绘制Rational方程据/h1>
已经有账户了?据一种Href="//www.parkandroid.com/account/login/?next=/wiki/graphing-rational-equations/" class="ax-click" data-ax-id="clicked_signup_modal_login" data-ax-type="link">这里登录。据/a>
测验据/h4>
有关……据/h4>
- 代数据/span>>据/span>
检查孔据/h2>
- 根据分解分子和分母,找到任何常见因素。比如说据span class="katex">
y据/span>=据/span>G据/span>(据/span>X据/span>)据/span>H据/span>(据/span>X据/span>)据/span>F据/span>(据/span>X据/span>)据/span>H据/span>(据/span>X据/span>)据/span>,在那里据span class="katex">
F据/span>(据/span>X据/span>)据/span>和据span class="katex">
G据/span>(据/span>X据/span>)据/span>没有任何常见因素。据/li>
- 对于任何据span class="katex"> X据/span>*据/span>值,这样据span class="katex"> H据/span>(据/span>X据/span>*据/span>)据/span>=据/span>0.据/span>,我们会有一个洞据span class="katex"> X据/span>=据/span>X据/span>*据/span>相应的据span class="katex"> y据/span>=据/span>G据/span>(据/span>X据/span>*据/span>)据/span>F据/span>(据/span>X据/span>*据/span>)据/span>.记录下来。据/li>
- 约去公因式,求出据span class="katex"> y据/span>=据/span>G据/span>(据/span>X据/span>)据/span>F据/span>(据/span>X据/span>)据/span>.据/li>
- 对于任何据span class="katex"> X据/span>*据/span>值,这样据span class="katex"> H据/span>(据/span>X据/span>*据/span>)据/span>=据/span>0.据/span>,我们会有一个洞据span class="katex"> X据/span>=据/span>X据/span>*据/span>相应的据span class="katex"> y据/span>=据/span>G据/span>(据/span>X据/span>*据/span>)据/span>F据/span>(据/span>X据/span>*据/span>)据/span>.记录下来。据/li>
例如,图表据span class="katex">
y据/span>=据/span>(据/span>X据/span>-据/span>1据/span>)据/span>(据/span>X据/span>-据/span>2据/span>)据/span>X据/span>-据/span>1据/span>与的图表相似吗据span class="katex">
y据/span>=据/span>X据/span>-据/span>2据/span>1据/span>尖端有个洞据span class="katex">
(据/span>1据/span>那据/span>-据/span>1据/span>)据/span>.据/p>
从此,我们将绘制图表据span class="katex">
y据/span>=据/span>G据/span>(据/span>X据/span>)据/span>F据/span>(据/span>X据/span>)据/span>,在那里据span class="katex">
G据/span>光盘据/span>(据/span>F据/span>(据/span>X据/span>)据/span>那据/span>G据/span>(据/span>X据/span>)据/span>)据/span>=据/span>1据/span>.据/p>
发现拦截据/h2>
确定据span class="katex"> y据/span>拦截。据/h3>
替代据span class="katex">
X据/span>=据/span>0.据/span>代入方程据span class="katex">
y据/span>-Intercept将是据span class="katex">
G据/span>(据/span>0.据/span>)据/span>F据/span>(据/span>0.据/span>)据/span>.据/p>
这发生在据span class="katex">
y据/span>=据/span>0.据/span>,或者分子为0时。解出据span class="katex">
F据/span>(据/span>X据/span>)据/span>=据/span>0.据/span>.据/p>
确定据span class="katex">
X据/span>拦截。据/h3>
确定正/负无穷大的行为据/h2>
执行据一种Href="//www.parkandroid.com/wiki/partial-fractions-linear-factors/" class="wiki_link" title="部分分数GydF4y2Ba" target="_blank">部分分数据/a>:据/p>
G据/span>(据/span>X据/span>)据/span>F据/span>(据/span>X据/span>)据/span>=据/span>一种据/span>(据/span>X据/span>)据/span>+据/span>C据/span>(据/span>X据/span>)据/span>B.据/span>(据/span>X据/span>)据/span>那据/span>
在哪里据span class="katex">
deydF4y2BaG据/span>B.据/span>(据/span>X据/span>)据/span>据据/span>deydF4y2BaG据/span>C据/span>(据/span>X据/span>)据/span>.然后,对于大值据span class="katex">
