Now we'll look at one more example
of determining the equation of a rational function
from the graph of the function.
And we'll find our rational function in factored form
or this form here, where a is the constant,
and then we'll have factors in the numerator
and factors in the denominator
based upon the properties of the graph.
So looking at the graph, let's start
by identifying our intercepts.
Notice how the x-intercept here is positive one,
this would be the point (1,0),
but notice how at this x-intercept,
the graph doesn't cross the x-axis, it touches it,
and then bounces back.
This will affect the factors in our rational function.
And then we also have a y-intercept here
at negative two, so the function contain the point (0,-2).
Now let's take a look at our vertical asymptotes.
We have one here at x equal two.
We have another one, it's a little bit more difficult
to see, but it is x equals negative one.
And I do want to point out that notice
that x equals two, on one side, the graphs approaches upward
toward the vertical asymptote,
and the other side it approaches downward,
but at x equals negative one,
notice how on both sides, the graph is approaching
downward toward the vertical asymptote.
This is going to affect, again,
the factors of our rational function.
We also need to recognize that the knowing the equations
of the vertical asymptotes, gives us information
about the factors of the denominator,
and knowing the x-intercept, or the zero of the function,
gives us information about the factors of numerator.
And then we have this extra point here,
the y-intercept, which we can use
to determine this constant value, or the value of a.
So let's begin to set this up.
We'll have f of x
equals a times a rational function
based upon the properties of the graph.
We'll find a in the second step.
So let's start by considering the vertical asymptotes.
Remember, we find the vertical asymptotes
by finding the zeros of the denominator,
which means if x equals negative one
is a vertical asymptote, then x plus one
must be a factor in the denominator.
But there's more to it than just that.
Because of how the graph behaves here,
meaning it approaches the vertical asymptote
downward on both sides, we're going to have
to have two factors of x plus one.
And then because we have a vertical asymptote
at x equals two, we must have a factor of x minus two.
Again, because it approaches up on one side
and down on the other, we only need
one factor of x minus two.
Now let's consider the factors of the numerator.
Because of the x-intercept (1,0),
we know f of one equals zero,
which means one is a zero of the function,
and the zeros of the function come from
where the numerator is equal to zero.
So if one is a zero of the function, the numerator
must contain a factor of x minus one.
But again, there's more to it than just that,
because the graph touch at x equals one,
but doesn't cross the x-axis,
we need two factors of x minus one.
Now for the last step, we'll determine the value of a
using the y-intercept or point (0,-2),
we know that f of zero must equal negative two.
So we'll find f of zero by substituting zero for x,
and this will allow us to find the value of a.
So we'll have a times,
again, if x is zero, we would have negative one squared
in the numerator, and then we'd have positive one squared
in the denominator, and also a negative two.
And this must equal negative two.
So looking at this fraction here,
we'd have positive one over negative two,
so we have negative one-half a equals negative two.
Multiplying both sides by the reciprocal
of negative one-half,
that would be negative two over one,
or just negative two if we want.
This simplifies to a.
On the right we have positive four,
which means our rational function is
f of x equals, if a is four,
we'll have four times the quantity x minus one squared
divided by two factors of x plus one,
and one factor of x minus two.
And this the function that we're looking for.
Notice how the degree of the numerator is two,
the degree of the denominator would be three,
and because the degree of the numerator is less
than the degree of the denominator,
we would have a horizontal asymptote of y equals zero.
Looking at our graph, y equals zero is the x-axis.
Notice how the graph is approaching the x-axis
to the left and to the right.
Okay, I hope you found this helpful.