Problem 22 - Brief 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 is no y-intercept.

Step 1: Recognize the equation's form

The given equation is of the form \( x = c \), where \( c \) is a constant. This represents a vertical line.

Step 2: Determine the slope of the line

A vertical line has no slope (or it is undefined) because the change in \(y\) is non-zero while the change in \(x\) is zero, which makes the division for slope, \( \frac{\Delta y}{\Delta x} \), undefined.

Step 3: Identify the y-intercept

For vertical lines, there is no \(y\)-intercept because the line is parallel to the \(y\)-axis and doesn't cross it at any point.

Step 4: Graph the equation

Plot the line \(x = -3\) on a coordinate plane by drawing a vertical line passing through \(x = -3\). The line will extend infinitely in the positive and negative \(y\)-directions while maintaining \(x = -3\).

Key Concepts

Slope of a Line

In the world of coordinate geometry, the slope of a line is a crucial concept. The slope is a measure of how steep a line is. We calculate it by determining the change in the y-value (vertical change) divided by the change in the x-value (horizontal change). This can be written as \( m = \frac{\Delta y}{\Delta x} \). Here, "\(m\)" stands for slope, "\(\Delta y\)" is the change in y-values, and "\(\Delta x\)" is the change in x-values.

When graphed, a positive slope means the line goes up as it moves from left to right, while a negative slope means it goes down. A zero slope indicates a horizontal line, and an undefined slope, which happens when the change in x is zero, signifies a vertical line.

However, for the equation \(x = -3\), the line is vertical, as we'll discuss later. Thus, it does not have a defined slope according to the formula because the denominator becomes zero, making \( \frac{\text{some number}}{0} \) undefined.

Graphing Equations

Graphing equations means visualizing mathematical expressions on a coordinate system. It's a way to see their behavior and interactions nicely laid out on paper or a screen. To graph an equation, we must first understand its form and then plot points that satisfy it.

For instance, in the equation \(x = a\), like \(x = -3\), you know you're dealing with a vertical line. The graph of this equation is simple. You won’t worry about having a y-intercept as it pertains to vertical line situations. All points on the line will have the same x-coordinate value, in this example, \(-3\), with y-values that can vary.

To graph this, find \(-3\) on the x-axis, plot a dot, and draw a straight line upwards and downwards through it. It’ll form a vertical line, clearly demarcating the spot where all x-values are \(-3\). This helps us visualize that all y-values from \(-\infty\) to \(+\infty\) lie on this line, albeit all with the same x-value.

Vertical Line

A vertical line in a coordinate system is quite unique. It runs parallel to the y-axis and is defined by equations of the form \(x = c\), where \(c\) is any constant number. In the case of \(x = -3\), this specifies that every point through which the vertical line runs has \(x = -3\).

One important feature of vertical lines is their "undefined" slope. Why? Because the change in x-values (\(\Delta x\)) is zero, and dividing by zero in mathematics is undefined. Simply put, while y-values change up and down the line, there’s no horizontal change to measure the slope against, essentially making a division by zero.

Moreover, vertical lines cannot have a y-intercept because they do not cross the y-axis. Instead, they either run to the left or right of it. For practical graphing, simply draw a straight line up and down the point where x equals your constant \(c\). This ensures an accurate representation of any vertical line equation like \(x = -3\).