X据/span>那据span class="katex">
C据/span>(据/span>X据/span>)据/span>B.据/span>(据/span>X据/span>)据/span>将接近0。据/p>
因此,我们知道大量值,据span class="katex">
y据/span>≈据/span>一种据/span>(据/span>X据/span>)据/span>.然后我们需要确定大的正(和大负)值据span class="katex">
X据/span>,如果据span class="katex">
y据/span>将在上面或以下据span class="katex">
一种据/span>(据/span>X据/span>)据/span>.通过检查迹象很容易确定据span class="katex">
B.据/span>(据/span>X据/span>)据/span>和据span class="katex">
C据/span>(据/span>X据/span>)据/span>.据/p>
你可能对这个很熟悉:据/p>
让据span class="katex">
deydF4y2BaG据/span>F据/span>=据/span>m据/span>和据span class="katex">
deydF4y2BaG据/span>G据/span>=据/span>N据/span>.据/p>
情况1:据span class="katex">
m据/span>据据/span>N据/span>.据B.r>该图将具有水平渐近的据span class="katex">
y据/span>=据/span>0.据/span>.据/p>
案例2:据span class="katex">
m据/span>=据/span>N据/span>.让领先的系数据span class="katex">
F据/span>(据/span>X据/span>)据/span>是据span class="katex">
F据/span>的前导系数据span class="katex">
G据/span>(据/span>X据/span>)据/span>是据span class="katex">
G据/span>.然后,曲线的水平渐近线为据span class="katex">
y据/span>=据/span>G据/span>F据/span>.据/p>
案例3:据span class="katex">
m据/span>>据/span>N据/span>.据B.r>没有水平渐近线。据span class="katex">
□据/span> 这包括在上述分析中:据/p>
案例1将给我们据span class="katex">
一种据/span>(据/span>X据/span>)据/span>=据/span>0.据/span>那据B.r>案例2将给我们据span class="katex">
一种据/span>(据/span>X据/span>)据/span>=据/span>G据/span>F据/span>是一个常数,据B.r>情形3会给出这个据span class="katex">
一种据/span>(据/span>X据/span>)据/span>不是常数。据/p>
寻找和检查垂直渐近据/h2>
当分母为0时,会发生垂直渐近斑点。解决据span class="katex">
G据/span>(据/span>X据/span>)据/span>=据/span>0.据/span>.据/p>
对于每个解决方案据span class="katex">
X据/span>*据/span>,我们需要确定图表在左侧和右侧的样子据span class="katex">
X据/span>*据/span>.我们知道它将倾向于无限(因此是渐近的),但需要弄清楚它是否倾向于积极的无穷大或负无穷大。据/p>
为此,我们检查分解术语的迹象是什么,然后将它们乘以确定最终标志。据/p>
绘制信息和草图据/h2>
上面的分析为你提供了很多关于图表的信息。是时候把这些信息写在纸上了。请参阅下面的示例了解如何做到这一点。据/p>
我们如何画出它的图像据span class="katex"> y据/span>=据/span>X据/span>-据/span>4.据/span>2据/span>X据/span>+据/span>3.据/span>还是据/h3>
步骤0:据B.r>检查孔。由于分子和分母没有共同因素,因此没有孔。据/p>
步骤1:据B.r>确定据span class="katex"> y据/span>拦截。这发生在据span class="katex"> X据/span>=据/span>0.据/span>,这给了我们据span class="katex"> y据/span>=据/span>0.据/span>-据/span>4.据/span>2据/span>×据/span>0.据/span>+据/span>3.据/span>=据/span>4.据/span>-据/span>3.据/span>.据/p>
步骤2:据B.r>确定据span class="katex"> X据/span>拦截。这是分子为0时的情况据span class="katex-display"> 0.据/span>=据/span>X据/span>-据/span>4.据/span>2据/span>X据/span>+据/span>3.据/span>⇒据/span>2据/span>X据/span>+据/span>3.据/span>=据/span>0.据/span>⇒据/span>X据/span>=据/span>-据/span>2据/span>3.据/span>.据/span>
第3步:据B.r>确定无穷远处的行为。因为分子和分母都是线性多项式(相同次),我们知道我们将得到一条水平渐近线据span class="katex"> X据/span>=据/span>1据/span>2据/span>=据/span>2据/span>.据/p>
