https://file.aiquickdraw.com/mathbot/file/1728009847893zbqll2h6.jpg

The problem provided in the image asks for the result of translating the function \( f(x) = 2(3x - 5) + 1 \) by a translation vector \( \left( \frac{-2}{3} \right) \).

### Step-by-step explanation:

1. **Original function**:  
   \( f(x) = 2(3x - 5) + 1 \)
   Simplifying it:
   \[
   f(x) = 6x - 10 + 1 = 6x - 9
   \]
   So, the function is now \( f(x) = 6x - 9 \).

2. **Translation vector**:  
   The translation vector \( \left( \frac{-2}{3} \right) \) suggests a translation in both the \(x\)-direction and the \(y\)-direction. The exact form is unclear from the image, but generally, a translation applies changes to both the \(x\) and \(y\) coordinates.

3. **Impact on the function**:
   A translation affects the function based on how the vector modifies \( x \) and \( y \). If we are translating the input \(x\)-value by \( \frac{-2}{3} \), this affects \(x\) in the following way:
   
   \( f(x) \) is transformed to \( f\left( x - \frac{-2}{3} \right) = f\left( x + \frac{2}{3} \right) \).

4. **Applying the translation**:
   We substitute \( x + \frac{2}{3} \) into the function:
   \[
   f\left( x + \frac{2}{3} \right) = 6\left( x + \frac{2}{3} \right) - 9
   \]
   Simplifying the expression:
   \[
   = 6x + 4 - 9 = 6x - 5
   \]

So, after the translation, the new function becomes \( f(x) = 6x - 5 \).

### Final Answer:
The image of the function \( f(x) \) under the translation by \( \left( \frac{-2}{3} \right) \) is \( f(x) = 6x - 5 \).

---

Do you want further details or have any questions?

Here are some related questions to expand on the topic:
1. How do different translations affect linear functions like this one?
2. What happens if the translation vector affects both \(x\) and \(y\)?
3. Can we apply translations to non-linear functions similarly?
4. How does a translation in the \(y\)-direction only affect a function?
5. What are the general rules for translating functions geometrically?

**Tip**: When applying translations, be careful to adjust the input \(x\)-value in accordance with the vector components.

Bayangan fungsi f(x) = 2(3x - 5) + 1 oleh translasi (-2/3) adalah?

Understand how to translate the function f(x) = 2(3x - 5) + 1 by the translation vector (-2/3). This problem covers key concepts in function transformation and translation, with step-by-step solutions. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation