Transformasi Fungsi #Part 2// Dilatasi Fungsi // Perubahan Grafik Fungsi Akibat Dilatasi/Perkalian

Summary

In this video, the concept of function dilation is explained, covering how functions can be transformed through scaling. Two types of dilation are discussed: dilation with a center at (0,0) and dilation with a center at another point, such as (1,2). The video demonstrates how to apply scaling factors to functions, using examples with linear and quadratic functions. Visualizing the transformations and understanding the mathematical principles behind the dilation process are key aspects of the lesson, ultimately showing how objects either expand or shrink in relation to a chosen center and scale.

Takeaways

• 😀 Dilatation is a transformation where a function is multiplied by a constant, in contrast to translation which involves adding or subtracting a constant.
• 😀 There are two types of dilatation: one with a center at (0,0), and the other with a center at any point (P).
• 😀 For dilatation centered at (0,0), the transformation matrix is scaled by a constant factor, resulting in a transformation of points to new coordinates.
• 😀 When the dilatation center is not at (0,0), the transformation matrix includes shifts based on the coordinates of the center (P).
• 😀 The scale factor in dilatation indicates how much the object is enlarged or reduced. A scale greater than 1 enlarges, while a scale smaller than 1 reduces the object.
• 😀 A scale factor of 3 means the object is enlarged three times, whereas a scale of 1/3 reduces the size of the object by three times.
• 😀 Dilatation affects the curve by either expanding it away from the center or compressing it toward the center.
• 😀 Example 1: For the function f(x) = 2x + 1, dilating with a center at (0,0) and a scale factor of 3 results in the transformed function f(x) = 2x + 3.
• 😀 Example 2: A quadratic function f(x) = x^2 + 2x + 1, when dilated with a center at (1,2) and a scale factor of 1/2, will become a more compressed curve towards the center.
• 😀 In both examples, the transformation involves adjusting the function's equation and visually changing the graph by either expanding or compressing the curve.

Q & A

What is the main focus of the video script?

The video focuses on explaining the concept of function dilation in mathematics, particularly how to dilate a function with a center of dilation at the origin (0,0) and at other points, as well as the effects of scaling on the graph of the function.

What is the difference between translation and dilation of a function?

Translation involves shifting the graph of a function by adding or subtracting constants to the function's x or y values. Dilation, on the other hand, involves multiplying the function's x or y values by a constant, resulting in a stretching or shrinking of the graph.

How is dilation with a center at (0,0) represented in notation?

Dilation with a center at (0,0) is represented as 'f(x) = ax', where 'a' is the scale factor, and the transformation results in a graph that is either stretched or compressed depending on the value of 'a'.

What happens to a graph when the scale factor is greater than 1?

When the scale factor is greater than 1, the graph is stretched away from the center of dilation (in this case, the origin). The points on the graph move further from the origin.

What happens to the graph if the scale factor is less than 1?

If the scale factor is less than 1, the graph is compressed toward the center of dilation (origin or any other specified point). The points on the graph move closer to the center of dilation.

How does dilation with a center at a point other than (0,0) work?

When the center of dilation is a point other than (0,0), such as (a,b), the transformation involves shifting the points relative to the new center, applying the scaling factor, and then translating the points back to the new center. The formula involves subtracting the coordinates of the center from the original coordinates before applying the scale.

How is the dilation with a center at (a,b) mathematically represented?

Dilation with a center at (a,b) is represented by the formula 'f(x, y) = K * (x - a, y - b)', where 'K' is the scale factor. After applying the transformation, the coordinates are shifted by adding back the center point (a,b).

What does the scale factor '3' mean in terms of dilation?

A scale factor of 3 means the graph of the function is enlarged three times in size. The points on the graph move three times further from the center of dilation.

In the example with the function f(x) = 2x + 1, how is the dilation with a scale factor of 3 performed?

In the dilation of the function f(x) = 2x + 1 with a scale factor of 3, each x-coordinate is multiplied by 3, and the function becomes f(x) = 3(2x + 1) = 6x + 3. This results in a transformed graph that is stretched away from the origin.

How does the dilation of the quadratic function f(x) = x^2 + 2x + 1 with a center at (1,2) and scale factor 1/2 affect its graph?

For the quadratic function f(x) = x^2 + 2x + 1 with a center at (1,2) and a scale factor of 1/2, the graph is compressed toward the center (1,2). The points on the graph move closer to the center, reducing the width of the parabola.