Problem 73 - Precalculus with Limits

Question

Consider a line with slope \(m\) and \(y\)-intercept (a) Write the distance \(d\) between the origin and the line as a function of \(m\). (b) Graph the function in part (a). (c) Find the slope that yields the maximum distance between the origin and the line. (d) Find the asymptote of the graph in part (b) and interpret its meaning in the context of the problem.

Step-by-Step Solution

Answer

The distance \(d\) between the origin and the line is always zero, regardless of the slope.

Step 1: Formulate Distance from Line to Origin

The distance \(d\) from the origin to the line with slope \(m\) and y-intercept \(b\) can be calculated using the formula: \(d = \frac{|b|}{\sqrt{1 + m^2}}\). Since the y-intercept is 0, the formula simplifies to \(d = \frac{0}{\sqrt{1 + m^2}}\), implying that d is always 0.

Step 2: Graphing the Distances

Given the equation for \(d\) as obtained above, it is found that \(d\) is always 0 whatever is the value of the slope \(m\) - This behavior is the same as the x-axis on a graph.

Step 3: Finding the Maximum Distance

Since \(d\) is always 0, there is no value of \(m\) that yields a maximum distance between the origin and the line. The distance remains 0 irrespective of the slope of the line.

Step 4: Asymptote Determination and Interpretation

Again, because \(d\) is always 0 irrespective of the slope \(m\), there is no asymptote in the graph. Asymptotes only occur when the graph approaches but never touches or crosses a line.

Key Concepts

Understanding Slope

The slope is a measure of how steep a line is on a graph. It tells us how much the line rises or falls as it moves from left to right. In mathematical terms, the slope (often denoted by the letter \(m\)) is the ratio of the vertical change (rise) to the horizontal change (run) between any two points on a line. This is given by the formula:

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

A positive slope means the line is going upwards, while a negative slope indicates that it is going downward. A slope of zero means the line is perfectly horizontal, while undefined slope means the line is vertical. In this exercise, the slope plays a key role in determining the distance from the origin to the line.

What is the Y-intercept?

The y-intercept is the point where a line crosses the y-axis of a graph. It represents the value of \(y\) when \(x\) is zero. In the equation of a line, which is commonly expressed as \(y = mx + b\), the y-intercept is represented by \(b\). Here, \(b\) is a constant that tells you where the line intersects the y-axis.

• If \(b = 0\), this means that the line passes through the origin. In the context of this problem, the y-intercept being zero simplifies the calculation of the distance between the origin and the line.

Understanding the y-intercept helps in easily plotting the line on a graph, as it provides a starting point.

Graphing Functions Made Simple

Graphing functions involves drawing a line or curve on a coordinate plane that represents the solutions to a particular equation. To graph a line, you typically start by plotting the y-intercept on the y-axis. From there, use the slope to determine the direction and steepness of the line.

• Move up or down according to the rise of the slope.
• Move left or right according to the run of the slope.

In the given exercise, graphing the distance function \(d = \frac{|b|}{\sqrt{1+m^2}}\) results in a horizontal line at \(d=0\) since the line's y-intercept is zero. This means that, regardless of the value of \(m\), the distance from the origin to the line will always be zero. Thus, the graph essentially represents the x-axis.

Breaking Down the Distance Formula

The distance formula helps determine the distance between a point and a line, or between two points in a plane. In this exercise, the formula given is

• \(d = \frac{|b|}{\sqrt{1 + m^2}}\)

It calculates the shortest distance from the origin \((0,0)\) to a line with a given slope \(m\) and y-intercept \(b\). Typically, the formula involves both the slope and the y-intercept. However, with a y-intercept \(b=0\), this renders the distance \(d\) to always be zero. This key idea shows the power of the distance formula in translating algebraic expressions into understanding of geometric spaces.

• It's important to note that for a line passing through the origin, the shortest direct line from the origin to the line is zero, confirming the results of the problem.