https://file.aiquickdraw.com/mathbot/file/1729553397615qapcpafw.jpg

features are: horizontal asymptotes, vertical asymptotes, coordinates of holes, x and y intercepts

Let's break down the given function \( f(x) = \frac{4(x - 1)}{3(x - 1)^3} \) and determine each feature of the graph as requested:

### 1. **Simplifying the function**:
   The function simplifies as follows:
   \[
   f(x) = \frac{4(x - 1)}{3(x - 1)^3} = \frac{4}{3(x - 1)^2}
   \]
   Since the \( (x - 1) \) in the numerator and denominator cancel out, we are left with the simplified function:
   \[
   f(x) = \frac{4}{3(x - 1)^2}
   \]

   This will help us identify key features.

### 2. **Vertical Asymptote**:
   A vertical asymptote occurs where the denominator equals zero and the function is undefined. Setting the denominator to zero:
   \[
   3(x - 1)^2 = 0
   \]
   This gives \( x = 1 \). Therefore, there is a **vertical asymptote at \( x = 1 \)**.

### 3. **Horizontal Asymptote**:
   To determine the horizontal asymptote, observe the behavior of the function as \( x \to \infty \) or \( x \to -\infty \). The term \( (x - 1)^2 \) in the denominator grows large, making the whole fraction approach zero. Therefore, the function approaches:
   \[
   \lim_{x \to \pm \infty} f(x) = 0
   \]
   Thus, there is a **horizontal asymptote at \( y = 0 \)**.

### 4. **Holes**:
   Holes occur when a factor cancels out from both the numerator and denominator, making the function undefined at that point. However, after simplifying the original function, we see that \( x = 1 \) does not result in a hole, but instead leads to a vertical asymptote. **There are no holes** in this graph.

### 5. **X-intercept**:
   The x-intercept is found by setting \( f(x) = 0 \). However, in the simplified function, the numerator is a constant (4), meaning the function never equals zero. **There is no x-intercept**.

### 6. **Y-intercept**:
   To find the y-intercept, we evaluate the function at \( x = 0 \):
   \[
   f(0) = \frac{4}{3(0 - 1)^2} = \frac{4}{3}
   \]
   Therefore, the **y-intercept is at \( \left(0, \frac{4}{3} \right) \)**.

### Summary of Features:
1. **Vertical Asymptote**: \( x = 1 \)
2. **Horizontal Asymptote**: \( y = 0 \)
3. **Holes**: None
4. **X-intercept**: None
5. **Y-intercept**: \( \left(0, \frac{4}{3} \right) \)

Would you like more details on any of these features?

Here are five related questions to expand this topic:
1. How would the function change if the numerator were \( 4(x - 2) \) instead of \( 4(x - 1) \)?
2. What would happen to the vertical asymptote if the denominator were \( (x - 2)^3 \)?
3. Can a function have multiple vertical asymptotes? Provide an example.
4. What is the significance of a hole in a function graph, and how does it differ from an asymptote?
5. How can we find oblique asymptotes for rational functions, and when do they occur?

**Tip**: Always start by simplifying a rational function to better understand its behavior and features like asymptotes and intercepts.

Determine each feature of the graph of the function f(x) = (4(x - 1))/(3(x - 1)^3), including horizontal asymptotes, vertical asymptotes, coordinates of holes, x and y intercepts.

Analyze the rational function f(x) = 4(x-1) / 3(x-1)^3 to determine vertical asymptotes, horizontal asymptotes, x and y intercepts, and holes. Understand key algebraic concepts related to rational functions, including asymptotes and intercepts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation