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Chapter 3: Problem 7

Determine the slope and \(y\) -intercept. $$ y=3 $$

Short Answer

Expert verified
Slope is 0; y-intercept is (0, 3).

Step by step solution

01

Identify the Form of the Equation

Notice that the equation given is of the form \(y = c\), which is a horizontal line. The equation \(y = c\) represents a line where all the y-values are constant and equal to \(c\).
02

Determine the Slope

The slope of a horizontal line is 0 because there is no change in the y-value as the x-value changes. No matter what x-value is chosen, the y-value remains constant. Thus, the slope \(m = 0\).
03

Determine the Y-Intercept

The y-intercept is the point where the line crosses the y-axis. For the equation \(y = 3\), the line crosses the y-axis at \(y = 3\). Thus, the y-intercept is \((0, 3)\).

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Key Concepts

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

Horizontal Line Equation
A horizontal line is a special type of linear graph where all y-values remain the same, regardless of the x-values. This means that no matter where you are on the x-axis, the y-coordinate is constant. The general formula for a horizontal line is given by the equation \(y = c\). Here, \(c\) is a constant and represents the specific y-value where the line sits. This is why a horizontal line appears flat, running parallel to the x-axis.
  • Each point on the line has the same y-coordinate.
  • Horizontal lines show no upward or downward slope.
  • The line is visually a straight line that does not rise or fall.
An example would be \(y = 3\), which is a horizontal line where every point on the line has a y-coordinate of 3.
Slope Calculation
The slope is a measure of how steep a line is. It is calculated as the "rise over run," or the change in y for a unit change in x. For most lines, you would use the formula \(m = \frac{\Delta y}{\Delta x}\). However, for horizontal lines, things are a bit different.In the case of a horizontal line:
  • There is no vertical change (\(\Delta y= 0\)) as you move along the line.
  • The formula simplifies as the numerator becomes zero, \(m = 0\).
This means the slope of a horizontal line is always 0. It reflects the fact that the line does not rise or fall as the x-coordinate changes.
Y-Intercept Determination
The y-intercept is the point where a line crosses the y-axis. For horizontal lines, determining the y-intercept is straightforward because the line is already expressed in the form \(y = c\), where \(c\) is the constant y-value of all points on the line.For the equation \(y = 3\):
  • The line crosses the y-axis at \(y = c\).
  • Substitute \(c\) to find the intercept: \((0, 3)\).
Thus, the y-intercept of the line \(y = 3\) is \((0, 3)\), indicating where the line touches the y-axis. This highlights that for any horizontal line, the y-intercept is simply \((0, c)\).

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Most popular questions from this chapter

Come up with your own correspondence between real-world sets. Explain why it does or does not represent a function.

Find at least five ordered pair solutions and graph. $$ x=-3 $$

Is a horizontal line a function? What are the domain and range of a horizontal line?

Calculate the midpoint between the given points. (6,-3) and (-8,-11)

Graph using the slope and \(y\) -intercept. $$ y=0 $$
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