1. The graph of a quadratic function (a parabola) has x-intercepts -1 and 3 and a range consisting of all number less than or equal to 4. Determine an expression for the function. 2. none 3. I do not know how to use the intercepts and the range and manipulate it in to a function. Can you show me step by step? Thank You!
The parabola has formula y=A(x+B)^{2}+C You got the intercepts i.e y=0, x_{1}=-1, x_{2}=3 Have you ever learned about Vieta's formulas? x_{1}+x_{2}=-b/a x_{1}*x_{2}=c/a where x^{2}+(b/a)x+c/a=0 You already got the solutions of the quadratic equation.
In general, the graph of a parabola is given by the equation y = ax^{2} + bx + c, which can also be written as y = a(x - r_{1})(x - r_{2}). Since you are given the two x-intercepts, you should be able to figure out what r_{1} and r_{2} are in the second equation I showed. So all you need to do is determine that value of a. Since the range is all numbers <= 4, the parabola opens downward. The vertex of a parabola is a point on the parabola that is lower than all others (for a parabola that opens upward) or higher than all others (for one that opens downward). The vertex is always midway between the two x-intercepts, provided that there are two x-intercepts. With this information you should be able to find the coordinates of the vertex, and from them the value of a in the parabola's equation.