Problem 19 - Brief Applied Calculus

Question

For each equation, find the slope \(m\) and \(y\) -intercept \((0, b)\) (when they exist) and draw the graph. \(y=4\)

Step-by-Step Solution

Answer

Slope: 0; y-intercept: (0, 4); the graph is a horizontal line at y=4.

Step 1: Identify the Equation Type

Notice that the given equation is in the form of a horizontal line, expressed as \(y = 4\). This means the line is parallel to the x-axis and passes through all points where \(y = 4\).

Step 2: Determine the Slope

For a horizontal line, the change in the y-value (rise) is zero as there is no vertical change for any change in x. Thus, the slope \(m\) is \(0\).

Step 3: Find the y-Intercept

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

Step 4: Summarize Equation Characteristics

For the equation \(y = 4\): the slope \(m = 0\), and the y-intercept is \((0, 4)\), representing a horizontal line in the Cartesian plane.

Key Concepts

Understanding the Slope of a Line

The slope of a line is a key concept in the Cartesian coordinate system that describes how slanted or steep a line is. It tells us how much the line rises or falls for each step it takes horizontally. In technical terms, slope is defined as the "rise" over the "run." Mathematically, it can be expressed as:\[ m = \frac{\Delta y}{\Delta x} \]where:

• \( \Delta y \) is the change in the y-values (vertical change).
• \( \Delta x \) is the change in the x-values (horizontal change).

If the slope \( m \) is positive, the line inclines upwards as it moves from left to right. A negative slope means the line declines. When \( m \) is zero, as in the equation \( y = 4 \), there is no vertical rise, indicating a perfectly flat line.

Exploring the y-Intercept

The y-intercept is a point where a line crosses the y-axis. This point is often expressed as \((0, b)\), where \(b\) is the intercept value. The y-intercept provides a tangible visualization of where on the y-axis the line meets when the x value is zero. Consider it as the starting point of a line on the graph assuming the line extends infinitely in both directions.In our exercise, the line from the equation \( y = 4 \) intersects the y-axis at \( (0, 4) \). This means that when x is zero, y remains constant at 4. This intercept is easy to identify, particularly in horizontal lines where every point along the line shares the same y-coordinate value.

Characteristics of a Horizontal Line

A horizontal line in the Cartesian Coordinate System is one where all points on the line have the same y-coordinate. This is why its equation looks like \( y = c \), where \( c \) is a constant representing the fixed y-value of the line.Some key characteristics include:

• The slope of a horizontal line is always zero, as there is no vertical rise; it simply runs parallel to the x-axis.
• Horizontal lines are represented by equations that do not contain \"+\" terms for x, confirming there is no slope component connected to the x-variable, unlike diagonal or vertical lines.

Understanding these characteristics helps to quickly identify horizontal lines and to visually recognize them on graphs. In essence, all horizontal lines reflect stability in the y-coordinate, regardless of changes in x.