We're given f of t equals the quantity
t squared minus two t plus six
times u sub two of t
and we're asked to find the Laplace transform of f of t.
Notice how f of t is a function of t times
a unit step function,
or a heavy side step function,
where a unit step function in the form
of u sub c of t equals zero of t is less than c
and equals one if t is greater than or equal to c.
So because we have u sub two of t
this is equal to zero of t is less than two
and it's equal to one if t is greater than
or equal to two.
Let's first look at the graph of f of t.
Notice how f of t is zero
when t is less than two and f of t is equal
to t squared minus two t plus six
given by the parabola when t is greater than or equal to two
because u sub two of t would be equal to one.
To find the Laplace transform of f of t
we'll be using this formula here.
Where the Laplace transform of f of the quantity
t minus c times u sub c of t
equals e raised to the power of negative cs
times the Laplace transform of f of t.
There's a couple of important things to notice about this
formula on the left the values of c
must be the same in order to apply this formula
and on the left we have f of the quantity t minus c
on the right we have f of t.
So beginning with the given function
f of t equals the quantity t squared
minus two t plus six times u sub two of t.
We'll take the Laplace transform
of both sides of the equation.
So we'll have a Laplace transform of f of t
equals the Laplace transform of
the quantity t squared minus two t plus six
times u sub two of t.
Well the Laplace transform of f of t
is equal to big f of s.
Now looking at the right side
and the formula here because we have u sub two of t
we know that c
must equal two.
And that means this function here
must be f of the quantity t minus two.
In order to apply this formula though
notice how we need to find f of t.
So let's work on determining f of t.
If we know f of the quantity t minus two must equal
t squared minus two t plus six
we can determine f of t
by taking this function and replacing
or substituting t plus two for t.
Notice t plus two minus two
would simplify to t.
Which means f of t must be equal to
the quantity t plus two squared
minus two times the quantity t plus two.
Plus six.
So now we'll go ahead and expand this
and then simplify to find f of t.
F of t equals t squared plus four t plus four.
And then minus two t minus four plus six
which means f of t is equal to t squared.
Then we have four t minus two t that's plus two t
and then we have four minus four plus six,
so we have plus six.
Now that we know f of t if we wanted to,
we could also rewrite f of the quantity
t minus two in a different form
by substituting t minus two for t
in our function f of t.
If we did that we'd have the quantity
t minus two squared plus two times the quantity
t minus two plus six.
Let's go ahead and write this as
big f of s equals the Laplace transform of,
let's write f of the quantity t minus two
in this form here.
We'd have the quantity t minus two squared
plus two times the quantity t minus two
plus six all times u sub two of t.
Notice in this form this fits our formula perfectly.
Where we have f written in the form
of f of the quantity t minus two.
Which means big f of s equals
again if c equals two
this Laplace transform is equal to e
raised to the power of negative two s times
the Laplace transform of f of t,
which we know is equal to t squared
plus two t plus six.
When we have f of the quantity t minus two
written in this form we can easily see
this is f of t by replacing t minus two with t.
So big f of s is equal to e
raised to the power of negative two s times,
let's quickly review our Laplace transform formulas.
We'll be using this formula
as well as this formula.
The Laplace transform of t raised to the power of n
is equal to n factorial divided by s
raised to the power of n plus one,
and the Laplace transform of one
is equal to one over s.
So the Laplace transform of t squared
is going to be equal to two factorial two
divided by s to the third.
We'll have plus two times the Laplace transform of t
which will be two divided by s squared,
plus the Laplace transform of six
which is six divided by s.
Let's go ahead and leave big f of s
in this form here.
I hope you found this helpful.