Transformasi Fungsi Y = 6x - 15: Solusi Lengkap!

Alright, teman-teman! Let's dive into the exciting world of function transformations! We've got a function, y = 6x - 15, and we're going to see what happens when we apply different transformations to it. Get ready, because we're about to embark on a mathematical journey that will test our skills and boost our understanding of how functions behave under various operations.

Memahami Transformasi Fungsi

Before we jump into the specific transformations, let's take a moment to understand what transformations are all about. Transformations are operations that change the position, size, or shape of a function's graph. They allow us to manipulate functions in predictable ways, which is super useful in various fields like computer graphics, physics, and engineering. Understanding these transformations will give you a powerful toolkit for analyzing and manipulating functions.

Jenis-Jenis Transformasi

There are several types of transformations we can apply to a function, including:

• Translasi (Translation): Moving the function horizontally or vertically without changing its shape.
• Refleksi (Reflection): Flipping the function over a line, like a mirror image.
• Rotasi (Rotation): Turning the function around a point.
• Dilatasi (Dilation): Stretching or compressing the function, changing its size.

Now that we have a basic understanding of transformations, let's tackle each part of the question step-by-step.

a) Translasi Horizontal dan Rotasi

Translasi Horizontal 5 Satuan ke Kanan

First, we're translating the function horizontally 5 units to the right. What does this mean? It means we're shifting the entire graph of the function 5 units to the right along the x-axis. To achieve this, we replace x with (x - 5) in the original function. So, our new function becomes:

y = 6(x - 5) - 15

y = 6x - 30 - 15

y = 6x - 45

So, after the horizontal translation, the function is now y = 6x - 45. Easy peasy, right? This transformation simply slides the graph to the right, changing the x-intercept but keeping the slope the same. It's a fundamental concept in understanding how functions move around the coordinate plane.

Rotasi Berlawanan Arah Jarum Jam

Next, we need to rotate the function counter-clockwise. But wait! We need more information to perform a rotation. Specifically, we need to know:

• The center of rotation: Around which point are we rotating the function?
• The angle of rotation: By how many degrees are we rotating the function?

Without this information, we can't give a specific equation for the rotated function. However, let's talk about the general process. Rotating a function usually involves using rotation matrices, which are a bit complex. If we were rotating around the origin (0,0) by an angle θ, we'd need to apply the following transformations:

x' = x * cos(θ) - y * sin(θ)

y' = x * sin(θ) + y * cos(θ)

But since we don't have the angle and center of rotation, we can't proceed further with this step. If you can provide the angle and center of rotation, I can help you complete this part.

b) Refleksi terhadap Garis y = 2 dan Dilatasi Vertikal

Refleksi terhadap Garis y = 2

Now, let's reflect the original function y = 6x - 15 over the line y = 2. Reflecting over a horizontal line involves changing the y-values in a specific way. Here’s how we do it:

1. Find the distance from the original y-value to the line of reflection (y = 2).

   Distance = |y - 2| = |(6x - 15) - 2| = |6x - 17|

2. Subtract twice this distance from the line of reflection to get the new y-value.

   y' = 2 - 2(6x - 17)

   y' = 2 - 12x + 34

   y' = -12x + 36

So, after reflecting over the line y = 2, the function becomes y' = -12x + 36. Reflecting a function over a horizontal line essentially flips the graph vertically across that line. The new function has a different slope and y-intercept compared to the original, but it maintains the same basic linear form.

Dilatasi Vertikal dengan Faktor Skala 5

Next, we're applying a vertical dilation with a scale factor of 5. This means we're stretching the function vertically by a factor of 5. To do this, we multiply the entire function by 5:

y'' = 5 * (-12x + 36)

y'' = -60x + 180

Thus, after the vertical dilation, the function is now y'' = -60x + 180. A vertical dilation stretches the graph away from the x-axis, making it steeper if the scale factor is greater than 1. In this case, the slope becomes significantly steeper, and the y-intercept is also multiplied by the scale factor.

c) Refleksi terhadap Sumbu Y dan Translasi

Refleksi terhadap Sumbu Y

To reflect the original function y = 6x - 15 over the y-axis, we replace x with -x. This flips the function horizontally across the y-axis. So, our new function becomes:

y = 6(-x) - 15

y = -6x - 15

After reflecting over the y-axis, the function is now y = -6x - 15. The slope of the function changes its sign, but the y-intercept remains the same. This transformation essentially mirrors the graph across the vertical axis.

Translasi

Finally, we need to translate the function. But again, the question doesn't specify the direction or distance of the translation. To perform a translation, we need to know:

• Horizontal translation: How many units to the left or right?
• Vertical translation: How many units up or down?

If we have this information, we can apply the translation as follows:

• Horizontal translation by h units: Replace x with (x - h). If h is positive, it's a translation to the right; if h is negative, it's a translation to the left.
• Vertical translation by k units: Add k to the function. If k is positive, it's a translation upward; if k is negative, it's a translation downward.

For example, if we translate the function 2 units to the right and 3 units up, the new function would be:

y' = -6(x - 2) - 15 + 3

y' = -6x + 12 - 15 + 3

y' = -6x

So, the final transformed function would be y' = -6x. Remember, without specific translation values, we can only provide the general process. With the translation details, getting the final equation will be a breeze.

Kesimpulan

We've successfully navigated through the various transformations of the function y = 6x - 15. We've seen how translations, reflections, and dilations affect the function's equation and graph. While some parts required additional information (like the rotation angle and center, and the translation values), we covered the general principles and steps involved. Keep practicing these transformations, and you'll become a master of function manipulation in no time!

Remember: practice makes perfect! The more you work with these transformations, the more comfortable and confident you'll become. So, keep exploring, keep experimenting, and keep having fun with math!