Question

How do you find the quadratic function with vertex (5,12) and point (7,15)?

Answer 1

Please see the explanation.

Explanation:

There are two equations that can be written.

1. Uses the vertex form:
   `y = a(x - h)^2 + k`
   where h and k correspond to the vertex point `(h, k)` and you solve for a using the specified point.
2. Uses the vertex form:
   `x = a(y - k)^2 + h`
   where h and k correspond to the vertex point `(h, k)` and you solve for a using the specified point.

Substitute the vertex `(5,12)` into both forms:

`y = a(x - 5)^2 + 12" [1]"`
`x = a(y - 12)^2 + 5" [2]"`

Using the specified point `(7,15)`, substitute 7 for x and 15 for y into equations [1] and [2]:

`15 = a(7 - 5)^2 + 12`
`7 = a(15 - 12)^2 + 5`

Solve for for a :

`15 = a(7 - 5)^2 + 12`
`7 = a(15 - 12)^2 + 5`

`15 = a(2)^2 + 12`
`7 = a(3)^2 + 5`

`4a = 3`
`9a = 2`

`a = 3/4`
`a = 2/9`

Substitute `3/4` for a into equation [1} and `2/9` for a into equation [2]:

`y = 3/4(x - 5)^2 + 12" [3]"`
`x = 2/9(y - 12)^2 + 5" [4]"`

Here is a graph showing that both equations [3] and [4] have the specified vertex and pass through the specified point:

However, you asked for a function, Equation [3] is the only function. Equation [4] is not a function.