 4 \), the \( y \)-intercept is simple to identify. The line crosses the \( y \)-axis exactly at the point \((0, 4)\).
  • **Finding the \( y \)-Intercept:** Set \( x \) to zero in the equation and solve for \( y \).
  • **Graphical Representation:** The \( y \)-intercept gives you a starting point to draw the line.
  • **Importance in Graphing:** This point helps determine the placement of the line on the graph, especially in combination with the slope.

Mastering the concept of the \( y \)-intercept enables students to more easily graph and understand linear equations, laying a groundwork for more complex mathematical analysis.

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Most popular questions from this chapter

Write an equation in slope-intercept form for each table of values. $$\begin{array}{|r|r|r|r|r|} \hline x & -3 & -1 & 1 & 3 \\ \hline y & 7 & 5 & 3 & 1 \\ \hline \end{array}$$

In \(2005,\) a movie actor was to pay 10 percent commission to his management company for work negotiated on his behalf. The commission amount totaled \(\$ 660,000\). How much money did he earn from his movies?

Graph each equation using the slope and \(y\) -intercept. $$-2 x+y=-1$$

Write an equation in slope-intercept form for each line. slope \(=\frac{2}{5}, y\) -intercept \(=0\)

Use the following information. The altitude in feet \(y\) of a hang glider who is slowly landing can be given by \(y=300-50 x,\) where \(x\) represents the time in minutes. Graph the equation using the slope and \(y\) -intercept.
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