Understanding graph of this rational function

Question

I have this equation: $(4 - 3x - x^2)/(x - 3)$ and I am learning Pre-Calculus from coolmath and using desmos to graph.

I can easily calculate:

Y-intercept: $f(0) = (0,-4/3)$

X-intercepts: $4 - 3x - x^2 = 0$ ==> $(1,0)$ and $(-4,0)$.

Vertical Asymptote: $x - 3 = 0$ => $(3,0)$

No Horizontal Asymptote

Oblique Asymptote: $(4 - 3x - x^2)/(x - 3)$ ==> $-x - 6$

Graph came out to be like this:

Problem #1: I thought graph has 3 parts but it got 2 (there is nothing in the middle)

Problem #2: How to deduce the shape of the graph from all these calculations. I mean the the turns and curves and directions without plotting the points. With points anyone can plot it but without plotting, it requires some Mathematical-Thinking and that is the thing I want to develop.

Answer 1

Regarding your first problem:
You can guess the function has "2 parts" because it is continuous everywhere except $x=3$. so you know the line $x=3$ will divide the 2 parts where the function is continuous.

Regarding your second problem:
When graphing the function, look at:

1. points of discontinuity
2. roots of the functions (where the function equals 0)
3. minimum and maximum
4. notice the numerator is of degree 2 and the denominator is of degree 1, so the function acts like a linear function as x goes further from the origin

Answer 2

What you were doing there is some kind of curve sketiching but without the powerful tools of calculus itself. But the things you denoted there are quite useful aswell.

First of all you know something about the asymptotic behavior of the function in two ways. So you can just say that the function will go to infinity while the x-value is decreasing and it will go to negative infinity while the x-value is increasing. These you can just derivate from the oblique asymptote you figured out. So now you know how the function will develop with the x argument.

As a next step concentrate on the oblique asymptote. You know the y-value will reach infinity in the near of this asymptote. When you put this together with the roots of the polynomial you can go further with the form of the graph.

Both roots are smaller than the x-value of the oblique asymptote. Also the function goes for infinity for big negative x-values. So the graph will fall while you going to the first root but right after this it have to rise against since the graph has to reach the next root AND has to go for infinity at the oblique asymptote.

On the other hand the right side of the graph never hits the x-axis hence it goes from negative infinity up while the x-value is decreasing. So if the function would now go for infinity again there would be another interception with the x-axis. So it goes to negative infinity.

To put it all together: From the asymptotes you know how the function behaves at critical points such as infinity, negative infinity and at poles. Now you just focus on the values of the roots and compare them with the poles. From there on you can sketch the graph in a way of thinking how it develops at several x-values.

Answer 3

Your function $$ y = (4−3x−x^2)/(x−3) $$

Can be rewritten as \begin{align} 0 &= (x-3) y + x^2 + 3x - 4 \\ &= xy - 3y + x^2 + 3x - 4 \\ &= x^2 + xy + 3x - 3y - 4 \\ \end{align} which is an instance of a conic section $$ A x^2 + B xy + Cy^2 + D x + E y + F = 0 $$ This is an ancient study subject. There is much known how the coefficients will select what solution curves.

In your case it is a hyperbola.