Course: 9.2 parabolas | The Way To Learn

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抛物体

When working with parabolas in the past you probably used vertex form and analyzed the graph by finding its roots and intercepts. There is another way of defining a parabola that turns out to be more useful in the real world. One of the many uses of parabolic shapes in the real world is satellite dishes. In these shapes it is vital to know where the receptor point should be placed so that it can absorb all the signals being reflected from the dish.
::与 parabolas 一起工作时, 你可能会使用顶点形式, 并通过找到其根部和截取器来分析图形。还有另一种方法可以定义在现实世界中更有用的抛物线。在现实世界中抛物线形状的多种用途之一是卫星天线。在这些形状中, 关键是要知道受体点应放置在哪里, 以便吸收从天线上反射的所有信号。

Where should the receptor be located on a satellite dish that is four feet wide and nine inches deep?
::四英尺宽九英寸深的卫星天线上的受体应位于何处?

Graphing Parabolas
::图示图

The definition of a parabola is the collection of points equidistant from a point called the focus and a line called the directrix .
::抛物线的定义是从一个称为焦点的点和直线的直线上收集等距点。

Notice how the three points $\begin{aligned} P_{1}, P_{2}, P_{3} \end{aligned}$ are each connected by a blue line to the focus point $\begin{aligned} F \end{aligned}$ and the directrix line $\begin{aligned} L \end{aligned}$ .
::请注意三点P1、P2、P3通过一条蓝线与焦点点F和直线L连接的方式。

\begin{aligned} \bar{F P_{1}} & = \bar{P_{1} Q_{1}} \\ \bar{F P_{2}} & = \bar{P_{2} Q_{2}} \\ \bar{F P_{3}} & = \bar{P_{3} Q_{3}} \end{aligned}

::P1P1Q1F2P2Q2F3P3Q3

There are two graphing equations for parabolas that will be used in this concept. The only difference is one equation graphs parabolas opening vertically and one equation graphs parabolas opening horizontally. You can recognize the parabolas opening vertically because they have an $\begin{aligned} x^{2} \end{aligned}$ term . Likewise, parabolas opening horizontally have a $\begin{aligned} y^{2} \end{aligned}$ term.
::此概念中将使用两种parabolas的图形方程式。唯一的区别是, 一种是垂直打开的 parabolas 方程式, 另一种是水平打开的 parapolas 方程式。您可以识别垂直打开的 parabolas , 因为它们有一个 x2 术语。同样, 水平打开的parbolas 也有一个 y2 术语。

The general equation for a parabola opening vertically is $\begin{aligned} (x - h)^{2} = \pm 4 p (y - k) \end{aligned}$ . The general equation for a parabola opening horizontally is $\begin{aligned} (y - k)^{2} = \pm 4 p (x - h) \end{aligned}$ .
::垂直打开的抛物线的一般方程是 (x-h)24p(y-k),水平打开的抛物线的一般方程是 (y-k)24p(x-h) 。

Note that the vertex is still $\begin{aligned} (h, k) \end{aligned}$ . The parabola opens upwards or to the right if the $\begin{aligned} 4 p \end{aligned}$ is positive. The parabola opens down or to the left if the $\begin{aligned} 4 p \end{aligned}$ is negative. The focus is just a point that is distance $\begin{aligned} p \end{aligned}$ away from the vertex. The directrix is just a line that is distance $\begin{aligned} p \end{aligned}$ away from the vertex in the opposite direction. You can sketch how wide the parabola is by noting the focal width is $\begin{aligned} | 4 p | \end{aligned}$ .
::注意顶点仍然 (h,k) 。抛物线向上或向右打开, 如果 4p 是正的。抛物线向下或向左打开, 如果 4p 是负的, 则向下或向左打开。焦点只是一个离顶点距离 p 。直线只是一条线, 距离顶点在相反方向。您可以通过注意焦宽 +4p 来绘制抛物线的宽度。

Once you put the parabola into this graphing form you can sketch the parabola by plotting the vertex, identifying $\begin{aligned} p \end{aligned}$ and plotting the focus and directrix and lastly determining the focal width and sketching the curve.
::一旦您将抛物线放入此图形形式, 您可以通过绘制顶点、识别 p 、绘制焦点和直线, 最后确定焦距宽度和绘制曲线来绘制抛物线。

