Final answer:
The student's question regarding the characteristics of the quadratic function f(x) = (x-2)(x+3) is answered by identifying the x-intercepts at x = 2 and x = -3, and the axis of symmetry being x = -0.5.
Step-by-step explanation:
The characteristics of the quadratic function f(x) = (x-2)(x+3) can be identified by looking for the x-intercepts and the axis of symmetry. The x-intercepts are the points where the function intersects the x-axis, which occur where the function equals zero. Therefore, for the given function, the x-intercepts can be found by setting f(x) to zero and solving for x:
- 0 = (x - 2)(x + 3)
- x - 2 = 0 or x + 3 = 0
- x = 2 or x = -3
The x-intercepts are at x = 2 and x = -3. The axis of symmetry for a quadratic function is a vertical line that passes through the vertex of the parabola, which is the midpoint between the x-intercepts. The axis of symmetry can be found by averaging the x-intercepts:
Axis of symmetry: x = (2 + (-3)) / 2 = -1 / 2
The axis of symmetry for this function is the line x = -0.5.