Final answer:
The quadratic function in question appears to have its vertex as a minimum at (3,9), implying that the function is increasing after this point. Hence, the function f(x) is increasing in the interval (3, infinity), which corresponds to Option b.
Step-by-step explanation:
The question asks which statement about a quadratic function f(x) that passes through the points (-1,9), (3, 9) and (4, 14) is true. To determine the correct option, we need to understand the behavior of quadratic functions and how they are graphed based on given points.
A quadratic function generally has the form f(x) = ax^2 + bx + c. The points given are enough to imply that the vertex of the parabola (because a quadratic function graphs to a parabola) is (3,9) since it is the point through which the parabola passes and then changes direction. This vertex is a maximum or a minimum based on the value of 'a' in the quadratic equation. If 'a' is positive, the parabola opens upwards and the vertex is a minimum; if 'a' is negative, the parabola opens downwards and the vertex is a maximum. In this question, since the function values at both (-1,9) and (3,9) are equal, and the function value at (4,14) is higher, this suggests that the vertex is a minimum and not a maximum, and therefore, the function is increasing past the vertex (x=3).
Let's look at the options:
- Option a: This implies the function is decreasing from left to right through these points, which is not true since the function value at (4, 14) is higher than at (3, 9).
- Option b: As we established earlier, since the vertex (3, 9) appears to be a minimum, the function f(x) would be increasing beyond this point. Therefore, the function is increasing in the interval (3, infinity).
- Option c: With the information provided, we cannot determine the x-intercepts without the exact quadratic equation.
- Option d: We've established the vertex at (3,9) is likely a minimum, and therefore, f(x) does not have a maximum at x = 1.
The correct statement about f(x) is therefore Option b: f(x) is increasing in the interval (3, infinity).