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.

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.