https://file.aiquickdraw.com/mathbot/file/1728008980656nvj3yebz.jpg

The problem asks for the image of the function \( f(x) = 2(3x - 5) + 1 \) under the translation given by the vector \( \left( \frac{-2}{3}, 3 \right) \).

To solve this:

### Step 1: Understand the function
The function is:
\[
f(x) = 2(3x - 5) + 1 = 6x - 10 + 1 = 6x - 9
\]
So the function simplifies to:
\[
f(x) = 6x - 9
\]

### Step 2: Apply the translation
The translation vector is \( \left( \frac{-2}{3}, 3 \right) \), which means the transformation will shift the graph of the function by \( \frac{-2}{3} \) units in the x-direction and 3 units in the y-direction.

1. **For the x-direction**: We substitute \( x + \frac{-2}{3} \) into the function to translate it horizontally:
   \[
   f\left(x + \frac{-2}{3}\right) = 6\left(x + \frac{-2}{3}\right) - 9
   \]
   Simplifying:
   \[
   f\left(x + \frac{-2}{3}\right) = 6x + 6\left(\frac{-2}{3}\right) - 9 = 6x - 4 - 9 = 6x - 13
   \]

2. **For the y-direction**: We add 3 to the result to translate it vertically:
   \[
   f_{\text{translated}}(x) = (6x - 13) + 3 = 6x - 10
   \]

### Final Result:
The image of the function under the translation is:
\[
f_{\text{translated}}(x) = 6x - 10
\]

Would you like further details or have any questions?

Here are 5 related questions you might find helpful:
1. How do you apply translations to functions?
2. What is the general formula for translating a linear function?
3. How does a vertical shift affect the function’s graph?
4. What is the effect of a horizontal translation on the graph of a linear function?
5. Can translations be applied to any type of function?

**Tip**: When translating functions, always apply horizontal shifts to the variable \(x\) inside the function and vertical shifts outside the function.

Bayangan fungsi f(x) = 2(3x - 5) + 1 oleh translasi \left( \frac{-2}{3}, 3 \right) adalah

Learn how to apply a translation to the linear function f(x) = 6x - 9 with a detailed explanation. This example covers translating functions horizontally and vertically, essential 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