Short Answer 
The function f(x) is defined as f(x) = ax¬≤ + bx + c, with x-intercepts at (7,0) and (-3,0). It can be factored as f(x) = a(x Р7)(x + 3), leading to the expression for b as b = -4a, and resulting calculations show that for integer values of a greater than 1, the sums a + b yield -6, -9, and -12 for a = 2, 3, and 4 respectively.
Understand the Function’s Structure
The function f is defined as f(x) = ax² + bx + c, where a, b, and c are constants. The graph of this function intersects the x-axis at the points (7,0) and (-3,0), indicating these are the x-intercepts of the function. Thus, we can establish that f(7) = 0 and f(-3) = 0.
Transform to Factored Form
Using the x-intercepts, we rewrite the function in factored form. This can be expressed as f(x) = a(x Р7)(x + 3). Expanding this gives us f(x) = a(x¬≤ Р4x Р21). From this expression, we can identify that b = -4a, allowing us to express b in terms of a.
Calculate Values of a, b, and Their Sum
Since a must be an integer greater than 1, we can select values for a and compute the corresponding values of b and a + b. The calculations yield:
- For a = 2: b = -8, a + b = -6.
- For a = 3: b = -12, a + b = -9.
- For a = 4: b = -16, a + b = -12.