Final answer:
The quadratic function that passes through the origin, (4, 2), and (-4, 2) is f(x) = 1/8 x². To find this, we used the points to create and solve a system of equations for the coefficients a and b, given the general quadratic form f(x) = ax² + bx + c.
Step-by-step explanation:
Given that the quadratic function passes through the origin (0,0), the point (4,2), and the point (-4,2), we can determine the equation by using the general form of a quadratic function, which is f(x) = ax² + bx + c. Because the function passes through the origin, we know that c = 0. We can then utilize the other two points to set up a system of equations to solve for a and b.
Using the point (4,2), we get: 2 = a(4)² + b(4). Using the point (-4,2), we get: 2 = a(-4)² + b(-4). Notice that a(4)² = a(-4)², which simplifies our work because the equations for the points (4,2) and (-4,2) differ only in the sign of the b-term.
Since a(16) + 4b = 2 and a(16) - 4b = 2, adding these two equations will cancel out b, allowing us to solve for a. We get 2a(16) = 4, which simplifies to a = 1/8. Substituting a back into one of the original equations gives us b = 0. Therefore, the equation of the quadratic function is f(x) = ⅛ x².