Finding the function of a Parabola

[whitehorsey]

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 into a function. Can you show me step by step? Thank You!

 

[Дьявол]

The parabola has formula y=A(x+B)²+C
You got the intercepts i.e y=0, x₁=-1, x₂=3

Have you ever learned about Vieta's formulas?

x₁+x₂=-b/a
x₁*x₂=c/a

where

x²+(b/a)x+c/a=0

You already got the solutions of the quadratic equation.

 

[Mark44]

In general, the graph of a parabola is given by the equation y = ax² + bx + c, which can also be written as y = a(x - r₁)(x - r₂). Since you are given the two x-intercepts, you should be able to figure out what r₁ and r₂ 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.

 

[whitehorsey]

Ok Thank You!