What is the equation of the parabola that has a vertex at ## (14, -9) ## and passes through point ## (0, 2) ##?

##y=11/196(x-14)^2-9##

The equation of a parabola in ##color(blue)”vertex form”## is

##color(red)(|bar(ul(color(white)(a/a)color(black)(y=a(x-h)^2+k)color(white)(a/a)|)))## where (h ,k) are the coordinates of the vertex and a, is a constant.

here h = 14 and k = – 9, so we can write a partial equation

##y=a(x-14)^2-9##

To find a, substitute the coordinates of (0 ,2) a point on the parabola, into the partial equation.

##rArra(0-14)^2-9=2rArr196a=11rArra=11/196##

##rArry=11/196(x-14)^2-9″ is equation in vertex form”##

The equation may be expressed in ##color(blue)”standard form”##

That is ##y=ax^2+bx+c## by distributing the bracket and simplifying.

##rArry=11/196(x^2-28x+196)-9=11/196x^2-11/7x+2##

graph{11/196(x-14)^2-9 [-20, 20, -10, 10]}