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Chapter 1: Problem 26

Find the slope and \(y\) -intercept (if possible) of the equation of the line. Sketch the line. $$ y+4=0 $$

Short Answer

Expert verified
The slope of the line is 0 and the y-intercept is -4. The line is a horizontal line intersecting the y-axis at -4.

Step by step solution

01

Rearrange the Given Equation

Rearrange the given equation y + 4 = 0 to the slope-intercept form y = mx + b. In this case, the equation simplifies to y = -4.
02

Identify the Slope and Y-Intercept

Now that the equation is in slope-intercept form, identify the slope (m) and the y-intercept (b). In the simplified equation y = -4, no x term is present hence the slope m = 0. The y-intercept b, which is the constant term, is -4.
03

Sketch the Line

The given line is horizontal since its slope (m) is 0, and it intersects the y-axis at -4. Draw a horizontal line across the graph that passes through the point (0, -4).

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Key Concepts

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

Slope-Intercept Form
The slope-intercept form of a linear equation is a fundamental way of expressing linear relationships, commonly written as \( y = mx + b \). This form is incredibly useful because it clearly reveals two key characteristics of a line:
  • The slope, represented by \( m \), which describes the steepness or tilt of the line.
  • The y-intercept, represented by \( b \), which indicates where the line crosses the y-axis.
In the exercise you encountered, the equation was \( y + 4 = 0 \). By rearranging this equation to the form \( y = mx + b \), it becomes \( y = -4 \). Notice how there is no \( x \) term, meaning it has a slope \( m = 0 \). The equation in slope-intercept form allows for straightforward identification of its slope and y-intercept, essential for sketching or understanding the behavior of the line.
Slope
The slope is a measure of how a line rises or falls as it moves along the x-axis. It is the "m" in the slope-intercept form equation \( y = mx + b \). The slope can be understood through the following points:
  • A positive slope means the line rises as it moves from left to right.
  • A negative slope means the line falls as it moves from left to right.
  • If the slope is zero, the line is horizontal, indicating no rise or fall.
In the given exercise, the equation \( y = -4 \) does not have an \( x \) term. This absence indicates a slope of \( m = 0 \). A slope of zero signifies a perfectly horizontal line, consistently maintaining its y-value as \( x \) changes. The geometric implication of a zero slope is critical, as it shows that any change along the x-axis does not affect the y-value.
Y-Intercept
The y-intercept of a line is the point where it crosses the y-axis, which is found in the slope-intercept form \( y = mx + b \) as the constant \( b \). This value can be directly read from the equation when it is in this form:
  • The y-intercept \( b \) represents the value of \( y \) when \( x = 0 \).
  • It provides a starting point for graphing the line on a coordinate plane.
In this exercise, the equation \( y = -4 \) clearly indicates a y-intercept at \( b = -4 \). This means that the line crosses the y-axis at the point \( (0, -4) \). Since the slope is zero, the y-intercept helps describe the entire line 鈥� in this case, a horizontal line that stretches endlessly in both directions at the y-value of -4.

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