Problem 22 - Applied Calculus

Question

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

Step-by-Step Solution

Answer

The slope is undefined and there's no y-intercept for the vertical line x = -3.

Step 1: Identify the Equation Type

The given equation is \(x = -3\). This is in the form of a vertical line equation. In a vertical line, every point on the line has the same \(x\) value.

Step 2: Determine the Slope

The slope \(m\) of a vertical line is undefined, because the run (change in \(x\)) is zero, making the slope division \(\frac{\text{rise}}{\text{run}} = \frac{\Delta y}{0}\) undefined.

Step 3: Find the Y-intercept

A vertical line does not intersect the \(y\)-axis unless it coincides with it. Since our line \(x = -3\) is parallel and distinct from the \(y\)-axis, there's no \(y\)-intercept \((0, b)\) for this line.

Step 4: Draw the Graph

To graph \(x = -3\), draw a vertical line passing through \(x = -3\) on the \(x\)-axis. This line will extend infinitely in both the positive and negative directions of the \(y\)-axis.

Key Concepts

Slope

The concept of slope is central in understanding the behavior of lines in coordinate geometry. Slope, often denoted by the letter \(m\), essentially measures the steepness or the incline of a line. It tells us how much a line rises or falls as we move along the horizontal axis, known as the run. The general formula to calculate slope is given by:

• \( m = \frac{\Delta y}{\Delta x} \)

"Delta" denotes the change in values, where \( \Delta y \) represents the change in the vertical direction, and \( \Delta x \) represents the change along the horizontal direction.
Vertical lines, like \(x = -3\), have a unique case. Their slope is undefined, because the run \(\Delta x\), which describes the horizontal change, is zero. Division by zero is undefined in mathematics, leading to the conclusion that any vertical line has an undefined slope.
In comparison, horizontal lines, exhibiting no change in \(y\) as \(x\) changes, have a slope of zero. Understanding the slope helps us grasp how lines behave and how they might interact in a graph.

Y-intercept

The \(y\)-intercept is a vital aspect of understanding how a line interacts with the graph's axes. The \(y\)-intercept occurs where a line crosses the \(y\)-axis, at a point we refer to as \((0, b)\). The \(y\)-value at this point is termed \(b\), known as the intercept itself. It gives us a starting point to graph the line from the vertical perspective.
For a standard line with the equation \(y = mx + b\), the \(y\)-intercept is easy to identify because it is explicitly provided by the constant \(b\). However, in the case of vertical lines, such as \(x = -3\), we find that the \(y\)-axis is never intersected. This is because a vertical line is perfectly parallel to the \(y\)-axis but doesn't cross it unless it's the line defined by \(x = 0\). Therefore, vertical lines like \(x = -3\) do not possess a \(y\)-intercept.
In contrast, horizontal lines and most sloped lines have clear \(y\)-intercepts where the line meets the \(y\)-axis. Identifying the \(y\)-intercept when plotting different types of lines is crucial for unraveling the dynamics their equations depict.

Graphing Linear Equations

Graphing linear equations involves plotting points represented by the equation to visualize their line on a coordinate plane. This process helps us see the behavior, direction, and position of lines, providing a deeper understanding of their geometrical properties.
Given an equation, we first identify its type: a standard line will take the form \(y = mx + b\), while a vertical line will look like \(x = c\), where \(c\) is a constant.

• Standard lines are straightforward to graph. The slope \(m\) indicates the line's angle, while the \(y\)-intercept \(b\) indicates where the line would meet the \(y\)-axis.
• Vertical lines, such as \(x = -3\), are graphed by drawing a line through all points with \(x\)-values of \(-3\), extending parallel to the \(y\)-axis.

Once the graphing process is initiated, ensuring every point on the line meets the equation is key. For vertical lines, we visualize them as extending infinitely up and down at a constant \(x\)-coordinate.
Graphing such equations, despite the infinite range, remains tangible and executable, promoting a wealth of insights into the dynamics of linear equations.