Translasi Vertikal Hal 1-13 Bab 1 TRANSFORMASI FUNGSI Kelas 12 SMA SMK Kurikulum Merdeka

Ruang Pintar

22 Jul 202314:27

Summary

TLDRThis educational video script focuses on the concept of function transformation, specifically vertical translation, in the context of the Indonesian Merdeka curriculum for 12th-grade students. It explains the mathematical process of translating points on a graph by a certain distance, using the example of a linear function y = 2x being translated upwards to y = 2x + 3. The script also discusses the effects of vertical translation on the graph of quadratic functions and provides practical examples, such as modeling the growth of bacteria and the price change of masks during the COVID-19 pandemic. It concludes with exercises using GeoGebra to visualize the translations.

Takeaways

📚 The lesson focuses on vertical transformations of functions, specifically translations, as part of the 'Merdeka' curriculum for 12th-grade high school students.
🔍 Translation is defined as a transformation that moves points in a certain direction and distance, which can be represented as a shift in the x and y coordinates.
📈 In matrix form, the translation of a point (x, y) by (a, b) results in a new point (x', y') where x' = x + a and y' = y + b.
📉 For linear functions, a vertical shift upwards or downwards changes the equation by adding or subtracting the shift value to the original equation.
📊 The script provides examples of vertical shifts for linear functions, such as changing from y = 2x to y = 2x + 3 when shifted upwards by 3 units.
📐 The script explains how to determine whether a graph has shifted upwards or downwards by comparing the original and new equations of the functions.
📘 It also discusses the vertical translation of quadratic functions, using the example of y = x² + 1 being translated to y = x² - 2 by subtracting 3 from the original function.
📌 The script uses GeoGebra to illustrate the graphical representation of function translations, showing the shift of the graphs in the coordinate plane.
🔢 The lesson includes practical examples, such as modeling the growth of bacteria with the function y = 2^x and its translation to y = 2^x + 1 after treatment.
📝 The script provides step-by-step instructions on how to translate a given linear equation, like 8X - 4y + 16 = 0, by a certain amount and determine the new equation and graph.
🤔 The importance of critical thinking is emphasized through the script, encouraging students to observe and analyze changes in function graphs after translations.

Q & A

What is the main topic of the video script?
-The main topic of the video script is the concept of function transformation, specifically focusing on vertical translations in the context of mathematical functions.
What is a translation in the context of mathematical functions?
-A translation in the context of mathematical functions is a transformation that moves points in a certain direction and distance, also known as a shift.
How is a point translated in a function?
-A point is translated by shifting it by a certain distance 'a' horizontally and 'b' vertically, resulting in a new point with coordinates (x', y') where x' = x + a and y' = y + b.
What is the effect of vertical translation on the equation of a linear function?
-Vertical translation affects the constant term in the equation of a linear function. For example, if the original function is y = 2x and it is translated upwards by 3 units, the new equation becomes y = 2x + 3.
How does a positive vertical translation change the graph of a function?
-A positive vertical translation moves the graph of a function upwards. For instance, translating y = 2x upwards by 5 units results in the graph of y = 2x + 5.
What happens to the graph of a function when it is translated vertically by a negative value?
-When a function is translated vertically by a negative value, its graph moves downwards. For example, translating y = 2x by -3 units results in the graph of y = 2x - 3.
What is the relationship between the original function y = 2x + 4 and its translated version y = 2x + 6?
-The translated version y = 2x + 6 is obtained by adding 2 to the original function y = 2x + 4, indicating a vertical translation upwards by 2 units.
How does the script explain the vertical translation of quadratic functions?
-The script explains that a quadratic function, such as y = x^2 + 1, can be translated vertically by subtracting a value from it, resulting in a new function like y = x^2 - 2, which represents a downward shift by 3 units.
What is the significance of the term 'b' in the context of vertical translation of functions?
-The term 'b' in the equation y = f(x) + b represents the vertical shift of the function. If 'b' is positive, the graph shifts upwards, and if 'b' is negative, the graph shifts downwards.
Can you provide an example from the script where a real-world scenario is modeled using vertical translation of functions?
-Yes, the script provides an example of a real-world scenario where the growth of bacteria after treatment is modeled. The original model is y = 2^x, and after treatment, it becomes y = 2^x + 1, indicating a vertical translation upwards by 1 unit.
How does the script use the concept of vertical translation to solve a problem related to a mask offer during a pandemic?
-The script models the increasing demand for masks during the COVID-19 pandemic with the linear equation 8x - 4y + 16 = 0. After 8 days, the model changes due to translation, resulting in the new equation y = 2x + 4 + 8, which represents a vertical translation upwards by 8 units.
What is the new equation obtained after translating the function y = x^2 - 2x - 8 by 4 units upwards?
-After translating the function y = x^2 - 2x - 8 upwards by 4 units, the new equation becomes y' = x^2 - 2x - 4, which is the result of adding 4 to the original function.

