TRANSFORMASI FUNGSI (MENCARI FUNGSI HASIL TRANSLASI)

Introduction

This tutorial will guide you through the process of transforming functions by finding the resulting function after translation. This concept is essential in understanding how functions behave under transformations, which is a significant topic in algebra and calculus. By the end of this guide, you will be able to apply these transformations to various functions effectively.

Step 1: Understanding Function Translation

Function translation involves shifting a function horizontally or vertically on a graph. Here are the key concepts:

• Horizontal Translation: This occurs when you add or subtract a value from the input (x) of the function.

  • Example: If you have a function f(x), the translation f(x - h) shifts the graph to the right by h units, while f(x + h) shifts it left by h units.

• Vertical Translation: This happens when you add or subtract a value from the output (f(x)).

  • Example: f(x) + k shifts the graph up by k units, while f(x) - k shifts it down by k units.

Practical Tips

• Always note the direction of the shift: left/right for horizontal and up/down for vertical.
• Keep in mind that adding a positive number shifts the graph up/right, while subtracting shifts it down/left.

Step 2: Applying Horizontal Translation

To apply a horizontal translation to a given function, follow these steps:

1. Identify the original function f(x).
2. Decide the value of h you want to use for translation.
3. Write the new function using the horizontal translation formula:
   • For a shift to the right: f(x - h)
   • For a shift to the left: f(x + h)

Example

If f(x) = x^2 and you want to shift it right by 3 units:

• New function: f(x - 3) = (x - 3)^2

Step 3: Applying Vertical Translation

To perform a vertical translation, follow these steps:

1. Start with your original function f(x).
2. Choose a value for k.
3. Write the new function using the vertical translation formula:
   • For a shift upward: f(x) + k
   • For a shift downward: f(x) - k

Example

Using the same function f(x) = x^2 and shifting it up by 2 units:

• New function: f(x) + 2 = x^2 + 2

Step 4: Combining Translations

You can combine both horizontal and vertical translations in one function. Here’s how:

1. Start with the original function f(x).
2. Apply horizontal translation first, followed by vertical translation.
3. Combine the transformations into one function:
   • New function: f(x - h) + k

Example

Starting with f(x) = x^2, shifting right by 3 and up by 2:

• New function: f(x - 3) + 2 = (x - 3)^2 + 2

Conclusion

In this tutorial, you learned how to transform functions through horizontal and vertical translations. Remember to identify the original function, decide on your translation values, and apply the transformations step by step. With practice, you’ll be able to visualize these changes on a graph and apply them to various functions easily. For further learning, explore more complex functions and practice combining multiple translations.