我们有据span class="katex"> y据/span>=据/span>X据/span>-据/span>4.据/span>2据/span>X据/span>+据/span>3.据/span>=据/span>2据/span>+据/span>X据/span>-据/span>4.据/span>1据/span>1据/span>.因此,这告诉我们据span class="katex-display"> 作为据/span>X据/span>→据/span>∞据/span>那据/span>y据/span>→据/span>2据/span>+据/span>和据/span>作为据/span>X据/span>→据/span>-据/span>∞据/span>那据/span>y据/span>→据/span>2据/span>-据/span>.据/span>
第四步:据B.r>求任意的垂直渐近线。这是分母为0时的情况。据/p>
解决据span class="katex"> X据/span>-据/span>4.据/span>=据/span>0.据/span>,我们得到据span class="katex"> X据/span>=据/span>4.据/span>作为唯一的垂直渐近。让我们确定行为:据span class="katex-display"> 作为据/span>X据/span>→据/span>4.据/span>+据/span>那据/span>y据/span>作为据/span>X据/span>→据/span>4.据/span>-据/span>那据/span>y据/span>=据/span>X据/span>-据/span>4.据/span>2据/span>X据/span>+据/span>3.据/span>〜据/span>+据/span>+据/span>因此据/span>y据/span>→据/span>+据/span>∞据/span>.据/span>=据/span>X据/span>-据/span>2据/span>2据/span>X据/span>+据/span>3.据/span>〜据/span>-据/span>+据/span>因此据/span>y据/span>→据/span>-据/span>∞据/span>.据/span>
第5步:据B.r>绘制信息。你应该得到以下内容。圆圈表示我们使用的步骤编号。据/p>
最后,将信息连接起来,画出图。你应该能够画出这条曲线(粗黑线)。据/p>
工作的例子据/h2>
绘制图表据span class="katex"> y据/span>=据/span>X据/span>2据/span>+据/span>2据/span>X据/span>-据/span>3.据/span>X据/span>2据/span>-据/span>4.据/span>X据/span>+据/span>3.据/span>.据/h3>
步骤0:0据B.r>我们可以把图分解成据span class="katex"> (据/span>X据/span>-据/span>1据/span>)据/span>(据/span>X据/span>+据/span>3.据/span>)据/span>(据/span>X据/span>-据/span>3.据/span>)据/span>(据/span>X据/span>-据/span>1据/span>)据/span>,所以有一个洞据span class="katex"> X据/span>=据/span>1据/span>.的据span class="katex"> y据/span>-Value将是据span class="katex"> 1据/span>+据/span>3.据/span>1据/span>-据/span>3.据/span>=据/span>4.据/span>-据/span>2据/span>=据/span>-据/span>0.据/span>.据/span>5.据/span>.据/p>
我们想要的是据span class="katex"> y据/span>=据/span>X据/span>+据/span>3.据/span>X据/span>-据/span>3.据/span>那据/span>X据/span>据/span>=据/span>1据/span>.据/p>
步骤1:据span class="katex"> y据/span>拦截据B.r> X据/span>=据/span>0.据/span>那据/span>y据/span>=据/span>-据/span>3.据/span>3.据/span>=据/span>-据/span>1据/span>.据/p>
步骤2:据span class="katex"> X据/span>拦截据B.r>记住,我们现在只和据span class="katex"> X据/span>+据/span>3.据/span>X据/span>-据/span>3.据/span>而不是据span class="katex"> X据/span>2据/span>+据/span>2据/span>X据/span>-据/span>3.据/span>X据/span>2据/span>-据/span>4.据/span>X据/span>+据/span>3.据/span>.因此,这是据span class="katex"> X据/span>拦截时据span class="katex"> X据/span>-据/span>3.据/span>=据/span>0.据/span>⇒据/span>X据/span>=据/span>3.据/span>.据/p>
步骤3:无限的行为据B.r>我们在线性多项式上具有线性多项式,因此有一个地平线渐近的据span class="katex"> y据/span>=据/span>1据/span>1据/span>=据/span>1据/span>.用部分分式,我们有据span class="katex"> y据/span>=据/span>X据/span>+据/span>3.据/span>X据/span>-据/span>3.据/span>=据/span>1据/span>-据/span>X据/span>+据/span>3.据/span>6.据/span>.据/p>
作为据span class="katex"> X据/span>→据/span>∞据/span>那据span class="katex"> -据/span>X据/span>+据/span>3.据/span>6.据/span>〜据/span>(据/span>-据/span>)据/span>+据/span>+据/span>=据/span>-据/span>那据/span>所以据span class="katex"> y据/span>→据/span>1据/span>-据/span>.据B.r>作为据span class="katex"> X据/span>→据/span>-据/span>∞据/span>那据span class="katex"> -据/span>X据/span>+据/span>3.据/span>6.据/span>〜据/span>(据/span>-据/span>)据/span>-据/span>+据/span>=据/span>+据/span>那据/span>所以据span class="katex"> y据/span>→据/span>1据/span>+据/span>.据/p>
