Properties of the Denominator

Oct 13, 2025 by magicalceylontours.com 121 views

Understanding the Graph of $y = \frac{10}{x^2}$

The graph of a function is a visual representation of its behavior, and understanding the properties of a graph is crucial in mathematics. In this article, we will explore the graph of the function $y = \frac{10}{x^2}$ and explain why no part of it appears below the $x$ -axis.

The Function $y = \frac{10}{x^2}$

The function $y = \frac{10}{x^2}$ is a rational function, which means it is the ratio of two polynomials. In this case, the numerator is a constant, 10, and the denominator is a quadratic expression, $x^2$ . To understand the behavior of this function, we need to consider the properties of the denominator.

Properties of the Denominator

The denominator of the function, $x^2$ , is always positive for all non-zero values of $x$ . This is because the square of any real number is always non-negative. In other words, $x^2 \geq 0$ for all $x \neq 0$ .

The Sign of the Function

Since the denominator is always positive, the sign of the function $y = \frac{10}{x^2}$ is determined by the numerator, 10. The numerator is a positive constant, which means that the function will always have a positive value for all non-zero values of $x$ .

Why No Part of the Graph Appears Below the $x$ -Axis

Now that we have established that the function $y = \frac{10}{x^2}$ is always positive, we can explain why no part of its graph appears below the $x$ -axis. The $x$ -axis represents the values of $y$ that are equal to zero. Since the function is always positive, it will never intersect the $x$ -axis, and therefore, no part of its graph will appear below the $x$ -axis.

In conclusion, the graph of the function $y = \frac{10}{x^2}$ does not appear below the $x$ -axis because the function is always positive. The denominator of the function, $x^2$ , is always positive for all non-zero values of $x$ , and the numerator, 10, is a positive constant. Therefore, the function will always have a positive value for all non-zero values of $x$ , and its graph will never intersect the $x$ -axis.

The correct answer to the question is B, "The $y$ -value is always positive." This is because the function $y = \frac{10}{x^2}$ is always positive for all non-zero values of $x$ . The denominator is always positive, and the numerator is a positive constant, which means that the function will always have a positive value.

The other options are incorrect because they do not accurately describe the behavior of the function $y = \frac{10}{x^2}$ . Option A, "The $y$ -value is always negative," is incorrect because the function is always positive. Option C, "The $y$ -value is always zero," is incorrect because the function will never intersect the $x$ -axis. Option D is not provided.

In our previous article, we explored the graph of the function $y = \frac{10}{x^2}$ and explained why no part of it appears below the $x$ -axis. In this article, we will answer some frequently asked questions about the graph of this function.

Q: What is the domain of the function $y = \frac{10}{x^2}$ ?

A: The domain of the function $y = \frac{10}{x^2}$ is all real numbers except zero. This is because the denominator of the function, $x^2$ , cannot be equal to zero.

Q: Why is the denominator $x^2$ always positive?

A: The denominator $x^2$ is always positive because the square of any real number is always non-negative. In other words, $x^2 \geq 0$ for all $x \neq 0$ .

Q: What is the range of the function $y = \frac{10}{x^2}$ ?

A: The range of the function $y = \frac{10}{x^2}$ is all positive real numbers. This is because the function is always positive for all non-zero values of $x$ .

Q: How does the graph of $y = \frac{10}{x^2}$ behave as $x$ approaches zero?

A: As $x$ approaches zero, the graph of $y = \frac{10}{x^2}$ increases without bound. This is because the denominator of the function, $x^2$ , approaches zero, and the numerator, 10, remains constant.

Q: How does the graph of $y = \frac{10}{x^2}$ behave as $x$ approaches infinity?

A: As $x$ approaches infinity, the graph of $y = \frac{10}{x^2}$ approaches zero. This is because the denominator of the function, $x^2$ , increases without bound, and the numerator, 10, remains constant.

Q: Can the graph of $y = \frac{10}{x^2}$ intersect the $x$ -axis?

A: No, the graph of $y = \frac{10}{x^2}$ cannot intersect the $x$ -axis. This is because the function is always positive for all non-zero values of $x$ .

Q: What is the $y$ -intercept of the graph of $y = \frac{10}{x^2}$ ?

A: The $y$ -intercept of the graph of $y = \frac{10}{x^2}$ is undefined. This is because the function is not defined at $x = 0$ .

Q: Can the graph of $y = \frac{10}{x^2}$ be vertical?

A: No, the graph of $y = \frac{10}{x^2}$ cannot be vertical. This is because the function is always positive for all non-zero values of $x$ .

In conclusion, the graph of the function $y = \frac{10}{x^2}$ is a positive function that never intersects the $x$ -axis. The denominator of the function, $x^2$ , is always positive for all non-zero values of $x$ , and the numerator, 10, is a positive constant. Therefore, the function will always have a positive value for all non-zero values of $x$ , and its graph will never intersect the $x$ -axis.

In this article, we have answered some frequently asked questions about the graph of the function $y = \frac{10}{x^2}$ . We hope that this article has provided you with a better understanding of the graph of this function and its properties.