Math Problem Statement

Diketahui fungsi f(x) = 2x + 3, tentukan bayangan fungsi f(x) dari a. Translasi oleh (2,3), b. Refleksi terhadap garis x = 1, c. Dilatasi dengan pusat (0,0) dan skala 4

Solution

We are given the function f(x)=2x+3f(x) = 2x + 3 and need to determine its transformed images based on three transformations:

  1. Translation by the vector (23)\begin{pmatrix} 2 \\ 3 \end{pmatrix}.
  2. Reflection about the line x=1x = 1.
  3. Dilation with center at (0,0)(0,0) and scale factor of 4.

Let’s go through each part in detail:

a. Translation by (23)\begin{pmatrix} 2 \\ 3 \end{pmatrix}:

A translation shifts the function horizontally by 2 units and vertically by 3 units.

For a translation (hk)\begin{pmatrix} h \\ k \end{pmatrix}, the function becomes: f(xh)+kf(x-h) + k Substituting h=2h = 2 and k=3k = 3, the transformed function is: f(x2)+3f(x-2) + 3 Now, substituting f(x)=2x+3f(x) = 2x + 3 into this expression: f(x2)=2(x2)+3=2x4+3=2x1f(x-2) = 2(x-2) + 3 = 2x - 4 + 3 = 2x - 1 Adding 3: f(x2)+3=2x1+3=2x+2f(x-2) + 3 = 2x - 1 + 3 = 2x + 2 Thus, the new function after translation is 2x+22x + 2.

b. Reflection about the line x=1x = 1:

Reflection about a vertical line x=cx = c changes the function as follows: f(x)f(2cx)f(x) \rightarrow f(2c - x) For reflection about x=1x = 1, this becomes: f(2×1x)=f(2x)f(2 \times 1 - x) = f(2 - x) Substituting f(x)=2x+3f(x) = 2x + 3: f(2x)=2(2x)+3=42x+3=72xf(2 - x) = 2(2 - x) + 3 = 4 - 2x + 3 = 7 - 2x So, the reflected function is 72x7 - 2x.

c. Dilation with center (0,0) and scale factor 4:

For a dilation with center (0,0)(0,0) and scale factor kk, the transformation is given by: f(x)f(xk)f(x) \rightarrow f\left(\frac{x}{k}\right) For k=4k = 4, the function becomes: f(x4)f\left(\frac{x}{4}\right) Substituting f(x)=2x+3f(x) = 2x + 3: f(x4)=2(x4)+3=12x+3f\left(\frac{x}{4}\right) = 2\left(\frac{x}{4}\right) + 3 = \frac{1}{2}x + 3 Thus, the new function after dilation is 12x+3\frac{1}{2}x + 3.

Summary of Results:

  • a) Translated function: 2x+22x + 2
  • b) Reflected function: 72x7 - 2x
  • c) Dilated function: 12x+3\frac{1}{2}x + 3

Would you like more details on any step or clarification?

Here are five related questions to explore:

  1. How does translating a function affect its graph in general?
  2. Can you describe how reflection over different lines (like y=0y=0) changes a function?
  3. What would happen if the dilation had a different center point?
  4. How would the transformations change if the function were quadratic, like f(x)=x2+3f(x) = x^2 + 3?
  5. How can you combine multiple transformations into a single operation?

Tip: Reflections and dilations around points other than the origin often involve more complex transformations, but they follow similar principles by adjusting coordinates accordingly.

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Math Problem Analysis

Mathematical Concepts

Linear Functions
Transformation
Translation
Reflection
Dilation

Formulas

Translation formula: f(x - h) + k
Reflection formula: f(2c - x) for reflection over vertical line x = c
Dilation formula: f(x/k) for dilation with scale factor k

Theorems

Basic Transformations in Function Graphs

Suitable Grade Level

Grades 10-12