Given 2 points in a parabola, say (1,0) and (2,4) and axis the y-axis. How do you find the equation?

Question

Given 2 points on a parabola A(1,0) and B(2,4) and having the y-axis as its axis. How do you find the equation?

Answer 1

Start with 

 

y = a(x - h)² + k

 

There are 3 unknowns a, h, and k, so we need 3 pieces of information.  The 2 points provide 2 pieces.  The fact that the parabola is on the y-axis provides another.  For the parabola to be along the y-axis, the vertex must be on the y-axis, so h = 0.

 

So far we have

 

y = ax² + k

 

Now use the 2 points to find a and k.

 

0 = a(1)² + k ⇒ a + k = 0

4 = a(2)² + k ⇒ 4a + k = 4

 

Solve for a and k from

 

a + k = 0

4a + k = 4

 

and get

 

a = 4/3

k = -4/3

 

Final answer:

 

y = (4/3)x² - 4/3

 

Test the answer.