Finding a quadratic function using given information: แบบฝึกหัด

Which of the following functions represents a quadratic relationship whose graph has a vertex at the origin \((0, 0)\) and passes through the point \((2, 12)\)?

The graph of a quadratic function passes through the points \((-1, 0)\), \((3, 0)\), and \((0, -6)\). What is the equation of this function in the form \(f(x) = ax^2 + bx + c\)?

A quadratic function \(f(x)\) has its vertex at the point \((1, 4)\). The graph of the function passes through the point \((3, 0)\). Determine the expression for \(f(x)\) in the form \(ax^2 + bx + c\).

Find the quadratic function \(f(x) = ax^2 + bx + c\) that has its vertex at \((2, -3)\) and passes through the point \((5, 15)\). Give your answer in expanded form.

A quadratic function has its vertex at the point \((2, -9)\) and passes through the point \((5, 0)\).

(a) Find the quadratic function in the form \(y = a(x - h)^2 + k\).
(b) Expand your answer from part (a) to write the function in the form \(y = ax^2 + bx + c\).
(c) State the coordinates of the other \(x\)-intercept.

A quadratic function f(x) has x-intercepts at (-3, 0) and (5, 0). The graph of the function passes through the point (0, -30).

(a) Find the equation of the function in the form f(x) = a(x - p)(x - q).
(b) Expand the result from part (a) to write the function in the form f(x) = ax^2 + bx + c.
(c) Determine the coordinates of the y-intercept and the vertex of the graph.