1. ## Curve Sketching

For my project, I have to pick an equation and sketch it.
I picked (x^2-3x+4)/(x^2+2x-1) and I found that there is a horizontal asymptote at y=1. But when I plug in 1 in to the equation there is a point at (1,1). How does that happen?

2. Hello, dimpleyy!

A common misunderstanding . . .
(I went through it myself many years ago.)

For my project, I have to pick an equation and sketch it.
I picked: .$\displaystyle y \:=\:\frac{x^2-3x+4}{x^2+2x-1}$
and I found that there is a horizontal asymptote at $\displaystyle y=1.$
But when I plug in 1 in to the equation there is a point at (1,1).
How does that happen?

A horizontal asymptote indicates the behavior of $\displaystyle f(x)$ for large values of $\displaystyle x.$

That is, as we go to the far right (or far left), the graph approaches $\displaystyle y = 1.$

However, locally, the function can have $\displaystyle y = 1.$

The graph might look like this . . .
Code:
                |
|        *
|     *    *
|   *       *
|  *           *
| *                 *
|                          *
- - - - - +o- - - - - - - - - - - - - - - -
|
-----------*------------------------------------
|
|
*|
|
* |
*  |

You see, the graph can cross the horizontal asymptote.
. . The asymptote takes over on extreme values only.