Final answer:
To find the quadratic function with the given roots and value, we use the form f(x) = a(x - r1)(x - r2), expand it, and solve for a using the given function value. The function results in f(x) = -4(x - 3)^2 + 8.
Step-by-step explanation:
To write a quadratic function given the properties f(3 √2 )=f(3− √2 )=0, f(1) = −8, first consider that the roots of the function are 3 + √2 and 3 − √2. Using the fact that a quadratic equation with roots r1 and r2 can be written as f(x) = a(x - r1)(x - r2), we can write the base form f(x) = a(x - (3 + √2))(x - (3 - √2)). Expanding this, we get f(x) = a[(x - 3) - √2][(x - 3) + √2], which simplifies to f(x) = a(x - 3)^2 - 2a. To determine the coefficient a, we use the property f(1) = -8: -8 = a(1 - 3)^2 - 2a, which simplifies to -8 = 4a - 2a. From this, we find that a = -4. Thus, the quadratic function is f(x) = -4(x - 3)^2 + 8.