10.2 在(h、k)处带有静脉排的帕拉波拉斯(Perabolas)

Section outline

• Select section General

  Collapse Expand

  General

  Collapse all Expand all

  • Select activity 新闻通告

    

    新闻通告 Forum
• Select section 在(h, k) 时使用静脉排(Vetex)

  Collapse Expand

  在(h, k) 时使用静脉排(Vetex)

  播放段落

  Your homework assignment is to find the focus of the parabola $\begin{array}{r}(x+4{)}^2=-12(y-5)\end{array}$ . You say the focus is $\begin{array}{r}(-4,5)\end{array}$ . Banu says the focus is $\begin{array}{r}(0,-3)\end{array}$ . Carlos says the focus is $\begin{array}{r}(-4,2)\end{array}$ . Which one of you is correct?
  ::您的作业任务是找到抛物线( x+4) 2\\\\\\\\\\\\\\\\\\5\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

  播放段落

  Parabolas with Vertex at (h, k)
  ::在(h, k) 时使用静脉排(Vetex)

  播放段落

  You have already learned that don’t always have their vertex at $\begin{array}{r}(0,0)\end{array}$ . In this concept, we will address parabolas where the vertex is $\begin{array}{r}(h,k)\end{array}$ , learn how to find the focus, directrix and graph.
  ::您已经知道, 他们的顶点并不总是在 0,0 。 在这个概念中, 我们将解决顶点所在的parabolas (h,k) , 学习如何找到焦点、 方向和图表 。

  播放段落

  Recall that the equation of a parabola is $\begin{array}{r}{x}^2=4py\end{array}$ or $\begin{array}{r}{y}^2=4px\end{array}$ and the vertex is on the origin. Also, recall that the  vertex form of a parabola is $\begin{array}{r}y=a(x-h{)}^2+k\end{array}$ . Combining the two, we can find the vertex form for conics.
  ::回顾抛物线的方程是 x2=4py 或 y2= 4px, 顶点在原位。 另外, 提醒注意抛物线的顶点形式是 y=a( x-h) 2+k。 将两者结合起来, 我们能找到二次曲线的顶点形式 。

  播放段落

  $$\begin{array}{rlrl}y& =a(x-h{)}^2+kand{x}^2=4py& & \text{Solve the first for}(x-h{)}^2.\\ (x-h{)}^2& =\frac{1}{a}(y-k)& & \text{We found that}4p=\frac{1}{a}.\\ (x-h{)}^2& =4p(y-k)\end{array}$$

  
  ::y=a (x-h) 2+k 和 x2= 4pySolve 的第一个 (x-h) 2. (x-h) 2= 1a(y-k) 我们发现 4p= 1a. (x-h) 2= 4p(y-k)

  播放段落

  If the parabola is horizontal, then the equation will be $\begin{array}{r}(y-k{)}^2=4p(x-h)\end{array}$ . Notice, that even though the orientation is changed, the $\begin{array}{r}h\end{array}$ and $\begin{array}{r}k\end{array}$ values remain with the $\begin{array}{r}x\end{array}$ and $\begin{array}{r}y\end{array}$ values, respectively.
  ::如果抛物线是水平的,则方程将是 (y-k)2=4p(x-h) 。 注意, 即使方向有所改变, h 和 k 值仍分别与 x 和 y 值相同 。

  播放段落

  Finding the focus and directrix are a little more complicated. Use the extended table below to help you find these values.
  ::查找焦点和指向比较复杂。 使用下面的扩展表格来帮助您查找这些值 。

  

  播放段落

  Notice that the way we find the focus and directrix does not change whether $\begin{array}{r}p\end{array}$ is positive or negative.
  ::注意我们找到焦点和指针的方式 不会改变 p 是正是负

  播放段落

  Let's analyze the equation $\begin{array}{r}(y-1{)}^2=8(x+3)\end{array}$ . We'll find the vertex, axis of , focus, and directrix. Then, we'll determine if the function opens up, down, left or right.
  ::让我们分析方程 (y- 1) 2=8( x+ 3) 。 我们会找到顶点、 轴、 焦点和指向。 然后, 我们会确定函数向上、 向下、 向左或向右打开 。

