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Chapter 1: Problem 63

Find the slope and \(y\)-intercept of the line, and draw its graph. \(y=4\)

Short Answer

Expert verified
Slope: 0; y-intercept: (0, 4)

Step by step solution

01

Identify the equation and simplify if necessary

The given equation of the line is \(y = 4\). This is already in a well-understood form and does not require any further simplification.
02

Understand the equation type

The equation \( y = 4 \) represents a horizontal line where \( y \) is always equal to 4 for any value of \( x \).
03

Identify the slope

The slope of a horizontal line is always zero because there is no change in the \( y \)-value regardless of the change in the \( x \)-value. Thus, the slope \( m = 0 \).
04

Identify the y-intercept

Since the line \( y = 4 \) is horizontal, it crosses the \( y \)-axis at \( y = 4 \). Therefore, the \( y \)-intercept is \( (0, 4) \).
05

Draw the graph

To graph the line, draw a straight horizontal line passing through all points where \( y = 4 \) and \( x \) can be any real number. This line will cut the \( y \)-axis at 4.

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Key Concepts

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

Horizontal Line
A horizontal line is one of the simplest types of lines in a Cartesian coordinate system. It is defined by an equation of the form \( y = c \), where \( c \) is a constant. This means the value of \( y \) does not change no matter what \( x \) value is used. Such a line runs parallel to the \( x \)-axis.
  • **Constant \( y \)-Value:** For a horizontal line, the \( y \) value is always constant. For example, in the equation \( y = 4 \), the line is always four units above the \( x \)-axis.
  • **Parallel to \( x \)-Axis:** The horizontal line will never intersect with the \( x \)-axis unless the equation is \( y = 0 \). Otherwise, it remains at the height determined by \( y = c \).
This characteristic makes it easy to graph horizontal lines, as all you need to do is draw a line parallel to the \( x \)-axis at the specified \( y \) value.
Slope
The slope of a line is a measure of its steepness or incline. It is calculated as the "rise" over the "run," or the change in \( y \)-value over the change in \( x \)-value. For any horizontal line, like \( y = 4 \), the slope is unique.
  • **Zero Slope:** The slope of a horizontal line is always \( 0 \). This is because there is no vertical change regardless of the horizontal movement. In mathematical terms, the rise is always \( 0 \), leading to a slope \( m = \frac{0}{x_2 - x_1} = 0 \).
  • **Flat Line Indicator:** A slope of zero signifies a perfectly flat line, which means it's entirely level with no increase or decrease as you move along the \( x \)-axis.
Understanding that horizontal lines have a slope of zero is crucial when analyzing the type and equation of a line.
y-Intercept
The \( y \)-intercept is the point where the line crosses the \( y \)-axis. It's an essential component of the slope-intercept form of a line equation, \( y = mx + b \), where \( b \) denotes the \( y \)-intercept.
  • **Position on \( y \)-Axis:** For a horizontal line like \( y = 4 \), the line intersects the \( y \)-axis at \( y = 4 \). This means the \( y \)-intercept is \( (0, 4) \).
  • **Intersection Point:** The \( y \)-intercept occurs at the point where \( x = 0 \). Consequently, for the equation \( y = 4 \), the line meets the \( y \)-axis precisely at the point 4 on the \( y \)-axis.
Knowing the \( y \)-intercept is not only vital for graphing but also for understanding the starting point of a line on the \( y \)-axis. It tells you where the line will be when \( x \) is zero, providing a foundational reference for the graph.

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Most popular questions from this chapter

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