Question

What is the quadratic function that has a vertex of (2, 3) and passes through the point (0,-5)?

Answer 1

The function is `y = -2(x-2)^2+3`

Explanation:

Because you asked for a function, I shall use only the vertex form:

`y = a(x-h)^2+k" [1]"`

where `(x,y)` is any point on the described parabola, `(h,k)` is the vertex of the parabola, and `a` is an unknown value that is found using the given point that is not the vertex.

NOTE: There is a second vertex form that can be used to make a quadratic:

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

But it is not a function, therefore, we shall not use it.

Substitute the given vertex, `(2,3)`, into equation [1]:

`y = a(x-2)^2+3" [1.1]"`

Substitute the given point `(0,-5)` into equation [1.1]:

`-5 = a(0-2)^2+3`

Solve for a:

`-8=4a`

`a = -2`

Substitute `a = -2` into equation [1.1]:

`y = -2(x-2)^2+3" [1.2]"`

Here is a graph of the parabola and the two points: