latihan 1.2 transformasi fungsi matematika kelas XII

Introduction

In this tutorial, we will explore various transformations of mathematical functions as discussed in the video "latihan 1.2 transformasi fungsi matematika kelas XII." We will cover reflection over axes, transformations, and graphical representations. This guide will help you understand how to manipulate functions algebraically and visually, which is essential in advanced mathematics.

Step 1: Transforming the Function y = x² + 2x – 1

To start, we will determine the resulting functions when transformations are applied to the function y = x² + 2x – 1.

Transformations:

1. Reflection over the x-axis:

   • The function becomes:
     y = -f(x)
     Therefore, the new function is:
     y = -(x² + 2x – 1) = -x² - 2x + 1

2. Reflection over the y-axis:

   • The function becomes:
     y = -f(-x)
     Therefore, the new function is:
     y = -((-x)² + 2(-x) – 1) = -x² + 2x + 1

Step 2: Graphing the Function f(x) = x(x + 1)(x – 2)

Given the function f(x) = x(x + 1)(x – 2), we will graph the transformations.

Transformations to Graph:

1. Graph of 1 + f(-x):

   • Substitute -x into the function:
     f(-x) = -x(-x + 1)(-x - 2)
     Simplifying gives you a new function.
   • Then you add 1 to this result.

2. Graph of -f(x + 5):

   • Substitute (x + 5) into the function:
     f(x + 5) = (x + 5)((x + 5) + 1)((x + 5) - 2)
     Simplify this expression, then negate it to get -f(x + 5).

Step 3: Reflecting the Function y = (x + 2)⁴ + 1

To determine the results of reflections on the function:

Steps for Reflection:

1. Horizontal Reflection:

   • Replace x with -x:
     The resulting function is:
     y = ((-x) + 2)⁴ + 1 = (2 - x)⁴ + 1

2. Vertical Reflection:

   • Negate the function:
     The resulting function is:
     y = -((x + 2)⁴ + 1) = -(x + 2)⁴ - 1

Step 4: Additional Transformations

Now we will determine the equations after applying specified transformations.

Transformations:

1. Reflect y = 2x⁴ over the x-axis:

   • Resulting function:
     y = -2x⁴

2. Reflect y = 2x + 1 – 3 over the y-axis:

   • Resulting function:
     y = 2(-x) + 1 – 3 = -2x - 2

3. Reflect y = 3 – x + 1 over the x-axis:

   • Resulting function:
     y = - (3 - x + 1) = -3 + x - 1 = x - 4

Conclusion

In this tutorial, we covered how to transform and reflect functions mathematically. We explored transformations such as reflections over the x-axis and y-axis, and practiced graphing transformed functions. Understanding these concepts is crucial for higher-level mathematics and can be applied in various fields, including physics and engineering. For further practice, try applying these transformations to other functions and sketch their graphs.