Problem 20 - Brief Applied Calculus

Question

For each equation, find the slope \(m\) and \(y\) -intercept \((0, b)\) (when they exist) and draw the graph. \(y=-3\)

Step-by-Step Solution

Answer

Slope: 0, y-intercept: (0, -3). Graph is a horizontal line at y = -3.

Step 1: Identifying the Equation Type

The equation given is in the form of a horizontal line. It is written as \( y = -3 \), where \( y \) is a constant and does not depend on \( x \). This indicates the graph is a horizontal line.

Step 2: Determining the Slope

For a horizontal line of the form \( y = c \), where \( c \) is a constant, the slope \( m \) is always 0 because the line does not rise or fall as \( x \) changes.

Step 3: Finding the y-intercept

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

Step 4: Graphing the Line

To graph \( y = -3 \), draw a horizontal line across the coordinate plane that passes through all points where \( y \) is \(-3\). This line will be parallel to the \( x \)-axis and will have no incline.

Key Concepts

Horizontal Line: Understanding its Characteristics

In mathematics, a horizontal line is special because it has a constant value for its vertical coordinate across all its points. This means no matter what value we have for the x-coordinate, the y-value stays the same. When you see an equation like \( y = -3 \), it tells you that every point on the line has a y-coordinate of -3. This leads to the formation of a line that runs parallel to the x-axis.

Key features of a horizontal line include:

• The line never inclines or declines; it is perfectly flat.
• All points on this line have the same y-value.
• Horizontal lines are identified by the equation form \( y = c \), where \( c \) is a constant.

Recognizing these features can help quickly identify and graph horizontal lines.

Slope of a Line: Exploring its Meaning and Value

The slope of a line is crucial in understanding the incline or decline between two points on the line. It is defined as the rise over run, or mathematically, the change in y divided by the change in x. For a horizontal line, however, things are a bit different. The equation \( y = c \), such as \( y = -3 \), is peculiar as the slope \( m \) for such a line is always 0. This happens because there is no vertical change between any two points along the line.

When determining the slope:

• Slope \( m \) indicates steepness.
• For horizontal lines, \( m = 0 \) because there is no rise.
• A slope of 0 creates a perfectly flat line parallel to the x-axis.

Understanding that the absence of rise results in a zero slope makes analyzing horizontal lines much easier.

y-intercept: Where the Line Meets the y-axis

The y-intercept of a line is the specific point where the line crosses the y-axis on a coordinate plane. This is a fundamental aspect of linear equations as it provides a starting point for the line's path across the chart. In the context of a horizontal line, like the one expressed by the equation \( y = -3 \), the y-intercept is straightforward to identify. It is the point (0, -3), indicating where the constant value intersects the y-axis.

Key aspects of the y-intercept include:

• It is a point of intersection between the line and the y-axis.
• For equations in the form \( y = c \), the y-intercept will always be \( (0, c) \).
• The position of the y-intercept helps in easily plotting and graphing the equation.

Recognizing the y-intercept as a cross point helps in visualizing the line quickly and accurately on a graph.