### Video Transcript

Write the quadratic equation
represented by the graph shown. Give your answer in factored
form.

Letβs begin by examining the graph
weβve been given. We might first notice that the
vertex or turning point of this graph has coordinates one, negative nine. This gives us some idea of what the
completed square form equation of this graph might look like. An equation of the form π π₯ plus
π squared plus β has a vertex negative π, β. So we let negative π be equal to
one and β be equal to negative nine. And we see that the equation of our
graph is π¦ equals some constant π times π₯ minus one all squared minus nine.

So how do we find the value of
π? Well, in fact, we can choose the
coordinate of any point that lies on this curve and substitute that in. A really straightforward one is the
coordinate four, zero. The π₯-coordinate is four, and the
π¦-coordinate is zero. And so our equation becomes zero
equals π times four minus one squared minus nine. Well, four minus one squared is
three squared, which is nine. So our equation becomes zero equals
nine π minus nine. We add nine to both sides of this
equation. And finally, weβll divide through
by nine. And when we do, we find that π is
equal to one. Substituting this back into the
equation π times π₯ minus one all squared minus nine, and we find that the equation
of this quadratic is π¦ equals π₯ minus one squared minus nine.

Now, in fact, weβre told to give
this in factored form. So what next? Well, weβre simply going to
distribute the parentheses, simplify, and then factor. π₯ minus one all squared is π₯
minus one times π₯ minus one. Distributing the parentheses, and
we get π₯ squared minus π₯ minus π₯ plus one. And so, our equation becomes π¦
equals π₯ squared minus two π₯ minus eight. To factor this, we simply find a
pair of numbers that have a product of negative eight and sum to make negative
two. Thatβs negative four and two. And so, in factored form, the
quadratic equation represented by the graph shown is π¦ equals π₯ minus four times
π₯ plus two.