Problem 21 - Applied Calculus

Question

For each equation, find the slope \(m\) and \(y\) -intercept \((0, b)\) (when they exist) and draw the graph. $$ x=4 $$

Step-by-Step Solution

Answer

Slope is undefined; no y-intercept.

Step 1: Identify the equation type

The equation given is \( x = 4 \). This is an equation of a vertical line, where all points on the line have an \( x \)-coordinate of 4.

Step 2: Determine the slope

A vertical line does not have a defined slope because the change in \( y \) could be anything while the change in \( x \) is zero, which would lead to division by zero in the slope formula \( m = \frac{\Delta y}{\Delta x} \). Thus, the slope \( m \) is undefined.

Step 3: Identify the y-intercept

A vertical line such as \( x = 4 \) does not cross the \( y \)-axis. Therefore, there is no \( y \)-intercept for this equation since it does not intersect the \( y \)-axis at any point.

Step 4: Graph the equation

To graph \( x = 4 \), draw a vertical line on the coordinate plane that crosses the \( x \)-axis at \( x = 4 \). This line is parallel to the \( y \)-axis and does not cross the \( y \)-axis.

Key Concepts

Vertical line

When we talk about a vertical line in terms of graphing, it means that the line runs up and down on a graph. This line is parallel to the y-axis. In the equation form, a vertical line is straightforward, represented by an equation like \( x = a \). This equation indicates that the \( x \)-coordinate is a constant value, in this case, \( a \).

All points that lie on this line will share the same \( x \)-coordinate but can have different \( y \)-coordinates.

• For instance, in the equation \( x = 4 \), every point on this line will have \( x \) equal to 4, such as \((4, 1), (4, -2), (4, 6.5)\).

Understanding vertical lines is crucial since it helps clarify why they behave differently in terms of slope and intercepts.

Undefined slope

The concept of slope is fundamental in understanding how steep a line is. It's represented mathematically by \( m = \frac{\Delta y}{\Delta x} \), where \( \Delta y \) is the change in the \( y \)-coordinates, and \( \Delta x \) is the change in the \( x \)-coordinates.

However, with a vertical line, like \( x = 4 \), there is no change in the \( x \)-coordinate since it's constant. This results in \( \Delta x = 0 \), which makes the formula for the slope undefined because you cannot divide by zero.

• This is why the slope of a vertical line is undefined. It essentially means that the line is so steep it's not comparable to any other incline or slope.

Recognizing an undefined slope is important, ensuring you can quickly identify vertical lines just by looking at their equations.

Graphing equations

Graphing equations is an essential skill in understanding algebra and geometry. To graph an equation like \( x = 4 \), we plot all the points that make \( x \) equal to 4 on a coordinate plane. This involves plotting multiple points, such as \((4, 2), (4, -8)\), consistently positioning them in a vertical direction.

Once these points are plotted, draw a straight line through them; this is the graph of \( x = 4 \). A few important notes on graphing vertical lines:

• They will appear parallel to the y-axis.
• They will never intersect or cross the y-axis.

Graphing becomes much simpler when seeing these characteristics, offering a visual understanding of otherwise abstract math concepts.