Final answer:
To evaluate and simplify the expression (f(x+8)-f(x))/(8) for the given function f(x) = ax² + c, substitute x+8 into the function, subtract f(x), expand and simplify the expression to get 2ax + 8a.
Step-by-step explanation:
To evaluate and simplify the expression (f(x+8)-f(x))/(8) given f(x) = ax² + c, we first substitute x+8 into the function f(x) and subtract f(x). This gives us f(x+8) - f(x) = a(x+8)² + c - (ax² + c). Simplifying further, we get (f(x+8) - f(x))/(8) = ((a(x+8)² + c) - (ax² + c))/8.
Next, we expand the squared term and simplify the expression further. We can combine like terms and divide all terms by 8, resulting in ((a(x² + 16x + 64) + c) - (ax² + c))/8.
We can simplify further by canceling out the c terms and combining like terms in the numerator: ((ax² + 16ax + 64a + c) - (ax² + c))/8.
Simplifying even further by canceling out the ax² terms, we get (16ax + 64a)/8, which simplifies to 2ax + 8a.