Graph of the function $\frac{(x-1)(x-4)}{(x-2)(x-3)}$

Question

We are given the function $$\frac{(x-1)(x-4)}{(x-2)(x-3)}$$ and we have to draw graph of this.

We know its domain will be all real numbers except $x=2$ and $x=3$; also, it has a local minimum at $x=\frac52$.

But now how to proceed?

Answer 1

We can find out many more factoids about the graph: There is a zero at $x=1$ and one at $x=4$, both with a change of sign, the poles at $x=2$, $x=3$ are also with change of sign. We have a horizontal asymptote $y=1$ as $x\to\pm\infty$. Additionally, the graph is in fact symmetric to $x=\frac52$. These should be enough guiding hints to allow you making a good enough sketch of the graph:

Coming from negative infinity the graph is close to the line $y=1$ and falls down to intersect the $x$ axis at $x=1$ and then tend to $-\infty$ as $x\to 2^-$; to the right of $x=2$, the graph comes down again from $+\infty$, passes through the minimum at $x=\frac52$ 8and $y=9$, as you may compute) just do go up to $+\infty$ again as we approach $3$ from the left; to the right of $3$ the graph comes up again from $-\infty$, crosses the $x$ axis at $x=4$ and huddles against the line $y=1$ again.