How do you write the equation given vertex (0,1) and focus (0,5)?

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Answer 1 · 2016-11-24T22:32:55

Please see the explanation for steps leading to the equation #y = 1/16x^2 + 1#

The y coordinate changes between the vertex and the focus, therefore, we know that the equation is of the form:

#y = f(x)#

Note: If it were the x coordinate that changes between the vertex and the focus then the equation would be of the form #x = f(y)#

The vertex form for this type is:

#y = a(x - h)^2 + k#

where (h, k) is the focus and #a = 1/(4f)# where f is the vertical shift from the vertex to the focus.

Substitute 0 for h and 1 for k:

#y = a(x - 0)^2 + 1#

#y = ax^2 + 1#

The vertical shift from the vertex to the focus is:

#f = (5 - 1) = 4#

#a =1/(4(4))#

#a = 1/16#

Substitute #1/16# for a:

#y = 1/16x^2 + 1#