课程： 10.2 在(h、k)处带有静脉排的帕拉波拉斯(Perabolas)

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选择节在(h, k) 时使用静脉排(Vetex)

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

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

You have already learned that don’t always have their vertex at $\begin{aligned} (0, 0) \end{aligned}$ . In this concept, we will address parabolas where the vertex is $\begin{aligned} (h, k) \end{aligned}$ , 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。

Recall that the equation of a parabola is $\begin{aligned} x^{2} = 4 p y \end{aligned}$ or $\begin{aligned} y^{2} = 4 p x \end{aligned}$ and the vertex is on the origin. Also, recall that the vertex form of a parabola is $\begin{aligned} y = a (x - h)^{2} + k \end{aligned}$ . 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。

$\begin{aligned} y & = a (x - h)^{2} + k a n d x^{2} = 4 p y & Solve the first for (x - h)^{2} . \\ (x - h)^{2} & = \frac{1}{a} (y - k) & We found that 4 p = \frac{1}{a} . \\ (x - h)^{2} & = 4 p (y - k) \end{aligned}$

::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)

If the parabola is horizontal, then the equation will be $\begin{aligned} (y - k)^{2} = 4 p (x - h) \end{aligned}$ . Notice, that even though the orientation is changed, the $\begin{aligned} h \end{aligned}$ and $\begin{aligned} k \end{aligned}$ values remain with the $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ 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。

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。

Notice that the way we find the focus and directrix does not change whether $\begin{aligned} p \end{aligned}$ 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

Let's analyze the equation $\begin{aligned} (y - 1)^{2} = 8 (x + 3) \end{aligned}$ . 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。

First, because $\begin{aligned} y \end{aligned}$ 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 $\begin{aligned} p \end{aligned}$ is positive. Next, find the vertex. Using the general equation, $\begin{aligned} (y - k)^{2} = 4 p (x - h) \end{aligned}$ , the vertex is $\begin{aligned} (- 3, 1) \end{aligned}$ and the axis of symmetry is $\begin{aligned} y = 1 \end{aligned}$ . Setting $\begin{aligned} 4 p = 8 \end{aligned}$ , we have that $\begin{aligned} p = 2 \end{aligned}$ . Adding $\begin{aligned} p \end{aligned}$ to the $\begin{aligned} x \end{aligned}$ -value of the vertex, we get the focus, $\begin{aligned} (- 1, 1) \end{aligned}$ . Subtracting $\begin{aligned} p \end{aligned}$ from the $\begin{aligned} x \end{aligned}$ -value of the vertex, we get the directrix, $\begin{aligned} x = - 5 \end{aligned}$ .
::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 。

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。

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 $\begin{aligned} x = 5 \end{aligned}$ , then $\begin{aligned} y \end{aligned}$ is either -7 or 9. This means that the points $\begin{aligned} (5, - 7) \end{aligned}$ and $\begin{aligned} (5, 9) \end{aligned}$ 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。

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。

The vertex of a parabola is $\begin{aligned} (- 2, 4) \end{aligned}$ and the directrix is $\begin{aligned} y = 7 \end{aligned}$ . 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。

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 $\begin{aligned} p \end{aligned}$ will be negative (plot this on an $\begin{aligned} x - y \end{aligned}$ 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 ) 。

To find $\begin{aligned} p \end{aligned}$ , we can use the vertex, $\begin{aligned} (h, k) \end{aligned}$ and the equation for a horizontal directrix, $\begin{aligned} y = k - p \end{aligned}$ .
::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 。

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

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

Now, using the general form, $\begin{aligned} (x - h)^{2} = 4 p (y - k) \end{aligned}$ , 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。

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

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

Examples
::实例

Example 1
::例1

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。

This parabola is of the form $\begin{aligned} (x - h)^{2} = 4 p (y - k) \end{aligned}$ . From the table earlier in this lesson, we can see that the focus of a parabola of this form is $\begin{aligned} (h, k + p) \end{aligned}$ . 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。

If we compare $\begin{aligned} (x + 4)^{2} = - 12 (y - 5) \end{aligned}$ to $\begin{aligned} (x - h)^{2} = 4 p (y - k) \end{aligned}$ , 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 :

