https://file.aiquickdraw.com/mathbot/file/1730255496262nx55fnpa.jpg

The question provides the function \( f(x) = x^2 - 2x - 3 \) (black graph) and asks for the function \( g(x) \) (red graph), obtained by transforming \( f(x) \).

The red graph appears to be a horizontal shift of the black graph to the left. Let’s analyze the transformations:

1. **Determine the horizontal shift**: The vertex of \( f(x) = x^2 - 2x - 3 \) is at \( x = 1 \) (from \( x = -\frac{-2}{2 \cdot 1} = 1 \)). Observing the red graph, its vertex seems shifted to the left of \( x = 1 \), suggesting a shift by 2 units to the left (vertex now at \( x = -1 \)).

2. **Transform \( f(x) \) horizontally**: A horizontal shift by 2 units to the left can be applied by replacing \( x \) with \( x + 2 \) in \( f(x) \). This gives:
   \[
   g(x) = f(x + 2) = (x + 2)^2 - 2(x + 2) - 3
   \]

3. **Simplify \( g(x) \)**:
   \[
   g(x) = (x + 2)^2 - 2(x + 2) - 3
   \]
   \[
   = x^2 + 4x + 4 - 2x - 4 - 3
   \]
   \[
   = x^2 + 2x - 3
   \]

After simplifying, we find that the correct answer is:
\[
\boxed{g(x) = x^2 + 2x - 3}
\]
This matches option **e**.

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Would you like more details or have questions about transformations?

### Related Questions:
1. How do horizontal transformations affect the graph of a quadratic function?
2. What is the difference between shifting left and shifting right in function transformations?
3. How can we determine the vertex form of a quadratic function?
4. How do transformations affect the domain and range of a function?
5. What other types of transformations (like reflections) can be applied to quadratic functions?

#### Tip:
When shifting a function horizontally, replace \( x \) with \( x \pm h \) in the function’s equation, where \( h \) is the number of units shifted.

Diberikan grafik fungsi f(x) = x^2 - 2x - 3 pada bidang kartesius. Tentukan fungsi g(x) yang grafiknya diperoleh dengan mentransformasi grafik fungsi f(x).

Determine the function g(x) by applying a horizontal transformation to f(x) = x^2 - 2x - 3. Learn how to apply horizontal shifts to quadratic functions, including detailed analysis and simplification. Suitable for high school students in Grades 9-10.

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