Problem 20 Concept Check If the $x$ -inte... [FREE SOLUTION]

91影视

A Graphical Approach to College Algebra

John Hornsby,Margaret L. Lial,Gary Rockswold

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 1: Problem 20

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?

Short Answer

Expert verified

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

Step by step solution

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.

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 $x$.

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 $y = 0$.

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥檚 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鈥檚 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鈥檙e looking at a zero function!

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91影视

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?

Short Answer

Step by step solution

Understanding the Problem

Defining the Horizontal Line

Identifying the Solution Set

Special Name for the Equation

Key Concepts

Horizontal Line

Solution Set

Zero Function

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