Problem 20 - A Graphical Approach to College Algebra

Question

Concept Check If the \(x\) -intercept method leads to a horizontal line that coincides with the \(x\) -axis, what is the solution set of the equation? What special name is given to this kind of equation?

Step-by-Step Solution

Answer

The solution set is all real numbers; the equation is called the "x-axis".

Step 1: Understanding the Problem

We need to determine the solution set of an equation whose graph is a horizontal line coinciding with the \(x\)-axis. We should also identify the special name for such an equation.

Step 2: Defining the Horizontal Line

A horizontal line coinciding with the \(x\)-axis means the line's equation is \(y = 0\). This is because, for any point \((x, y)\) on this line, \(y = 0\) for all values of \(x\).

Step 3: Identifying the Solution Set

The line \(y = 0\) suggests that for any \(x\), \(y\) must be zero. Therefore, the solution set to the equation is all possible \(x\) values, since every \(x\) corresponds to a point on the \(x\)-axis where \(y = 0\).

Step 4: Special Name for the Equation

The special name given to the equation \(y = 0\), because it forms a line parallel to the \(x\)-axis and overlaps it, is the "\(x\)-axis ". This type of line is sometimes referred to as a 'zero function' in certain contexts.

Key Concepts

Horizontal Line

Imagine the graph of a line. Normally, it might slant up or down. But when you have a horizontal line, it runs straight across. It doesn't rise or fall which means it maintains a constant value of the vertical coordinate throughout. Specifically, a horizontal line that coincides with the x-axis has an equation like \(y = 0\). Here, no matter what \(x\) value you choose, \(y\) will always be zero.
Such a line looks like this:

• It’s flat and level, perfectly aligned with the horizontal plane.
• Every single point on the line has a \(y\)-value of zero.

For students, it's also important to understand that on a graph, the x-axis itself is a horizontal line. This means when we talk about a horizontal line like \(y = 0\), we're essentially describing the x-axis.

Solution Set

When we talk about a solution set in mathematics, we're referring to all the possible values that satisfy a given equation. For a horizontal line represented by \(y = 0\), the solution set includes every \(x\)-value you can imagine.
Here's the breakdown:

• All values of \(x\) can be part of the solution set because for every value of \(x\), the equation \(y = 0\) holds true.
• Visually, this means every point along the x-axis, which is the solution set for \(y = 0\).

This illustrates that while the \(y\)-value is fixed (at zero), the \(x\)-value isn't restrained. This concept helps us see that horizontal lines offer a multitude of solutions as they span the entire axis. The key takeaway is: any point on the x-axis is part of the solution set for \(y = 0\).

Zero Function

The term 'zero function' might sound a bit puzzling, but it's quite straightforward. A zero function is a function that always returns zero, regardless of the input. In the context of a line, the simplest illustration of a zero function is the equation \(y = 0\).
Here's how it breaks down:

• It’s called a 'zero' function because the output value for \(y\) is consistently zero.
• This has a straightforward graph, being directly on the x-axis with no deviation upwards or downwards.

This function serves important roles in many mathematical investigations and exercises. It can help simplify complex problems and provide a foundation for understanding other functions. Recognizing when a graph represents a zero function is crucial in solving equations that result in horizontal lines. Remember, if you ever see \(y = 0\), you’re looking at a zero function!