Take the conic:
::使用二次曲线 :

$\begin{aligned} 2 x^{2} + 16 x + y = 0 \end{aligned}$
::2x2+16x+y=0

This is a parabola because the $\begin{aligned} y^{2} \end{aligned}$ coefficient is zero.
::这是一个抛物线,因为 y2 系数是零。

\begin{aligned} x^{2} + 8 x & = - \frac{1}{2} y \\ x^{2} + 8 x + 16 & = - \frac{1}{2} y + 16 \\ (x + 4)^{2} & = - \frac{1}{2} (y - 32) \\ (x + 4)^{2} & = - 4 \cdot \frac{1}{8} (y - 32) \end{aligned}

::x2+8x_4x2}12y2+8x+16}12y+16(x+4)2}12(y-32)(x+4)2}4}18(y-32)

The vertex is (-4, 32). The focal length is $\begin{aligned} p = \frac{1}{8} \end{aligned}$ . This parabola opens down which means that the focus is at $\begin{aligned} (- 4, 32 - \frac{1}{8}) \end{aligned}$ and the directrix is horizontal at $\begin{aligned} y = 32 + \frac{1}{8} \end{aligned}$ . The focal width is $\begin{aligned} \frac{1}{2} \end{aligned}$ .
::顶点为 (4, 32) 。焦距为 p=18。此抛物线向下打开, 这意味着焦点在( - 4, 32 - 18) 上, 直线在 y= 32 +18 上水平。焦距为 12 。

Examples
::实例

Example 1
::例1

Earlier, you were asked where the receptor should be located on a satellite dish that is four feet wide and nine inches deep.
::早些时候,有人问过你,受体应放在四英尺宽九英寸深的卫星天线上的位置。

Since real world problems do not come with a predetermined coordinate system, you can choose to make the vertex of the parabola at (0, 0). Then, if everything is done in inches, another point on the parabola will be (24, 9). (Many people might mistakenly believe the point (48, 9) is on the parabola but remember that half this width stretches to (-24, 9) as well.) Using these two points, the focal width can be found.
::由于现实世界的问题并不带有预先确定的坐标系统,你可以选择在0,0时将抛物线的顶端设定为(0,0),然后,如果每件事情都以英寸为单位,抛物线上的另一个点将是(24,9)。 (许多人可能错误地认为抛物线上的点(48,9)是(48,9),但记住这一宽度的一半也伸展到(24,9))。利用这两个点,可以找到焦宽。

$\begin{aligned} (x - 0)^{2} & = 4 p (y - 0) \\ (24 - 0)^{2} & = 4 p (9 - 0) \\ \frac{24^{2}}{4 \cdot 9} & = p \\ 16 & = p \end{aligned}$

x-0)2=4p(y-0-0)(24-0)2=4p(9-0)24-24_9=p16=p)

The receptor should be sixteen inches away from the vertex of the parabolic dish.
::受体应该离抛物线盘的顶部16英寸

Example 2
::例2

Sketch the following parabola and identify the important pieces of information.
::绘制以下抛物线并查明重要信息。

$\begin{aligned} (y + 1)^{2} = 4 \cdot \frac{1}{2} \cdot (x + 3) \end{aligned}$
:y+1)2=412(x+3)

The vertex is at (-3, -1). The parabola is sideways because there is a $\begin{aligned} y^{2} \end{aligned}$ term. The parabola opens to the right because the $\begin{aligned} 4 p \end{aligned}$ is positive. The focal length is $\begin{aligned} p = \frac{1}{2} \end{aligned}$ which means the focus is $\begin{aligned} \frac{1}{2} \end{aligned}$ to the right of the vertex at (-2.5, -1) and the directrix is $\begin{aligned} \frac{1}{2} \end{aligned}$ to the left of the vertex at $\begin{aligned} x = - 3.5 \end{aligned}$ . The focal width is 2 which is why the width of the parabola stretches from (-2.5, 0) to (-2.5, -2).
::顶点在( 3, - 1) , 抛物线是侧面, 因为有一个 y2 术语。抛物线向右打开, 因为 4p 是正数。焦距为 p=12 , 这意味着焦点在 (- 2.5, - 1) 的顶点右侧是 12 , 直线在 x3.5 的顶点左侧是 12 。焦宽在 2 , 这就是为什么抛物线的宽度从 (- 2.5, 0) 到 (- 2.5, - 2) 。

Example 3
::例3

What is the equation of a parabola that has a focus at (4, 3) and a directrix of $\begin{aligned} y = - 1 \end{aligned}$ ?
::以(4, 3)为焦点的抛物线和 y1 的准点等式是什么?

