Another example simplifying slope of secant line

- [Voiceover] A secant
line intersects the graph

of f of x is equal to
x squared plus five x

at two points with x
coordinates three and t

where t does not equal three.

What is the slope of the
secant line in terms of t?

Your answer must be fully
expanded and simplified.

And my apologies ahead
of time if I'm a little

out of breath.

I just tried to do some
exercises in my office

to get some blood moving
and I think I'm still

a little out of breath.

Anyway, so we want to
find the slope of the

secant line

and they essentially give us two points

on the secant line.

They tell us what x is at
each of those two points

and if we know what an x is,

we are able to figure out what f of x is

at each of those points.

So we could make a little table here.

We know x and we know f of x.

So when x is equal to three

what is f of x?

Well it's going to be three squared

plus five times three.

Well this is going to be nine plus 15

which is 24.

So this is going to be 24

and when x is equal to t

what is f of t?

Well it is going to be
t squared plus five t.

And so we have two points
now that are on this line,

this is the secant line.

It intersects our function twice

so it has these two points on it.

So we just have to find
our change in y between

these two points.

Change in y and our
change in x, change in x.

And I'm assuming that
y is equal to f of x.

So our slope of our secant line is

change in y

over change in x.

Our change in y, if we
view this as our end point,

the second one with the
t in it is our end point

is going to be that minus that.

So it's going to be t squared plus five t

minus 24

and then in our denominator our ending x

minus our starting x is
going to be t minus three.

Now they tell us our answer must be fully

expanded and simplified.

So maybe there's a way to
simplify this a little bit.

Let's see, can I factor
the top into something

that involves

a t minus three?

All right,

so in the numerator, let's see,

negative three

times positive eight is negative 24.

Negative three plus
positive eight is five.

So we can rewrite this as

t plus eight

times t

times t minus three.

And so we could say this
is going to be equal to

if we cancel out the t
minus three or we divide

the numerator into the
denominator by t minus three

it's going to be equal to t plus eight.

Now if we wanted to be really strict,

mathematically strict,

this expression isn't
exactly the same as our

original expression right over here.

What makes them different?

Well they're going to
be true for all of t's

except where t equals three.

This thing right over here
is defined at t equals three.

In fact when t equals three
this expression is equal to 11.

But this thing up here was
not defined at t equals three

so if you wanted to be particular about it

you want this expression
to be the exact same thing

you would say

four t does not equal three.

Now this can take the
same inputs as this one

right over there.

But I'm assuming where
t does not equal three.

So you could view this
as maybe a little bit

redundant.

But this would be,

this is the slope of the
secant line in terms of t.