Question

Vertex form and standard form

Answer 1

Given:

The vertex of a quadratic function is (-5,-1) and it passes through the point (-2,2).

To find:

The vertex and standard form of the quadratic function.

Solution:

The vertex form of a quadratic function is:

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

Where, a is a constant, (h,k) is vertex.

The vertex of a quadratic function is (-5,-1). It means
`h=-5,k=-1`.

`f(x)=a(x-(-5))^2+(-1)`

`f(x)=a(x+5)^2-1` ...(i)

The quadratic function passes through the point (-2,2). Putting
`x=-2,f(x)=2` in (i), we get

`2=a(-2+5)^2-1`

`2+1=a(3)^2`

`3=9a`

`(1)/(3)=a`

Putting
`a=(1)/(3)` in (i), we get

`f(x)=(1)/(3)(x+5)^2-1`

Therefore, the vertex for of the quadratic function is
`f(x)=(1)/(3)(x+5)^2-1`.

The standard form of a quadratic function is:

`f(x)=Ax^2+Bx+C`

We have,

`f(x)=(1)/(3)(x+5)^2-1`

`f(x)=(1)/(3)(x^2+10x+25)-1`

`f(x)=(1)/(3)x^2+(10)/(3)x+(25)/(3)-1`

`f(x)=(1)/(3)x^2+(10)/(3)x+(22)/(3)`

Therefore, the standard form of a quadratic function is
`f(x)=(1)/(3)x^2+(10)/(3)x+(22)/(3)`.