It would probably be useful to graph the information that you have in order to reason about where the vertex is.
::或许应该用图表说明你掌握的信息,以便说明顶点在哪里。

The vertex must be halfway between the focus and the directrix. This places it at (4, 1). The focal length is 2. The parabola opens upwards. This is all the information you need to create the equation.
::顶点必须介于焦点和直线之间的一半。此处将它置于 (4, 1) 。焦距为 2 。抛物线向上打开。这是您创建方程式所需要的全部信息。

$\begin{aligned} (x - 4)^{2} = 4 \cdot 2 \cdot (y - 1) \end{aligned}$
:x-4)2=42(y-1)

OR $\begin{aligned} (x - 4)^{2} = 8 (y - 1) \end{aligned}$
:x-4)2=8(y-1)

Example 4
::例4

What is the equation of a parabola that opens to the right with focal width from (6, -7) to (6, 12)?
::向右开通的具有从(6,7)到(6,12)等距的焦距的抛物线等式是什么?

The focus is in the middle of the focal width. The focus is $\begin{aligned} (6, \frac{5}{2}) \end{aligned}$ . The focal width is 19 which is four times the focal length so the focal length must be $\begin{aligned} \frac{19}{4} \end{aligned}$ . The vertex must be a focal length to the left of the focus, so the vertex is at $\begin{aligned} (6 - \frac{19}{4}, \frac{5}{2}) \end{aligned}$ . This is enough information to write the equation of the parabola.
::焦点在焦点宽度的中间。焦点是 (6, 52) 。焦点宽度为 19 , 是焦点长度的四倍, 所以焦点长度必须是 194 。顶点必须是焦点左侧的焦点长度, 所以顶点是 (6-194, 52) 。这足以写出 parbola 的方程式。

$\begin{aligned} {(y - \frac{5}{2})}^{2} = 4 \cdot \frac{19}{4} \cdot (x - 6 + \frac{19}{4}) \end{aligned}$
:y-52)2=4194(x-6+194)

Example 5
::例5

Sketch the following conic by putting it into graphing form and identifying important information.
::将以下二次曲线拼贴成图表形式并识别重要信息,以图解如下曲线。

$\begin{aligned} y^{2} - 4 y + 12 x - 32 = 0 \end{aligned}$
::y2 - 4y+12x- 32=0

\begin{aligned} y^{2} - 4 y & = - 12 x + 32 \\ y^{2} - 4 y + 4 & = - 12 x + 32 + 4 \\ (y - 2)^{2} & = - 12 (x - 3) \\ (y - 2)^{2} & = - 4 \cdot 3 \cdot (x - 3) \end{aligned}

::y2 - 4y=12x+32y2- 4y+412x+32+4(y- 2)2°12(x-3)(y-2)2°43(x-3)

The vertex is at (3, 2). The focus is at (0, 2). The directrix is at $\begin{aligned} x = 6 \end{aligned}$ .
::顶点在(3,2),焦点在(0,2),直线在x=6。

Summary

A parabola is defined as the collection of points equidistant from a point called the focus and a line called the directrix.
::抛物线的定义是,从一个称为焦点的点和一个称为直线的线上收集等距点。

There are two graphing equations for parabolas: one for parabolas opening vertically (with an $\begin{aligned} x^{2} \end{aligned}$ term) and one for parabolas opening horizontally (with a $\begin{aligned} y^{2} \end{aligned}$ term).
::parabolas有两种图形式方程式:一种是垂直打开的parabolas(x2条件),一种是横向打开的parabolas(y2条件)。