第四步:垂直渐近线据B.r>分母是0何时据span class="katex"> X据/span>+据/span>3.据/span>=据/span>0.据/span>⇒据/span>X据/span>=据/span>-据/span>3.据/span>.据/p>
作为据span class="katex"> X据/span>→据/span>-据/span>3.据/span>+据/span>那据span class="katex"> y据/span>=据/span>X据/span>+据/span>3.据/span>X据/span>-据/span>3.据/span>〜据/span>+据/span>-据/span>=据/span>-据/span>那据/span>所以据span class="katex"> y据/span>→据/span>-据/span>∞据/span>.据B.r>作为据span class="katex"> X据/span>→据/span>-据/span>3.据/span>-据/span>那据span class="katex"> y据/span>=据/span>X据/span>+据/span>3.据/span>X据/span>-据/span>3.据/span>〜据/span>-据/span>-据/span>=据/span>+据/span>那据/span>所以据span class="katex"> y据/span>→据/span>+据/span>∞据/span>.据/p>
步骤5:绘制信息,并绘制图形。据/p>
绘制图表据span class="katex"> y据/span>=据/span>X据/span>+据/span>1据/span>X据/span>2据/span>+据/span>1据/span>.据/h3>
步骤0:没有洞。据/p>
步骤1:据span class="katex"> y据/span>拦截据B.r> X据/span>=据/span>0.据/span>那据/span>y据/span>=据/span>0.据/span>+据/span>1据/span>0.据/span>2据/span>+据/span>1据/span>=据/span>1据/span>.据/p>
步骤2:据span class="katex"> X据/span>拦截据B.r>解据span class="katex"> X据/span>2据/span>+据/span>1据/span>=据/span>0.据/span>,没有解决方案。因此没有据span class="katex"> X据/span>拦截。据/p>
步骤3:无限的行为据B.r>因为我们有一个2次多项式除以一个1次多项式,所以没有水平渐近线。相反,我们需要使用部分分式。据/p>
我们有据span class="katex"> y据/span>=据/span>X据/span>+据/span>1据/span>X据/span>2据/span>+据/span>1据/span>=据/span>X据/span>-据/span>1据/span>+据/span>X据/span>+据/span>1据/span>2据/span>.这意味着据span class="katex"> y据/span>≈据/span>X据/span>-据/span>1据/span>.据/p>
作为据span class="katex"> X据/span>→据/span>∞据/span>那据/span>X据/span>+据/span>1据/span>2据/span>→据/span>0.据/span>+据/span>那据/span>所以据span class="katex"> y据/span>≈据/span>X据/span>-据/span>1据/span>+据/span>.据B.r>作为据span class="katex"> X据/span>→据/span>∞据/span>那据/span>X据/span>+据/span>1据/span>2据/span>→据/span>0.据/span>-据/span>那据/span>所以据span class="katex"> y据/span>≈据/span>X据/span>-据/span>1据/span>-据/span>.据/p>
第4步:垂直渐近据B.r>让分母为0,我们得到据span class="katex"> X据/span>+据/span>1据/span>=据/span>0.据/span>⇒据/span>X据/span>=据/span>-据/span>1据/span>.据/p>
作为据span class="katex"> X据/span>→据/span>-据/span>1据/span>+据/span>那据span class="katex"> y据/span>=据/span>X据/span>+据/span>1据/span>X据/span>2据/span>+据/span>1据/span>〜据/span>+据/span>+据/span>=据/span>+据/span>那据/span>所以据span class="katex"> y据/span>→据/span>∞据/span>.据B.r>作为据span class="katex"> X据/span>→据/span>-据/span>1据/span>-据/span>那据span class="katex"> y据/span>=据/span>X据/span>+据/span>1据/span>X据/span>2据/span>+据/span>1据/span>〜据/span>+据/span>+据/span>=据/span>-据/span>那据/span>所以据span class="katex"> y据/span>→据/span>-据/span>∞据/span>.据/p>
第五步:绘制信息和图表。据/p>
