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# Slope of secant line challenge

Sal solves a challenging problem involving slopes of secant lines to a curve. Created by Sal Khan.

Video transcript

Consider the graph
of the function f of x that passes through
three points as shown. So these are the three
points, and this curve in blue is f of x. Identify which of the
statements are true. So they give us
these statements. Let's see. This first one says f of
negative a is less than 1 minus f of negative a over a. So this seems like some
type of a bizarre statement. How are we able to figure
out whether this is true from this right over here? So let's just go
piece by piece and see if something starts
to make sense. So f of negative a, where
do we see that here? Well, this is f of negative a. This is the point x
equals negative a. So this is negative
a, and this is y is equal to f of negative a. So this is f of negative
a right over here. And what we know about
f of negative a, based on looking at this graph,
is that f of negative a is between 0 and 1. So we can write that. 0 is less than f of negative
a, which is less than 1. So that's all I can deduce
about f of negative a right from the get-go. Now let's look at this
crazy statement, 1 minus f of negative a over a. What is this? Well, let's think
about what happens if we take the
secant line, if we're trying to find the slope
of the secant line, between this point
and this point, if we wanted to find the
average rate of change between the point negative a,
f of negative a, and the point 0, 1. If this is our endpoint,
our change in y is going to be 1
minus f of negative a. So 1 minus f of negative a
is equal to our change in y. And our change in x, going from
negative a to 0, so change in x is going to be equal to
0 minus negative a, which is equal to positive a. So this right over here
is essentially our change in x over our change in y
from this point to this point. It is our average rate of change
from this point to this point. So it is our average
rate of change, or you could say it's the
slope of the secant line. So the secant line would
look something like this. Slope of the secant line,
so this right over here is slope of secant
line between from f of negative a to 0 comma 1. So just looking at this
diagram right over here, what do we know
about this slope? And in particular, can
we make any statements about that slope
relative to, say, 0 or 1 or anything like that? Well, let's think about
what a line of slope 1 would look like. Well, a line of
slope 1, especially one that went through this
point right over here, would look something like this. A line of slope 1 would
look something like this. So this line right over
here that I've just drawn that goes from
negative 1 comma 0 to 0, 1, this has slope 1. So this slope is equal to 1. So if this green line
has a slope of 1, does this blue
line have a slope-- it clearly has a
different slope. Is that slope, is
this blue line steeper or less steep than
the green line? Well, it's pretty clear
that this secant line is steeper than the green line. It's increasing faster. So it's going to
have a higher slope. So this, looking at
it from this diagram, this blue line has a
slope higher than 1. Or the slope of the secant
line from negative a, f of negative a, to
0 comma 1, that is going to be greater than 1. So this thing right over
here is greater than 1. So we're able to deduce,
this thing right over here is less than 1. This thing over here
is greater than 1. So this thing is
less than that thing. So this must be true. Now let's look at this one. We're comparing the
slope of the secant line. We're comparing the
slope of the secant line that we just looked at. So this is the same
value right over here. So we're comparing this
slope right over here, to what's this? f
of a minus 1 over a. Well, this is the slope
of this secant line, this is the slope
of this secant line that I'm drawing in this--
let me do it in more contrast. Let me do it in orange. That is the slope
of this secant line. So which one has a higher slope? Well, it's pretty clear
that the blue secant line has a higher slope than
this orange secant line. But here, it's saying
that the blue's slope is lower than the orange. So this is not going to be true. So this is not true. Then finally, let's look at
this over here-- f of a minus f of negative a over 2a. So this is the slope. Let me draw this. So this right over
here, this is the slope of the secant line
between this point and this point right over here. Our change in y is f of
a minus f of negative a. Our change in x is a minus
negative a, which is 2a. So this is this secant
line right over here. So let me draw it. So this secant line
right over here, so they're comparing
that slope to this slope. f of a minus 1 is our change in
y, over a is our change in x. So we're comparing it to
that one right over there. And you could immediately
eyeball this kind of brownish,
maroon-- I guess it's kind of a brown color-- this
secant line that goes all the way from here to here is
clearly steeper than this one right over here. And we know that, that
the average rate of change from here to here is
going to be higher than the average rate of change
from here to here, because at least from
negative a to 0, we were increasing at
a much faster rate. And then we slowed
down to this rate. So the average over
the entire interval is definitely going to be more
than what we get from 0 to a. So this one is also not true. This has a higher-- we actually
know that this is false. Both of these would
have been true if we swapped
these signs around, if this was a greater
than sign, if this was a greater than sign. So this is the only
one that applies.