作图有理函数，N = M - 2题

我想绘制一些合理的功能，其中分子的程度等于分母的程度。下面是一个例子;我有一个理性的功能，其中分子作为二次和分母也是二次，所以我们有一个超过2度第一件事，我想要做的是，确定X拦截并渐近学位2。

The x intercepts come from the zeros in the numerator and since the numerator is factored, it’s very easy to see the two and 4s are zeros, and that gives me x intercepts of (2, 0) and (4, 0), and I’m going to plot those immediately, (2, 0) and (4, 0).

该垂直渐近线来自于零点，分母，x等于零和x等于3，水平渐近来自分子和分母的领先系数。So you could just mentally multiply out the numerator just far enough to know what the leading term is, x times x, x², and there’s other terms, x times x, again x² plus some other terms, and the leading coefficients are 1 and 1. So the horizontal asymptote, is going to be y equals 1/1, y equals 1.

让我把这些渐近线画出来趁我还记得。我们有x = 0，这是y轴。x = 3，就在这两个截距之间y = 1，就在这里。现在我要做的是画一些点，把渐近线和截距之间的情况画出来。我们从左边的区域开始。我们在平面的负部分画一些点。

让我们来试试-1。如果我插上-1我得到-3，-5倍，这将是15比-1倍-4，这是4，15比4这是3和¾。所以（-1，3¾），这是约在这里。我们将尝试一些远一点向左像-3。-3减2，-5，-3倍减去4，-7。-3倍-6，所​​以我们得到35/18。这不是一个数量很大，但它是非常，非常接近十八分之三十六这是2。所以，我会画出类似-3,2喜欢这里，有点小于2，但接近的地方。而这实际上是足够的信息来引起我们的函数的这个分支。它会涨这个渐近线，并向下接近这一个。类似的东西。

现在，让我们关注这两个垂直的渐近线之间的区域。We’ve got this intercept, it’s going to pass through the axis here but I want to know whether it’s below the x axis or above the x axis at this point, so let me plot x equals 1. I get 1 minus 2, -1, 1 minus 4, -3, and 1 minus 3, -2. So I have 3 over -2, -3/2. At 1 I have -3/2 at about here. Now what’s probably going to happen is, the graph is going to go down and approach this vertical asymptote. It’s going to have asymptotic behavior in this direction, and it has to have asymptotic behavior in this direction as well. So it’s going to have to go up like this, through the horizontal asymptote and just go up and approach the asymptote from the left something like that. Let’s graph this part.

What I want to do is figure out what happens to the right of this intercept, let’s plot, 1, 2, 3, , 4, 5, x equals 5. I get 5 minus 2, 3 times 5 minus 4, 1. 5 minus 3, 5 times 2. So this is 3 over 10, .3. It’s about a third. So at this point I am at 5.3 about here. I am going to approach this line asymptotically and this one asymptotically. That’s what my graph looks like. I’ve got a branch up here, I've got a piece that crosses the horizontal asymptote in the middle, and then this piece goes against this vertical asymptote and up, and approaches this horizontal asymptote from below.

请记住，我们可以跨越水平渐近线。一个图实际上可以跨越水平渐近线多次，但最终他们不得不他们接近右和左。这是垂直渐近线，因为函数不是在这些点定义永远不能越过;在这种情况下零和3。