The general equation for a parabola opening vertically is $\begin{aligned} (x - h)^{2} = \pm 4 p (y - k), \end{aligned}$ and for a parabola opening horizontally, it is $\begin{aligned} (y - k)^{2} = 4 p (x - h) . \end{aligned}$
- The vertex of the parabola is $\begin{aligned} (h, k), \end{aligned}$ and the parabola opens upwards or to the right if the value of $\begin{aligned} 4 p \end{aligned}$ is positive, and down or to the left if the value of $\begin{aligned} p \end{aligned}$ is negative.
  ::抛物线的顶点是(h,k),如果 4p 值为正值,抛物线向上或向右打开,如果 4p 值为负值,则向下或向左打开,如果 p 值为负值,则向下或向左打开。
- The focus is a point that is distance $\begin{aligned} p \end{aligned}$ away from the vertex,
  ::焦点是距离顶端的距离点,
- The directrix is a line that is distance $\begin{aligned} p \end{aligned}$ away from the vertex in the opposite direction
  ::直线指向是一条线距离正相反方向的顶部
- The focal width is $\begin{aligned} | 4 p | . \end{aligned}$
  ::焦宽为4p。
::垂直打开的抛物线的一般方程是 (x-h) 24p(y-k),横向打开的抛物线一般方程是 (y-k) 2= 4p(x-h) 。抛物线的顶点是 k , 而如果 4p 值为正值, 则抛物线向上或向右打开, 如果 4p 值为负值, 则向下或向左打开。焦点是距离顶端的距离点, 直线是一条与相反方向的顶点距离 p 。焦宽度是 +4p 。

Review
::回顾

1. What is the equation of a parabola with focus at (1, 4) and directrix at $\begin{aligned} y = - 2 \end{aligned}$ ?
::1. 以(1、4)为焦点的抛物线和以 y##2为焦点的指针等式是什么?

2. What is the equation of a parabola that opens to the left with focal width from (-2, 5) to (-2, -7)?
::2. 向左打开的具有从(-2)至(-2)至(-2)至(-2)至(7)等距的焦距的抛物线是什么等式?

3. What is the equation of a parabola that opens to the right with vertex at (5, 4) and focal width of 12?
::3. 以5、4和12焦距为顶层向右打开的抛物线的等式是什么?

4. What is the equation of a parabola with vertex at (1, 8) and directrix at $\begin{aligned} y = 12 \end{aligned}$ ?
::4. 以Y=12为顶点(1、8)和指针(y=12)为顶点的抛物线和顶点的等式是什么?

5. What is the equation of a parabola with focus at (-2, 4) and directrix at $\begin{aligned} x = 4 \end{aligned}$ ?
::5. 以(-2, 4)为焦点的抛物线和x=4的指针的等式是什么?

6. What is the equation of a parabola that opens downward with a focal width from (-4, 9) to (16, 9)?
::6. 向下开放的从(4、9)至(16、9)的焦距宽度从(4、9)到(16、9)的抛物线的等式是什么?

7. What is the equation of a parabola that opens upward with vertex at (1, 11) and focal width of 4?
::7. 在(1、11)和4焦距上方有顶部和焦距的抛物线是多少等式?

Sketch the following parabolas by putting them into graphing form and identifying important information.
::以图示形式和识别重要信息的形式,绘制以下图例。

8. $\begin{aligned} y^{2} + 2 y - 8 x + 33 = 0 \end{aligned}$
::8. y2+2y-8x+33=0

9. $\begin{aligned} x^{2} - 8 x + 20 y + 36 = 0 \end{aligned}$
::9. x2-8x+20y+36=0

10. $\begin{aligned} x^{2} + 6 x - 12 y - 15 = 0 \end{aligned}$
::10. x2+6x- 12y- 15=0

11. $\begin{aligned} y^{2} - 12 y + 8 x + 4 = 0 \end{aligned}$
::11. y2 - 12y+8x+4=0

12. $\begin{aligned} x^{2} + 6 x - 4 y + 21 = 0 \end{aligned}$
::12. x2+6x-4y+21=0

13. $\begin{aligned} y^{2} + 14 y - 2 x + 59 = 0 \end{aligned}$
::13. y2+14y-2x+59=0

14. $\begin{aligned} x^{2} + 12 x - \frac{8}{3} y + \frac{92}{3} = 0 \end{aligned}$
::14. x2+12x-83y+923=0

15. $\begin{aligned} x^{2} + 2 x - \frac{4}{5} y + 1 = 0 \end{aligned}$
::15. x2+2x-45y+1=0

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。

Section outline

General

抛物体

Graphing Parabolas ::图示图

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)