绘制图表据span class="katex"> y据/span>=据/span>(据/span>X据/span>+据/span>1据/span>)据/span>(据/span>X据/span>+据/span>3.据/span>)据/span>X据/span>-据/span>2据/span>.据/span>
步骤0:没有洞。据/p>
步骤1:据span class="katex"> y据/span>拦截据B.r> X据/span>=据/span>0.据/span>那据/span>y据/span>=据/span>(据/span>0.据/span>+据/span>1据/span>)据/span>(据/span>0.据/span>+据/span>3.据/span>)据/span>0.据/span>-据/span>2据/span>=据/span>3.据/span>-据/span>2据/span>.据/p>
第2步:X截距。据B.r> X据/span>-据/span>2据/span>=据/span>0.据/span>⇒据/span>X据/span>=据/span>2据/span>.据/p>
第3步:无限的行为。据B.r>因为我们有一个二次多项式上的线性多项式,水平渐近线是据span class="katex"> y据/span>=据/span>0.据/span>.据/p>
作为据span class="katex"> X据/span>→据/span>∞据/span>那据span class="katex"> y据/span>=据/span>(据/span>X据/span>+据/span>1据/span>)据/span>(据/span>X据/span>+据/span>3.据/span>)据/span>X据/span>-据/span>2据/span>〜据/span>(据/span>+据/span>)据/span>(据/span>+据/span>)据/span>+据/span>=据/span>+据/span>那据/span>所以据span class="katex"> y据/span>→据/span>0.据/span>+据/span>.据B.r>作为据span class="katex"> X据/span>→据/span>-据/span>∞据/span>那据span class="katex"> y据/span>=据/span>(据/span>X据/span>+据/span>1据/span>)据/span>(据/span>X据/span>+据/span>3.据/span>)据/span>X据/span>-据/span>2据/span>〜据/span>(据/span>-据/span>)据/span>(据/span>-据/span>)据/span>-据/span>=据/span>+据/span>那据/span>所以据span class="katex"> y据/span>→据/span>0.据/span>-据/span>.据/p>
第四步:垂直渐近线。据/p>
分母为0,我们有据span class="katex"> (据/span>X据/span>+据/span>1据/span>)据/span>(据/span>X据/span>+据/span>3.据/span>)据/span>=据/span>0.据/span>,这给了我们据span class="katex"> X据/span>=据/span>-据/span>1据/span>那据/span>X据/span>=据/span>-据/span>3.据/span>.据/p>
[据/span>X据/span>=据/span>-据/span>1据/span>]据/span>
作为据span class="katex"> X据/span>→据/span>-据/span>1据/span>+据/span>那据span class="katex"> y据/span>=据/span>(据/span>X据/span>+据/span>1据/span>)据/span>(据/span>X据/span>+据/span>3.据/span>)据/span>X据/span>-据/span>2据/span>〜据/span>(据/span>+据/span>)据/span>(据/span>+据/span>)据/span>-据/span>=据/span>-据/span>那据/span>所以据span class="katex"> y据/span>→据/span>-据/span>∞据/span>.据B.r>作为据span class="katex"> X据/span>→据/span>-据/span>1据/span>-据/span>那据span class="katex"> y据/span>=据/span>(据/span>X据/span>+据/span>1据/span>)据/span>(据/span>X据/span>+据/span>3.据/span>)据/span>X据/span>-据/span>2据/span>〜据/span>(据/span>-据/span>)据/span>(据/span>+据/span>)据/span>-据/span>=据/span>+据/span>那据/span>所以据span class="katex"> y据/span>→据/span>+据/span>∞据/span>.据/p>[据/span>X据/span>=据/span>-据/span>3.据/span>]据/span>
作为据span class="katex"> X据/span>→据/span>-据/span>3.据/span>+据/span>那据span class="katex"> y据/span>=据/span>(据/span>X据/span>+据/span>1据/span>)据/span>(据/span>X据/span>+据/span>3.据/span>)据/span>X据/span>-据/span>2据/span>〜据/span>(据/span>-据/span>)据/span>(据/span>+据/span>)据/span>-据/span>=据/span>+据/span>那据/span>所以据span class="katex"> y据/span>→据/span>+据/span>∞据/span>.据B.r>作为据span class="katex"> X据/span>→据/span>-据/span>3.据/span>-据/span>那据span class="katex">