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

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

Let’s go through each part in detail:

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

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

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

### b. **Reflection about the line** \( x = 1 \):

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

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

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

### Summary of Results:
- a) Translated function: \( 2x + 2 \)
- b) Reflected function: \( 7 - 2x \)
- c) Dilated function: \( \frac{1}{2}x + 3 \)

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 \( y=0 \)) 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 \( f(x) = x^2 + 3 \)?
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.

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

Explore how to transform the function f(x) = 2x + 3 using translation, reflection, and dilation. Learn the formulas for these transformations and apply them step by step. Suitable for students in Grades 10-12.

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