There are an infinite number of parabolas containing (1,0) and (0,3). Find two more and give their vertex form.

Question

How can I do this? I am very confused as to how to come up with a vertex form with only this. Please help. Thank you!

Answer 1

I find it easier to work your way from factored form, to generic form and then to the vertex form by completing the square.

1) The roots are 1 and Z(the second root is unknown)

So the factored form will be a(x-1)(x-z)

2) Open parenthesis

a(x² - (z+1)x +z)

3) For a=1

x² - (z+1)x +z

4) Plugin another point x=0 - you should get 3

So z=3

And the equation take the form

x² - 4x +3

5) Completing the square we get the vertex form:

(x-2)² - 4 + 3

(x-2)² - 1

6) repeat steps 3-5 for another a, say a=-1

and get

-(x+1)²+4