Question

What's the standard form, vertex, and factored​

Answer 1

1)
`f(x)=(x+1)(x-3)`

2)
`f(x)=(x-1)^2-4`

3)
`f(x)=x^2-2x-3`

Explanation:

So we have a graph and we know that its roots are at x=-1 and x=3.

We also know the vertex is at (1,-4). With that, we can figure out the three forms.

Factored Form:

The factored form, as given, is:

`f(x)=a(x-r_1)(x-r_2)`

We already know the roots of -1 and 3. So, substitute:

`f(x)=a(x-(-1))(x-3)`

Simplify:

`f(x)=a(x+1)(x-3)`

Now, we just need to figure out a. To do so, we can use the vertex. Since the vertex is at (1,-4), this means that f(1) is -4. So, substitute 1 for x and substitute -4 for f(x):

`-4=a(1+1)(1-3)`

Add and subtract:

`-4=a(2)(-2)`

Multiply:

`-4=-4a`

Divide both sides by -4:

`a=1`

So, our factored form is:

`f(x)=(x+1)(x-3)`

Vertex Form:

The vertex form is:

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

Where (h,k) is the vertex.

We already know the vertex is (1,-4), so substitute 1 for h and -4 for k.

We also previously determined that a is 1, so substitute that also. So:

`f(x)=(1)(x-(1))^2+(-4)`

Simplify:

`f(x)=(x-1)^2-4`

Standard Form:

To acquire the standard form, simply expand the factored or vertex form. I'm going to expand the factored form:

`f(x)=(x+1)(x-3)`

FOIL:

`f(x)=x^2-3x+x-3`

Combine like terms:

`f(x)=x^2-2x-3`

And we're done!