  播放段落

  First, because $\begin{array}{r}y\end{array}$ is squared, we know that the parabola will open to the left or right. We can conclude that the parabola will open to the right because 8 is positive, meaning that $\begin{array}{r}p\end{array}$ is positive. Next, find the vertex. Using the general equation, $\begin{array}{r}(y-k{)}^2=4p(x-h)\end{array}$ , the vertex is $\begin{array}{r}(-3,1)\end{array}$ and the axis of symmetry is $\begin{array}{r}y=1\end{array}$ . Setting $\begin{array}{r}4p=8\end{array}$ , we have that $\begin{array}{r}p=2\end{array}$ . Adding $\begin{array}{r}p\end{array}$ to the $\begin{array}{r}x\end{array}$ -value of the vertex, we get the focus, $\begin{array}{r}(-1,1)\end{array}$ . Subtracting $\begin{array}{r}p\end{array}$ from the $\begin{array}{r}x\end{array}$ -value of the vertex, we get the directrix, $\begin{array}{r}x=-5\end{array}$ .
  ::首先,因为y是正方形, 我们知道抛物线会向左或右打开。 我们可以得出结论, 抛物线会向右打开, 因为 8 是正值, 意味着 p 是正值 。 接下来, 找到顶点 。 使用一般方程, (y- k) 2= 4p( x- h) , 顶点是 (- 3, 1) , 而对称轴是 y= 1 。 设置 4p= 8, 我们有p=2 。 将 p 添加到顶点的 x 值上, 我们得到焦点, (-1) 。 从顶点的 x 值中减去 p , 我们得到直线, x = 5 。

  播放段落

  Let's graph the parabola from the problem above . Plot the vertex, axis of symmetry, focus, and directrix.
  ::让我们从上面的问题绘制抛物线图。 绘制顶点、 对称轴、 焦点轴和直线轴 。

  播放段落

  First, plot all the critical values we found earlier . Then, determine a set of symmetrical points that are on the parabola to make sure your curve is correct. If $\begin{array}{r}x=5\end{array}$ , then $\begin{array}{r}y\end{array}$ is either -7 or 9. This means that the points $\begin{array}{r}(5,-7)\end{array}$ and $\begin{array}{r}(5,9)\end{array}$ are both on the parabola.
  ::首先,绘制我们先前发现的所有关键值。 然后, 确定在抛物线上的一组对称点以确保您的曲线正确。 如果 x=5, 那么 y 是 - 7 或 9 。 这意味着这些点( 5, - 7) 和 ( 5, 9) 都在 抛物线上 。

  

  播放段落

  It is important to note that parabolas with a horizontal orientation are not functions because they do not pass the vertical line test.
  ::必须指出,横向方向的parabolas不是功能,因为它们没有通过垂直线测试。

  播放段落

  The vertex of a parabola is $\begin{array}{r}(-2,4)\end{array}$ and the directrix is $\begin{array}{r}y=7\end{array}$ . Let's find the equation of the parabola.
  ::抛物线的顶部是 (-2,4) , 直线是 Y= 7 。 让我们找到抛物线的方程 。

  播放段落

  First, let’s determine the orientation of this parabola. Because the directrix is horizontal, we know that the parabola will open up or down (see table/pictures above). We also know that the directrix is above the vertex, making the parabola open down and $\begin{array}{r}p\end{array}$ will be negative (plot this on an $\begin{array}{r}x-y\end{array}$ plane if you are unsure).
  ::首先,让我们来决定这个抛物线的方向。 由于直线是水平的,我们知道抛物线会向上或向下打开(见上文表格/图片 ) 。 我们还知道,直线线在顶端,使抛物线向下打开,p将是负的(如果不确定的话,在X-y平面上绘制这个图 ) 。

  播放段落

  To find $\begin{array}{r}p\end{array}$ , we can use the vertex, $\begin{array}{r}(h,k)\end{array}$ and the equation for a horizontal directrix, $\begin{array}{r}y=k-p\end{array}$ .
  ::要找到p, 我们可以使用顶点, (h, k) 和横向直线的方程, y=k- p 。

  播放段落

  $$\begin{array}{rl}7& =4-p\\ 3& =-p& & \text{Remember},p\text{is negative because of the downward orientation of the parabola}.\\ -3& =p\end{array}$$

