### Video Transcript

If π is the variation function for
π of π₯ is equal to π₯ squared minus four π₯ plus two, what is π of negative 0.2
when π₯ is equal to eight?

Weβre given a quadratic function π
of π₯ is π₯ squared minus four π₯ plus two and asked to find the value of the
variation function if π₯ changes from π₯ is equal to eight by an amount of negative
0.2. The first thing we need to do then
is to find the variation function π of β. We can do this using the definition
for a function π of π₯ where its variation function at π₯ is equal to π is given
by π of β is π of π plus β minus π of π. And thatβs where β is the change in
π₯. In our case, our function π of π₯
is π₯ squared minus four π₯ plus two. Our given value of π₯ is eight so
that π is equal to eight so that our variation function π of β is π of eight plus
β minus π of eight.

There are two ways we could
approach this problem. We know that our change in π₯ is
negative 0.2, and this means that β is negative 0.2. And we could substitute this
directly into our equation for π of β. Alternatively, we could first find
π of β, the variation function in terms of β, and then substitute β is negative
0.2. Weβre going to use the second
method to find the variation function π of β. And to do this, weβre first going
to substitute π₯ is equal to eight plus β into our function π of π₯. And this gives us eight plus β
squared minus four times π plus β plus two. Distributing our parentheses, this
gives us 64 plus 16β plus β squared minus 32 minus four β plus two. And collecting like terms, this
gives us eight squared plus 12β plus 34.

Now, to find π of eight, we
substitute π₯ equal to eight into our function π of π₯, which gives us eight
squared minus four times eight plus two. And this evaluates to 34. Now, with these two results into
our function π of β, we have eight squared plus 12β plus 34 minus 34, that is, β
squared plus 12β. And this is our variation function
π of β for π of π₯ at π₯ is equal to eight. Now, to find π of negative 0.2, we
substitute negative 0.2 in place of β. This gives us negative 0.2 squared
plus 12 times negative 0.2. That is 0.04 minus 2.4, which is
negative 2.36. π of negative 0.2 for π of π₯ is
equal to π₯ squared minus four π₯ plus two when π₯ is equal to eight is therefore
negative 2.36.