Question

Explain how you could write a quadratic function in factored form that would have a vertex with an x-coordinate of 3 and two distinct roots

Answer 1

Answer:

$f(x) = (x + 1)(x - 7)$

Step-by-step explanation:

For a quadratic function to have a vertex with an x-coordinate of 3, then

$3 = - \frac{b}{2a}$

Let a=1, then we have

$3 = - \frac{b}{2}$

$b = - 2 \times 3 = - 6$

So now our equation becomes:

$f(x) = {x}^{2} - 6x + c$

We now find two factors of c that add up to -6.

Let these factors be 1, and c.

Then

$c + 1 = - 6$

$c = - 6 - 1 = - 7$

Therefore the factors are :

1 and -7.

The function becomes:

$f(x) = {x}^{2} - 6x - 7$

The factored form is

$f(x) = (x + 1)(x - 7)$