  
  ::7=4-p3p remember, p是负的, 因为抛物线向下偏向。- 3=p

  播放段落

  Now, using the general form, $\begin{array}{r}(x-h{)}^2=4p(y-k)\end{array}$ , we can find the equation of this parabola.
  ::现在,使用(x-h)2=4p(y-k)的一般形式,我们可以找到这个抛物线的方程式。

  播放段落

  $$\begin{array}{rl}(x-(-2){)}^2& =4(-3)(y-4)\\ (x+2{)}^2& =-12(y-4)\end{array}$$

  
  :-(-2))2=4(-3-4(x+2)2)12(y-4)

  播放段落

  Examples
  ::实例

  播放段落

  Example 1
  ::例1

  播放段落

  Earlier, you were asked to determine which student is correct. 
  ::早些时候,有人要求你确定哪个学生是正确的。

  播放段落

  This parabola is of the form $\begin{array}{r}(x-h{)}^2=4p(y-k)\end{array}$ . From the table earlier in this lesson, we can see that the focus of a parabola of this form is $\begin{array}{r}(h,k+p)\end{array}$ . So now we have to find h , k , and p .
  ::此抛物线为表( x-h) 2= 4p(y-k) 。 从本课前面的表格中,我们可以看到这种抛物线的焦点是 (h, k+p) 。 所以我们现在必须找到 h, k 和 p。

  播放段落

  If we compare $\begin{array}{r}(x+4{)}^2=-12(y-5)\end{array}$ to $\begin{array}{r}(x-h{)}^2=4p(y-k)\end{array}$ , we see that:
  ::如果我们比较 (x+4) 212(y- 5) 到 (x- h) 2= 4p(y- k), 我们可以看到 :

  播放段落
  1. $\begin{array}{r}4=-h\end{array}$ or $\begin{array}{r}h=-4\end{array}$
     ::4 h 或 h 4
  播放段落
  2. $\begin{array}{r}-12=4p\end{array}$ or $\begin{array}{r}p=-3\end{array}$
     ::- 12=4p或p3
  播放段落
  3. $\begin{array}{r}5=k\end{array}$
     ::5=k 5=k

  播放段落

  From these facts we can find $\begin{array}{r}k+p=5+(-3)=2\end{array}$ .
  ::从这些事实中我们可以找到 k+p=5+(-3)=2。

  播放段落

  Therefore, the focus of the parabola is $\begin{array}{r}(-4,2)\end{array}$ and Carlos is correct.
  ::因此,抛物线的焦点是(-4,2),而Carlos的焦点是正确的。

  播放段落

  Example 2
  ::例2

  播放段落

  Find the vertex, focus, axis of symmetry and directrix of $\begin{array}{r}(x+5{)}^2=2(y+2)\end{array}$ .
  ::查找(x+5)2=2(y+2)的顶部、焦点、对称轴和直线轴。

  播放段落

  The vertex is $\begin{array}{r}(-5,-2)\end{array}$ and the parabola opens up because $\begin{array}{r}p\end{array}$ is positive and $\begin{array}{r}x\end{array}$ is squared. $\begin{array}{r}4p=2\end{array}$ , making $\begin{array}{r}p=\frac{1}{2}.\end{array}$  The focus is $\begin{array}{r}(-5,-2+2)\end{array}$ or $\begin{array}{r}(-5,0)\end{array}$ , the axis of symmetry is $\begin{array}{r}x=-5\end{array}$ , and the directrix is $\begin{array}{r}y=-2-\frac{1}{2}\end{array}$ or $\begin{array}{r}y=-2\frac{1}{2}\end{array}$ .
  ::顶点是(-5,-2),抛物线打开是因为p是正数,x是正数。 4p=2, 制成p=12。 焦点是(-5,-2+2)或(-5,0),对称轴是x+5, 直线是y2-12 或y212。

  播放段落

  Example 3
  ::例3

  播放段落

  Graph the parabola from Example 2.
  ::从例2中绘制抛物线图。

  

  播放段落

  Example 4
  ::例4

  播放段落

  Find the equation of the parabola with vertex $\begin{array}{r}(-5,-1)\end{array}$ and focus $\begin{array}{r}(-8,-1)\end{array}$ .
  ::找到有顶层(-5)和焦点(-8)的抛物线方程式。

