We are going to write an equation
for the graph of the rational function,
based upon the graph.
To do this, we will find the equation
of the rational function in factored form,
or this form here given below,
where A is a constant,
and we will have factors in the numerator,
and factors in the denominator.
Let's start by identifying the
key characteristics of the graph.
Let's start with our intercepts.
Notice how this graph has an X intercept here of -2.
This is the point, (-2,0).
It has a y-intercept of -2 as well.
This would be the point, (0,-2).
Now, let's find our vertical asymptotes.
We have a vertical asymptote here,
at x=-4,
and we have another vertical asymptote here, at x=+1.
This is enough information for us
to determine our rational function.
We do have to remember that if we know
we have vertical asymptotes at X equals -4,
and X equals +1, this gives us
information about the factors in the denominator.
If we have an X intercept of -2,
this gives us information about a factor in the numerator.
Then, using the Y intercept, or this point here,
we can determine the value of our constant, A.
Let's begin to set this up over here on the right.
We will have f(x)= --
We will find A in the second step,
so we will have a times a rational function.
Let's begin with our vertical asymptotes.
Again, knowing the vertical asymptotes
gives us information about the factors in the denominator.
Remember, the vertical asymptotes are found
by finding the zeros of the denominator,
or these are the values that make
the denominator equal to zero.
If we have a vertical asymptote at x=-4,
x+4 must be a factor in the denominator.
If we have a vertical asymptote of x=1,
x-1 must be a factor of the denominator.
Now, let's talk about the X intercept.
If the function contains the point (-2,0),
that means f(-2) must equal zero.
Well, rational functions are equal to zero
when the numerators are equal to zero.
X=-2 is a zero of the numerator,
Which means the numerator must contain a factor of x+2.
Now, I have everything except the value of A.
We can find the value of A by using
this point here, or our y-intercept.
If the function contains the point (0,-2),
that means f(0) must equal -2.
We will find f(0), and set it equal to -2,
and then solve for A.
We will have A times --
Again, X is equal to zero,
so we would have 2 in the numerator.
The denominator is going to be four times -1,
and this must equal -2.
This is going to be -2/4, or -1/2.
We have -1/2a=-2.
Now, I'm going to go ahead and multiply
both sides by the reciprocal of -1/2, or -2/1.
This simplifies to A, and -2 times -2 is +4,
which means our rational function, f(x),
Is equal to four times this fraction,
or 4(x+2), divided by
the quantity, (x+4)(x-1).
This function contains the
characteristics of the given graph.
Notice how the degree of the numerator is degree one,
the degree of the denominator is degree two.
Since the degree of the numerator is
less than the degree of the denominator,
we would have a horizontal asymptote of Y=0.
Looking at the graph, Y=0 is the x-axis,
which is our horizontal asymptote.
So, everything checks.
I hope you found this helpful.