find ﻿an ﻿equation ﻿with ﻿a ﻿vertical ﻿asymptote ﻿at ﻿x
=
1
,
a ﻿hole ﻿at ﻿x
=
4/3
,
and ﻿an ﻿oblique ﻿asymptote ﻿at
-
1/5
x
-
1

We are tasked with constructing a rational function that has:

1. A **vertical asymptote** at \( x = 1 \),
2. A **hole** at \( x = \frac{4}{3} \), and
3. An **oblique asymptote** at \( y = -\frac{1}{5}x - 1 \).

### Steps to construct the equation:

#### 1. Vertical Asymptote at \( x = 1 \)
A vertical asymptote at \( x = 1 \) suggests that the denominator of the rational function has a factor \( (x - 1) \).

#### 2. Hole at \( x = \frac{4}{3} \)
A hole at \( x = \frac{4}{3} \) implies that both the numerator and denominator of the function have a common factor \( \left( x - \frac{4}{3} \right) \), which cancels out and causes a hole.

#### 3. Oblique Asymptote at \( y = -\frac{1}{5}x - 1 \)
For an oblique asymptote, the degree of the numerator must be exactly one more than the degree of the denominator. The function will approach the asymptote as \( x \to \infty \) or \( x \to -\infty \). We will choose the numerator so that, upon long division with the denominator, the quotient matches the oblique asymptote \( y = -\frac{1}{5}x - 1 \).

### General Form of the Function
The general form of a rational function with these characteristics can be written as:

\[
f(x) = \frac{(x - \frac{4}{3})(Ax + B)}{(x - 1)(x - \frac{4}{3})}
\]

The factors \( (x - \frac{4}{3}) \) cancel out, leaving a hole at \( x = \frac{4}{3} \). We now simplify:

\[
f(x) = \frac{Ax + B}{x - 1}
\]

We need to determine \( A \) and \( B \) such that the oblique asymptote is \( y = -\frac{1}{5}x - 1 \). This means that as \( x \to \infty \), the function should behave like:

\[
f(x) \approx -\frac{1}{5}x - 1
\]

#### Long Division Setup
If we divide \( \frac{Ax + B}{x - 1} \), the quotient must equal \( -\frac{1}{5}x - 1 \). Thus, we set:

\[
\frac{Ax + B}{x - 1} \approx -\frac{1}{5}x - 1
\]

By comparing the coefficients, we see that \( A = -\frac{1}{5} \) and \( B = -1 \).

Thus, the rational function is:

\[
f(x) = \frac{-\frac{1}{5}x - 1}{x - 1}
\]

### Final Equation
The equation of the function is:

\[
f(x) = \frac{-\frac{1}{5}x - 1}{x - 1}
\]

This function has:
- A vertical asymptote at \( x = 1 \),
- A hole at \( x = \frac{4}{3} \) (due to the cancellation of terms, though the factor is no longer visible), and
- An oblique asymptote at \( y = -\frac{1}{5}x - 1 \).

Would you like further clarification or details on any part?

---

### Related Questions:
1. How can you find horizontal asymptotes for rational functions?
2. What is the process to determine slant asymptotes through polynomial division?
3. Can a function have more than one vertical asymptote? How is this determined?
4. What causes holes in a rational function and how can you identify them?
5. How do the degrees of the numerator and denominator affect the type of asymptote?

**Tip**: When finding an oblique asymptote, always perform polynomial long division to identify the behavior of the function as \( x \to \infty \).

find an equation with a vertical asymptote at x = 1, a hole at x = 4/3, and an oblique asymptote at -1/5x - 1

Learn how to find a rational function with a vertical asymptote at x = 1, a hole at x = 4/3, and an oblique asymptote at y = -1/5x - 1. This problem covers rational functions, asymptotes, and polynomial long division, suitable for students in Grades 11-12.

Math Problem Statement

find an equation with a vertical asymptote at x

1 , a hole at x

4/3 , and an oblique asymptote at

1/5 x

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