Which Of The Following Functions Has Vertical Asymptotes At $x = 9$ And $x = -5$, And X-intercepts At $(6, 0$\] And $(-7, 0$\]?A. $y = \frac{x^2 - 4x - 45}{x^2 + X - 42}$B. $y = \frac{x^2 - X - 42}{x^2 +

Vertical Asymptotes and X-Intercepts: A Mathematical Analysis

In mathematics, functions can be represented in various forms, and understanding their properties is crucial for solving problems and making predictions. One of the key concepts in function analysis is the presence of vertical asymptotes and x-intercepts. In this article, we will explore the functions that have vertical asymptotes at $x = 9$ and $x = -5$, and x-intercepts at $(6, 0)$ and $(-7, 0)$. We will analyze two given functions and determine which one satisfies the given conditions.

Vertical Asymptotes

A vertical asymptote is a vertical line that a function approaches but never touches. It occurs when the denominator of a rational function is equal to zero, and the numerator is not equal to zero. In other words, a vertical asymptote is a line that the function approaches as the input (x-value) gets arbitrarily close to a certain value.

X-Intercepts

An x-intercept is a point on the graph of a function where the function crosses the x-axis. It occurs when the y-value of the function is equal to zero. In other words, an x-intercept is a point on the graph where the function touches the x-axis.

We are given two functions:

A. $y = \frac{x^2 - 4x - 45}{x^2 + x - 42}$

B. $y = \frac{x^2 - x - 42}{x^2 + 9x + 20}$

To determine which function has vertical asymptotes at $x = 9$ and $x = -5$, and x-intercepts at $(6, 0)$ and $(-7, 0)$, we need to analyze the factors of the numerator and denominator of each function.

Function A

The numerator of function A is $x^2 - 4x - 45$, which can be factored as $(x - 9)(x + 5)$. The denominator of function A is $x^2 + x - 42$, which can be factored as $(x + 7)(x - 6)$.

Function B

The numerator of function B is $x^2 - x - 42$, which can be factored as $(x - 7)(x + 6)$. The denominator of function B is $x^2 + 9x + 20$, which can be factored as $(x + 5)(x + 4)$.

Determining Vertical Asymptotes

To determine the vertical asymptotes of a function, we need to find the values of x that make the denominator equal to zero. For function A, the denominator is equal to zero when $x = -7$ or $x = 6$. For function B, the denominator is equal to zero when $x = -5$ or $x = -4$.

Determining X-Intercepts

To determine the x-intercepts of a function, we need to find the values of x that make the numerator equal to zero. For function A, the numerator is equal to zero when $x = 9$ or $x = -5$. For function B, the numerator is equal to zero when $x = 7$ or $x = -6$.

Based on the analysis of the given functions, we can conclude that function A has vertical asymptotes at $x = 9$ and $x = -5$, and x-intercepts at $(6, 0)$ and $(-7, 0)$. Therefore, the correct answer is:

A. $y = \frac{x^2 - 4x - 45}{x^2 + x - 42}$

In our previous article, we explored the functions that have vertical asymptotes at $x = 9$ and $x = -5$, and x-intercepts at $(6, 0)$ and $(-7, 0)$. We analyzed two given functions and determined which one satisfies the given conditions. In this article, we will answer some frequently asked questions related to vertical asymptotes and x-intercepts.

Q: What is a vertical asymptote?

A: A vertical asymptote is a vertical line that a function approaches but never touches. It occurs when the denominator of a rational function is equal to zero, and the numerator is not equal to zero.

Q: What is an x-intercept?

A: An x-intercept is a point on the graph of a function where the function crosses the x-axis. It occurs when the y-value of the function is equal to zero.

Q: How do I determine the vertical asymptotes of a function?

A: To determine the vertical asymptotes of a function, you need to find the values of x that make the denominator equal to zero. You can do this by factoring the denominator and setting each factor equal to zero.

Q: How do I determine the x-intercepts of a function?

A: To determine the x-intercepts of a function, you need to find the values of x that make the numerator equal to zero. You can do this by factoring the numerator and setting each factor equal to zero.

Q: What is the difference between a vertical asymptote and an x-intercept?

A: A vertical asymptote is a line that a function approaches but never touches, while an x-intercept is a point on the graph of a function where the function crosses the x-axis.

Q: Can a function have both vertical asymptotes and x-intercepts?

A: Yes, a function can have both vertical asymptotes and x-intercepts. For example, the function $y = \frac{x^2 - 4x - 45}{x^2 + x - 42}$ has both vertical asymptotes at $x = 9$ and $x = -5$, and x-intercepts at $(6, 0)$ and $(-7, 0)$.

Q: How do I graph a function with vertical asymptotes and x-intercepts?

A: To graph a function with vertical asymptotes and x-intercepts, you need to plot the vertical asymptotes and x-intercepts on a coordinate plane. You can use a graphing calculator or software to help you with this process.

Q: What are some real-world applications of vertical asymptotes and x-intercepts?

A: Vertical asymptotes and x-intercepts have many real-world applications, including:

• Physics: Vertical asymptotes can represent the limits of a function, while x-intercepts can represent the points where a function crosses the x-axis.
• Engineering: Vertical asymptotes and x-intercepts can be used to design and analyze systems, such as electrical circuits and mechanical systems.
• Economics: Vertical asymptotes and x-intercepts can be used to model and analyze economic systems, such as supply and demand curves.

In this article, we answered some frequently asked questions related to vertical asymptotes and x-intercepts. We hope that this article has provided valuable insights into the analysis of functions and their properties. If you have any further questions, please don't hesitate to ask.