Outlines

00:00

📚 Introduction to Function Transformations

This paragraph introduces the topic of function transformations, specifically focusing on vertical translations. It explains the concept of translation in mathematics as a change in position or size of an object, which can be a point, line, curve, or area. The paragraph uses the example of a point 'a' being translated by 'AB' to result in a new point 'a'', and how this can be represented in matrix form with the equation x' = x + a and y' = y + b. It also illustrates the effect of vertical translation on the linear function y = 2x, showing how adding a constant 'b' results in a vertical shift upwards or downwards of the graph. The summary also covers the graphical representation of two different linear functions before and after translation, emphasizing the change in the equation and the corresponding shift in the graph.

05:02

📈 Vertical Translations and Their Impact on Graphs

The second paragraph delves deeper into vertical translations, providing examples of how linear and quadratic functions are affected by these transformations. It discusses the change in the equation of a linear function from y = 2x + 4 to y' = 2x + 6, indicating a two-unit upward shift. The paragraph also examines the translation of a quadratic function from y = x^2 + 1 to y' = x^2 - 2, which represents a downward shift. The summary explains the general rule that adding a positive value 'b' to the function results in an upward shift of the graph, while subtracting (or having a negative 'b') causes a downward shift. It also includes an example of modeling bacterial growth with the function y = 2^x and how subsequent treatment led to a new model y = 2^x + 1, indicating an upward shift in the graph.

10:04

📉 Geogebra Demonstrations and Example Problems

The final paragraph discusses the use of Geogebra to visualize the effects of vertical translations on graphs. It provides a step-by-step guide on how to plot the functions y = 2^x and y = 2^x + 1 using Geogebra, highlighting the one-unit upward shift of the graph. The summary also presents two example problems involving the translation of linear and quadratic equations. The first problem involves translating the equation 8X - 4y + 16 = 0 by eight units upwards, resulting in the new equation y' = 2x + 12. The second problem involves translating the quadratic equation y = x^2 - 2x - 8 by four units upwards, leading to the new equation y' = x^2 - 2x - 4. The paragraph concludes with a demonstration of these translations using Geogebra, showing the graphical representation of the original and translated equations.

Mindmap

Keywords

💡Transformation

Transformation refers to the process of altering the position, size, shape, or orientation of an object in space. In the context of the video, it specifically addresses the concept of function transformation, which is a mathematical operation applied to functions to modify their graphs. An example from the script is the vertical translation of the function y = 2x, resulting in y = 2x + 3, which graphically represents the original line moving upwards.

💡Translating

Translating, in the script, is a type of transformation that moves points in a specific direction and distance without altering their shape or size. It is a fundamental concept in the study of function transformations. For instance, the script describes the translation of a point 'a' to 'a'' by a vector 'AB', resulting in a new point 'a'' with coordinates (x', y') where x' = x + a and y' = y + b.

