Problem 16 - Calculus with Concepts in Calculus

Question

Find an equation of the line described. Then sketch the line. The line through \((0, \pi)\) with slope 0

Step-by-Step Solution

Answer

The equation of the line is \(y = \pi\). It is a horizontal line crossing the y-axis at \(\pi\).

Step 1: Understanding the Problem

We need to find the equation of a line that passes through the point \((0, \pi)\) and has a slope of 0. A line with a slope of 0 is horizontal.

Step 2: Identifying the Equation of a Horizontal Line

A horizontal line has an equation of the form \(y = c\), where \(c\) is the constant y-value for all points on the line.

Step 3: Substituting the Given Point

Since the line passes through \((0, \pi)\), the y-value for all points on this horizontal line is \(\pi\). Thus, the equation of the line is \(y = \pi\).

Step 4: Sketching the Line

To sketch the line \(y = \pi\), draw a horizontal line on a graph that crosses the y-axis at \(\pi\). This line remains horizontal across the graph.

Key Concepts

Horizontal Line

A horizontal line is a line that runs parallel to the x-axis on a graph. It has a unique feature where all points on the line share the same y-coordinate. This means that no matter what the x-value is, the y-value does not change. This characteristic is crucial in understanding why horizontal lines have a slope of zero.
In coordinate geometry, when a line is horizontal, it results in having no rise over the run. "Rise" refers to the change in the y-values, and "run" refers to the change in x-values, but since there is no change in y-values in a horizontal line, the slope is zero. The equation for a horizontal line in a 2D coordinate plane can be expressed as:

• y = c

Where "c" is the constant y-value through which the line passes.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometric figures using the coordinate plane. Using this system allows for precise definitions and descriptions of geometric shapes through algebra. A coordinate plane is defined by two perpendicular lines, the x-axis and y-axis, intersecting at the origin (0,0).
In coordinate geometry, each point is specified by an ordered pair (x, y). The ability to transform geometric problems into algebraic ones enables the calculation of distances, angles, and areas using algebraic equations. In the case of the given exercise, using coordinate geometry, the line through (0, π) with slope 0 is easily represented and visualized on a graph as:

• The x-coordinate remains variable, but the y-coordinate is fixed at π.

Slope

The slope of a line measures its steepness or incline. It is calculated as "rise over run," which means the amount the line goes up or down (rise) over the amount it goes horizontally (run). In mathematical terms, slope (m) is defined by the formula:

• \( m = \frac{{y_2 - y_1}}{{x_2 - x_1}} \)

When the slope is zero, as in this problem, it indicates that there is no vertical change as you move along the line. This is indicative of a horizontal line, where:

• The "rise" component is zero, and therefore, the slope is zero as well.

In practical terms, for any two points on this horizontal line, the difference in the y-values will always be zero regardless of the x-values.

Graphing

Graphing refers to visually plotting points, lines, shapes, and mathematical functions on a coordinate plane. It helps in understanding the relationships between variables and ungroups the information geometrically. In the exercise, graphing the equation \( y = \pi \) involves drawing a straight, horizontal line across the graph.

• This line will cross the y-axis at the point where y equals π, indicating that all points on this line have the same y-coordinate of π.

When graphing horizontal lines:

• Ensure the line remains consistent across the graph, not sloping up or down.
• Highlight the intersection with the y-axis to reinforce that the y-value is constant irrespective of x.
• Use clear markers to show distinct elements, like the point (0, π), which helps in verifying accuracy.

Graphing is a foundational tool in coordinate geometry that allows one to transform abstract numerical equations into visible and tangible shapes.