Answer:
When considering the quadratic function y = ax^2 + bx + c, where a and c remain constant while b varies, we can observe the behavior of the vertex of the quadratic as b changes!
(Long explanation ahead !!)
Explanation:
1. Vertex of the quadratic: The vertex of a quadratic function is given by the formula (-b/2a, f(-b/2a)), where f(x) represents the quadratic function. In this case, the x-coordinate of the vertex is always -b/2a, and the y-coordinate is f(-b/2a).
2. Path formed by the vertex: As b varies, the x-coordinate of the vertex, -b/2a, changes. This implies that the path formed by the vertex is a function of b. Let's call this function V(b), where V represents the x-coordinate of the vertex as a function of b.
3. Showing that V(b) is a quadratic function: To demonstrate that the path formed by the vertex is itself a quadratic function, we need to show that V(b) can be expressed as V(b) = px^2 + qx + r, where p, q, and r are constants.
4. Evaluating V(b): By substituting -b/2a for x in the quadratic function, we can find V(b) as follows: V(b) = a(-b/2a)^2 + b(-b/2a) + c. Simplifying this expression yields V(b) = ab^2/4a^2 - b^2/2a + c, which can be further simplified to V(b) = (-b^2 + 4ac)/4a!
5. Simplifying V(b): By dividing the numerator and denominator by 4, we obtain V(b) = -b^2/4a + ac/a. This expression matches the form px^2 + qx + r, where p = -1/4a, q = 0, and r = ac/a!
6. Vertex lies on the y-axis: The x-coordinate of the vertex, -b/2a, represents the line of symmetry for the quadratic. Since b varies while a and c remain constant, the line of symmetry is always x = -b/2a, which is the equation of the y-axis. Therefore, the vertex of the quadratic function formed by the path of the vertex always lies on the y-axis!
! : In conclusion, as b varies, the path formed by the vertex of the quadratic function y = ax^2 + bx + c is itself a quadratic function, V(b). The vertex of V(b) lies on the y-axis, which is represented by the equation x = -b/2a!!