  播放段落

  The vertex is $\begin{array}{r}(-5,-1)\end{array}$ , so $\begin{array}{r}h=-5\end{array}$ and $\begin{array}{r}k=-1\end{array}$ . The focus is $\begin{array}{r}(-8,-1)\end{array}$ , meaning that that parabola will be horizontal. We know this because the $\begin{array}{r}y\end{array}$ -values of the vertex and focus are both -1. Therefore, $\begin{array}{r}p\end{array}$ is added or subtracted to $\begin{array}{r}h\end{array}$ .
  ::顶点是 (-5, - 1), 所以 h5 和 k1。 焦点是 (-8, - 1) , 意指 parbola 将是水平的。 我们知道这一点, 因为顶点和焦点的 Y 值是 - 1 。 因此, p 被添加或减去到 h 。

  播放段落

  $\begin{array}{r}(h+p,k)\to (-8,-1)\end{array}$ we can infer that $\begin{array}{r}h+p=-8\to -5+p=-8\end{array}$ and $\begin{array}{r}p=-3\end{array}$
  :h+p,k)\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\-\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

  播放段落

  Therefore, the equation is $\begin{array}{r}(y-(-1){)}^2=4(-3)(x-(-5))\to (y+1{)}^2=-12(x+5)\end{array}$ .
  ::因此,等式是(y-(-1)2=4(-3)(x-(-5))__(y+1)2=12(x+5)。

  播放段落

  Review
  ::回顾

  播放段落

  Find the vertex, focus, axis of symmetry, and directrix of the parabolas below.
  ::找到以下抛物线的顶部、焦点、对称轴和直线。

  播放段落
  1. $\begin{array}{r}(x+1{)}^2=-3(y-6)\end{array}$
     :x+1)2%3(y-6)
  播放段落
  2. $\begin{array}{r}(x-3{)}^2=y-7\end{array}$
     :x-3-3)2=y-7
  播放段落
  3. $\begin{array}{r}(y+2{)}^2=8(x+1)\end{array}$
     :y+2)2=8(x+1)
  播放段落
  4. $\begin{array}{r}{y}^2=-10(x-3)\end{array}$
     ::y210(x- 3)
  播放段落
  5. $\begin{array}{r}(x+6{)}^2=4(y+8)\end{array}$
     :x+6)2=4(y+8)
  播放段落
  6. $\begin{array}{r}(y-5{)}^2=-\frac{1}{2}x\end{array}$
     :y-55)212x
  播放段落
  7. Graph the parabola from #1.
     ::从 # 1 绘制抛物线图。
  播放段落
  8. Graph the parabola from #2.
     ::从 # 2 绘制抛物线图。
  播放段落
  9. Graph the parabola from #4.
     ::从 # 4 绘制抛物线图。
  播放段落
  10. Graph the parabola from #5.
      ::从 # 5 绘制抛物线图。

  播放段落

  Find the equation of the parabola given the vertex and either the focus or directrix.
  ::根据顶点和焦点或指向来查找抛物线的方程。

  播放段落
  11. vertex: $\begin{array}{r}(2,-1)\end{array}$ , focus: $\begin{array}{r}(2,-4)\end{array}$
      ::顶点2,-1),重点2,-4)
  播放段落
  12. vertex: $\begin{array}{r}(-3,6)\end{array}$ , directrix: $\begin{array}{r}x=2\end{array}$
      ::顶点- 3, 6), 指向: x=2
  播放段落
  13. vertex: $\begin{array}{r}(6,10)\end{array}$ , directrix: $\begin{array}{r}y=9.5\end{array}$
      ::顶点: (6,10),指针:y=9.5
  播放段落
  14. Challenge focus: $\begin{array}{r}(-1,-2)\end{array}$ , directrix: $\begin{array}{r}x=3\end{array}$
      ::挑战焦点-1,-2), 指向:x=3
  播放段落
  15. Extension Rewrite the equation of the parabola, $\begin{array}{r}{x}^2-8x+2y+22=0\end{array}$ , in standard form by completing the square. Then, find the vertex.
      ::扩展名通过完成方形以标准格式重写抛物线( x2-8x+2y+22=0)的方程。然后找到顶点。

  播放段落

  Review (Answers)
  ::回顾(答复)

  播放段落

  Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
  ::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键 。