💡Vertical Translation

Vertical translation is a specific type of translating that moves a function or an object vertically up or down on a graph. It is a key concept in the video, as it demonstrates how the graph of a function changes when it is shifted along the y-axis. The script uses the example of the function y = 2x being translated upwards by 3 units to form y = 2x + 3.

💡Linear Function

A linear function is a mathematical function that represents a straight line when graphed. In the video, linear functions are used to illustrate the concept of vertical translation. For example, the script discusses the transformation of the linear function y = 2x + 4 into y = 2x + 6 by adding 2, which represents a vertical translation upwards.

💡Quadratic Function

A quadratic function is a polynomial function of degree two. Its graph is a parabola. The script mentions the quadratic function y = x^2 + 1, which is then translated to form y = x^2 - 2, demonstrating how the graph of a quadratic function can be shifted vertically.

💡Matrix Form

In the context of the video, the matrix form is used to represent the translation of points in a coordinate system. The script explains that the new coordinates (x', y') can be calculated from the original coordinates (x, y) by adding the translation vector components (a, b), which is represented as x' = x + a and y' = y + b.

💡Constant

In the script, constants are numerical values that remain unchanged during the transformation process. They are used to define the extent of the translation. For example, when the function y = 2x is translated by a constant 'a', the new function becomes y = 2x + a, where 'a' is the constant value added to the original function.

💡Variable

Variables in the context of the video represent the elements that can change within a function. They are crucial in understanding how functions are transformed. The script contrasts the variables in y = 2x with those in y = 2x + 5, where 'x' is the variable that remains consistent, while the transformation affects the constant term.

💡GeoGebra

GeoGebra is a dynamic mathematics software that is used for various mathematical tasks, including graphing functions and understanding geometric concepts. The script mentions using GeoGebra to illustrate the graphical representation of function transformations, such as the translation of y = 2^x to y = 2^x + 1.

💡Pandemic

The term pandemic is used in the script to describe a scenario where the demand for masks increases with the rising price during the COVID-19 pandemic. This real-world context is used to model a linear function that is then transformed, demonstrating the application of mathematical concepts to real-life situations.

💡Example Problem

The script provides example problems to illustrate the application of vertical translation in different mathematical contexts. These problems, such as translating the line y = x^2 - 2x - 8 by 4 units upwards, help to solidify the understanding of the concept by showing its practical use in solving mathematical questions.

Highlights

Introduction to the concept of function transformation with a focus on vertical translation.

Explanation of translation as a change in position or size of an object, including points, lines, curves, and areas.

Description of vertical translation as moving points by a certain direction and distance, also known as shifting.

Formula representation of vertical translation in matrix form as x' = x + a and y' = y + b.

Example of translating a linear function y = 2x upwards by 3 units to become y = 2x + 3.

Illustration of the graphical shift of a line in a function due to vertical translation.

Concept that vertical translation involves adding or subtracting a constant to the y-coordinate of each point on the function.

Demonstration of how the graph of y = 2x + 10 is shifted 10 units upwards compared to y = 2x.

Explanation of how a negative translation affects the function, as shown by y = 2x - 3.

Comparison of two linear functions y = 2x + 4 and y = 2x - y + 6 = 0 before and after vertical translation.

Graphical representation of the vertical translation of a quadratic function from y = x^2 + 1 to y = x^2 - 2.

Use of GeoGebra to visually demonstrate the effect of vertical translation on the graph of exponential functions.

Application of vertical translation in real-world scenarios, such as modeling the growth of bacteria after treatment.

Solution to an example problem involving the translation of a linear equation representing mask demand during a pandemic.

Explanation of how to translate a quadratic function y = x^2 - 2x - 8 by adding 4 to obtain y' = x^2 - 2x - 4.

GeoGebra demonstration of the graphical shift of a quadratic function due to vertical translation.

Conclusion summarizing the key points of vertical translation in function transformation.