Final answer:
The complete quadratic function with vertex (1, -4) and passing through (2, -3) is f(x) = x² - 2x - 3. However, this expression doesn't match any of the options provided, indicating there might be a typo in the question. The student should verify the options or select the one closest to the derived equation. The correct answer is option D.
Step-by-step explanation:
The student is asking for the complete quadratic function given the vertex (1, -4) and another point (2, -3). A quadratic function can be written in the vertex form, which is f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola. Here, h = 1 and k = -4, so the function starts as f(x) = a(x - 1)² - 4. We can use the other point (2, -3) to solve for 'a'. Substituting x = 2 and f(x) = -3, we get:
- -3 = a(2 - 1)² - 4
- -3 = a(1)² - 4
- -3 + 4 = a
- a = 1
Therefore, the quadratic equation is f(x) = (x - 1)² - 4 which simplifies to f(x) = x² - 2x + 1 - 4, which is f(x) = x² - 2x - 3. The correct answer is therefore option (d).
However, none of the given multiple-choice solutions exactly match this equation. It's possible there could be a typo in the options provided. In such cases, it's recommended to clarify with the source or choose the option that most closely matches the correct formula derived from the given vertex and point.