TRANSFORMASI FUNGSI TRANSLASI

Summary

The video explains how to translate mathematical equations and functions. It demonstrates shifting a linear equation, 2x - 3y = 4, one unit right and four units down, resulting in the new equation 2x - 3y = 18. The process involves substituting translated variables and simplifying the equation. Similarly, it shows translating a quadratic function, f(x) = x² - 4x + 3, two units left and five units up to obtain f'(x) = x² + 4. The tutorial emphasizes using substitution and careful simplification to derive the 'shadow' equation or function, illustrating translation techniques for both lines and parabolas in a clear, step-by-step approach.

Takeaways

• 📏 The equation of a line, such as 2x - 3y = 4, can be translated horizontally and vertically to produce a new 'shadow' line.
• ➡️ A horizontal shift of 1 unit to the right and a vertical shift of 4 units down transforms the original line to the shadow line 2x - 3y = 18.
• 🟦 Points on the original line (x, y) correspond to points on the shadow line (x', y') through a translation matrix.
• 🔄 The translation matrix allows calculation of new coordinates: x' = x + horizontal shift, y' = y + vertical shift.
• ✏️ To find the shadow line equation, replace original variables x and y with the translated variables x' and y' in the original equation.
• 🧮 After substitution, simplify the equation to get the final shadow line in standard form.
• 📐 The same method applies to functions like parabolas, e.g., f(x) = x² - 4x + 3, which remain the same shape but change position.
• ↔️ Horizontal and vertical shifts for functions are represented by adding or subtracting constants in the translation matrix (e.g., -2 units left, +5 units up).
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• 🔹 To derive the translated function, substitute x = x' + horizontal shift and y = y' - vertical shift into the original function.
• 🟢 Simplifying the resulting expression after substitution yields the shadow function in standard form, e.g., f'(x) = x² + 4.

Q & A

What is the original equation of the line discussed in the video?

The original equation of the line is 2x - 3y = 4.

How is a line translated horizontally and vertically according to the video?

A line is translated by adding or subtracting values to x and y coordinates using a translation matrix: x' = x + h for horizontal shift and y' = y + k for vertical shift, where h and k are the shifts.

What are the translation values used for the line in the example?

The line is translated 1 unit to the right and 4 units down.

How do you express the original variables in terms of the translated variables?

You solve the translation equations for the original variables: x = x' - h and y = y' - k. In the example, x = x' - 1 and y = y' + 4.

What is the equation of the translated (shadow) line after applying the translation?

The shadow line equation after translation is 2x - 3y = 18.

What is the original function of the parabola discussed in the video?

The original function of the parabola is f(x) = x² - 4x + 3.

What translation is applied to the parabola in the example?

The parabola is translated 2 units to the left and 5 units up.

How do you calculate the shadow function for a translated parabola?

Express the original variables in terms of the translated variables (x = x' + h, y = y' - k) and substitute these into the original function, then simplify to get the shadow function.

What is the resulting shadow function of the translated parabola?

The shadow function of the translated parabola is f'(x) = x² + 4.

Why is it necessary to substitute the translated variables back into the original equation or function?

Substituting the translated variables ensures that the new equation or function correctly represents the shifted line or curve, maintaining the correct relationship between x' and y'.

Can the method used for translating a line also be applied to other types of functions?

Yes, the method of using a translation matrix and substituting variables can be applied to any function or curve to calculate its translated (shadow) form.

What is the general formula for translating a point (x, y) by a vector (h, k)?

The general formula is x' = x + h and y' = y + k, where (h, k) are the horizontal and vertical translation distances.