$\begin{aligned} 4 = - h \end{aligned}$ or $\begin{aligned} h = - 4 \end{aligned}$
::4 h 或 h 4

$\begin{aligned} - 12 = 4 p \end{aligned}$ or $\begin{aligned} p = - 3 \end{aligned}$
::- 12=4p或p3

$\begin{aligned} 5 = k \end{aligned}$
::5=k 5=k

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

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

Example 2
::例2

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

The vertex is $\begin{aligned} (- 5, - 2) \end{aligned}$ and the parabola opens up because $\begin{aligned} p \end{aligned}$ is positive and $\begin{aligned} x \end{aligned}$ is squared. $\begin{aligned} 4 p = 2 \end{aligned}$ , making $\begin{aligned} p = \frac{1}{2} . \end{aligned}$ The focus is $\begin{aligned} (- 5, - 2 + 2) \end{aligned}$ or $\begin{aligned} (- 5, 0) \end{aligned}$ , the axis of symmetry is $\begin{aligned} x = - 5 \end{aligned}$ , and the directrix is $\begin{aligned} y = - 2 - \frac{1}{2} \end{aligned}$ or $\begin{aligned} y = - 2 \frac{1}{2} \end{aligned}$ .
::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。

Example 3
::例3

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

Example 4
::例4

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

The vertex is $\begin{aligned} (- 5, - 1) \end{aligned}$ , so $\begin{aligned} h = - 5 \end{aligned}$ and $\begin{aligned} k = - 1 \end{aligned}$ . The focus is $\begin{aligned} (- 8, - 1) \end{aligned}$ , meaning that that parabola will be horizontal. We know this because the $\begin{aligned} y \end{aligned}$ -values of the vertex and focus are both -1. Therefore, $\begin{aligned} p \end{aligned}$ is added or subtracted to $\begin{aligned} h \end{aligned}$ .
::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 。

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

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

Review
::回顾

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。

$\begin{aligned} (x + 1)^{2} = - 3 (y - 6) \end{aligned}$
:x+1)2%3(y-6)

$\begin{aligned} (x - 3)^{2} = y - 7 \end{aligned}$
:x-3-3)2=y-7

$\begin{aligned} (y + 2)^{2} = 8 (x + 1) \end{aligned}$
:y+2)2=8(x+1)

$\begin{aligned} y^{2} = - 10 (x - 3) \end{aligned}$
::y210(x- 3)

$\begin{aligned} (x + 6)^{2} = 4 (y + 8) \end{aligned}$
:x+6)2=4(y+8)

$\begin{aligned} (y - 5)^{2} = - \frac{1}{2} x \end{aligned}$
:y-55)212x

Graph the parabola from #1.
::从 # 1 绘制抛物线图。

Graph the parabola from #2.
::从 # 2 绘制抛物线图。

Graph the parabola from #4.
::从 # 4 绘制抛物线图。

Graph the parabola from #5.
::从 # 5 绘制抛物线图。

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。

vertex: $\begin{aligned} (2, - 1) \end{aligned}$ , focus: $\begin{aligned} (2, - 4) \end{aligned}$
::顶点2,-1),重点2,-4)

vertex: $\begin{aligned} (- 3, 6) \end{aligned}$ , directrix: $\begin{aligned} x = 2 \end{aligned}$
::顶点- 3, 6), 指向: x=2

vertex: $\begin{aligned} (6, 10) \end{aligned}$ , directrix: $\begin{aligned} y = 9.5 \end{aligned}$
::顶点: (6,10),指针:y=9.5

Challenge focus: $\begin{aligned} (- 1, - 2) \end{aligned}$ , directrix: $\begin{aligned} x = 3 \end{aligned}$
::挑战焦点-1,-2), 指向:x=3

Extension Rewrite the equation of the parabola, $\begin{aligned} x^{2} - 8 x + 2 y + 22 = 0 \end{aligned}$ , 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。

Review (Answers)
::回顾(答复)

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。

章节大纲

常规

在(h, k) 时使用静脉排(Vetex)

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

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Review ::回顾

Review (Answers) ::回顾(答复)

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

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Review
::回顾

Review (Answers)
::回顾(答复)