Start with the two -intercepts. The two zeros of the quadratic equation for this parabola are:, and.
(These are the -coordinates of the two -intercepts.)
By the factor theorem, (where is a real number) is a zero of a polynomial if and only if is a factor of that polynomial.
A quadratic equation is also a polynomial. In this case, the two zeros would correspond to the two factors.
A parabola could only have up to two factors. As a result, the power of these two factor should both be one. Hence, the equation for the parabola would be in the form
where is the leading coefficient that still needs to be found. Calculate the value of using the -intercept of this parabola. (Any other point on this parabola that is not one of the two -intercepts would work.)
Since the coordinates of the -intercept are , and . The equation becomes:
Solve for :
Hence the equation for this parabola:
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we know that
The equation of a vertical parabola in factored form is equal to
a is a coefficient
x_1 and x_2 are the roots or x-intercepts of the quadratic equation
In this problem we have
Find the value of a
we have the y-intercept (0,37.8)
substitute the value of x and the value of y of the y-intercept in the equation and solve for a
Convert to expanded form