Determine Each Feature Of The Graph Of The Given Function:$\[ F(x) = \frac{-5}{3x - 2} \\]1. Horizontal Asymptote: - \[$ Y = \$\] [blank] - No Horizontal Asymptote2. Vertical Asymptote: - \[$ X = \$\] [blank] - No

Mar 1, 2025 by ADMIN 222 views

Determine Each Feature of the Graph of the Given Function

In mathematics, graphing functions is a crucial aspect of understanding their behavior and characteristics. A function's graph can provide valuable information about its domain, range, and various asymptotes. In this article, we will focus on determining the features of the graph of a given function, specifically the horizontal and vertical asymptotes.

The given function is:

${ f(x) = \frac{-5}{3x - 2} }$

This is a rational function, which means it is the ratio of two polynomials. The function has a numerator of -5 and a denominator of 3x - 2.

A horizontal asymptote is a horizontal line that the graph of a function approaches as x goes to positive or negative infinity. To determine the horizontal asymptote of a rational function, we need to compare the degrees of the numerator and denominator.

In this case, the degree of the numerator is 0 (since it is a constant), and the degree of the denominator is 1 (since it is a linear polynomial). Since the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote is y = 0.

Therefore, the horizontal asymptote of the given function is:

${ y = 0 }$

A vertical asymptote is a vertical line that the graph of a function approaches as x goes to a specific value. To determine the vertical asymptote of a rational function, we need to find the values of x that make the denominator equal to zero.

In this case, the denominator is 3x - 2. To find the values of x that make the denominator equal to zero, we set 3x - 2 = 0 and solve for x:

${ 3x - 2 = 0 }$ ${ 3x = 2 }$ ${ x = \frac{2}{3} }$

Therefore, the vertical asymptote of the given function is:

${ x = \frac{2}{3} }$

In this article, we determined the features of the graph of the given function, specifically the horizontal and vertical asymptotes. We found that the horizontal asymptote is y = 0 and the vertical asymptote is x = 2/3.

It's worth noting that the graph of a rational function can have multiple vertical asymptotes if the denominator has multiple factors that equal zero. However, in this case, the denominator has only one factor that equals zero, so we only have one vertical asymptote.

In conclusion, determining the features of the graph of a given function is an essential aspect of understanding its behavior and characteristics. By analyzing the numerator and denominator of a rational function, we can determine the horizontal and vertical asymptotes, which provide valuable information about the function's graph.

Determine the horizontal and vertical asymptotes of the function:

${ f(x) = \frac{2}{x - 1} }$

Determine the horizontal and vertical asymptotes of the function:

${ f(x) = \frac{x + 1}{x^2 - 4} }$

The horizontal asymptote of the function is y = 0, and the vertical asymptote is x = 1.
The horizontal asymptote of the function is y = 0, and the vertical asymptote is x = 2.

Determine the horizontal and vertical asymptotes of the function:

${ f(x) = \frac{3}{2x + 1} }$

Determine the horizontal and vertical asymptotes of the function:

${ f(x) = \frac{x - 2}{x^2 + 1} }$

The horizontal asymptote of the function is y = 0, and the vertical asymptote is x = -1/2.
The horizontal asymptote of the function is y = 0, and there is no vertical asymptote.

In our previous article, we discussed how to determine the features of the graph of a given function, specifically the horizontal and vertical asymptotes. In this article, we will provide a Q&A section to help clarify any doubts and provide additional examples.

Q: What is a horizontal asymptote?

A: A horizontal asymptote is a horizontal line that the graph of a function approaches as x goes to positive or negative infinity.

Q: How do I determine the horizontal asymptote of a rational function?

A: To determine the horizontal asymptote of a rational function, you need to compare the degrees of the numerator and denominator. If the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote is y = 0.

Q: What is a vertical asymptote?

A: A vertical asymptote is a vertical line that the graph of a function approaches as x goes to a specific value.

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

A: To determine the vertical asymptote of a rational function, you need to find the values of x that make the denominator equal to zero.

Q: Can a rational function have multiple vertical asymptotes?

A: Yes, a rational function can have multiple vertical asymptotes if the denominator has multiple factors that equal zero.

Q: How do I determine the horizontal and vertical asymptotes of a function with a quadratic denominator?

A: To determine the horizontal and vertical asymptotes of a function with a quadratic denominator, you need to factor the denominator and find the values of x that make it equal to zero. Then, you can determine the horizontal asymptote by comparing the degrees of the numerator and denominator.

Q: Can a function have a horizontal asymptote and a vertical asymptote at the same time?

A: Yes, a function can have a horizontal asymptote and a vertical asymptote at the same time. For example, the function f(x) = 1/(x-1) has a horizontal asymptote at y = 0 and a vertical asymptote at x = 1.

Q: How do I determine the horizontal and vertical asymptotes of a function with a rational numerator and denominator?

A: To determine the horizontal and vertical asymptotes of a function with a rational numerator and denominator, you need to simplify the function by dividing the numerator and denominator by their greatest common factor. Then, you can determine the horizontal and vertical asymptotes using the methods discussed earlier.

Q: Can a function have a horizontal asymptote and no vertical asymptote?

A: Yes, a function can have a horizontal asymptote and no vertical asymptote. For example, the function f(x) = 1/x has a horizontal asymptote at y = 0 and no vertical asymptote.

Q: How do I determine the horizontal and vertical asymptotes of a function with a polynomial numerator and denominator?

A: To determine the horizontal and vertical asymptotes of a function with a polynomial numerator and denominator, you need to compare the degrees of the numerator and denominator. If the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote is y = 0. Then, you can determine the vertical asymptote by finding the values of x that make the denominator equal to zero.