节: 抛物体 | 11.3 抛物线

章节大纲

Introduction
::导言

One of the many uses of parabolic shapes in the real world is satellite dishes. With these shapes, it is vital to know where the receptor point should be placed so it can absorb all the signals being reflected from the dish.
::在现实世界中,抛物线形状的许多用途之一是卫星天线盘。有了这些天线盘,关键是要知道受体点应该放在哪里,这样它才能吸收从天线盘中反射的所有信号。

Where should the receptor be located on a satellite dish that is 4 feet wide and 9 inches deep?
::位于四英尺宽九英寸深的卫星天线上的受体应位于何处?

The Parabola
::帕拉波拉

Recall that a parabola is generated when a plane is tilted so that it is parallel to one generator and intersects the other generator and one nappe of the cone. The definition of a parabola is the collection of points equidistant from a point called the focus and a line called the directrix .
::回顾当飞机倾斜时会产生抛物线,使之与一个发电机平行,并交叉另一个发电机和锥形的一个环形。抛物线的定义是从一个称为焦点的点和直线的线上收集等距点。

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 $\begin{aligned} L \end{aligned}$ .
::请注意这三个点,即P1、P2、P3, 每一个点是如何通过一条蓝线与焦点点Fand直线L相连的。

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

::P1P1Q1F2P2Q2F3P3Q3

There are two graphing equations for parabolas that will be used in this concept. The only difference is that 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, 一种公式方程式的 parabolas 垂直打开, 另一种公式方程式的 parabolas 水平打开。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。

Standard Equation of Parabola
::帕拉波拉标准方程式

The general equation for a parabola opening vertically is $\begin{aligned} (x - h)^{2} = \pm 4 p (y - k) \end{aligned}$ .
::垂直开口的抛物线的一般方程为(x-h)2+4p(y-k)。

The general equation for a parabola opening horizontally is $\begin{aligned} (y - k)^{2} = \pm 4 p (x - h) \end{aligned}$ .
::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 a distance of $\begin{aligned} p \end{aligned}$ units away from the vertex. The directrix is a line that is a distance of $\begin{aligned} p \end{aligned}$ units 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。焦点只是一个点, 就是离顶点距离 p 单位的距离。直线是一条线, 离相反方向的顶点距离 p 单位的距离。您可以通过注意焦点宽度 +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, 您可以绘制抛物线, 绘制顶点, 识别 p, 绘制焦点和直线, 最后确定焦距宽度, 绘制曲线。

The following video defines a parabola and explains how to graph a parabola in standard form:
::以下视频定义了抛物线,并解释了如何以标准格式绘制抛物线图:

The following video explains how to graph a parabola given in general form by rewriting it in standard form:
::以下视频解释如何用标准格式重写一般形式的抛物线线图解:

Play, Learn, and Explore Parabolas:
::玩耍、学习和探索帕拉波拉斯:

Examples
::实例

Example 1
::例1

Identify the conic below. Then put it into graphing form and identify its vertex, focal length ( p ), focus, directrix, and focal width.
::识别下面的二次曲线。然后将其放入图形形式, 并确定它的顶部、焦距( p) 、焦点、直线和焦宽。

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

Solution:
::解决方案 :

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 length from the focus to the vertex 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 $\begin{aligned} \frac{1}{2} \end{aligned}$ .
::顶点是 (-4, 32) 。从焦点到顶点的长度是 p=18。此抛物线向下打开, 这意味着焦点在 (-4, 32 - 18) , 直线在 y= 32 + 18 水平。焦距为 12 。

Example 2
::例2

Sketch the following parabola and identify the important pieces of information:
::绘制下列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)。

Solution:
::解决方案 :

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 length from the focus to the vertex 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) , 抛物线是侧面的, 因为有一个 y2 术语。抛物线向右打开, 因为 4p 是正数。从焦点到顶点的长度是 p=12, 这意味着焦点在( 2.5, - 1) 的顶点右边是 12 , 直线在 x 3.5 的顶点左边是 12 。焦宽是 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?

Solution:
::解决方案 :

It would probably be useful to graph the information you have in order to reason about where the vertex is.
::或许应该用图表来说明你掌握的信息,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 length from the focus to the vertex 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。这将将其放置在( 4, 1) 上, 从焦点到顶点的长度是 2 。抛物线向上打开。这是您创建方程所需要的全部信息。

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

:x-4)2=42(y-1)(x-4)2=8(y-1)

Example 4
::例4

Recall the question from the Introduction: Where should the receptor be located on a satellite dish that is 4 feet wide and 9 inches deep?
::回顾导言中的问题:受体应位于四英尺宽九英寸深的卫星天线上何处?

Solution:
::解决方案 :

Convert feet to inches to get 48 inches, and then taking half for either side, use 24 in your calculations.
::将双脚转换为英寸,获得48英寸,然后将两侧的一半取出,在计算时使用24英寸。

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

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

The receptor should be 16 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 5
::例5

What is the equation of a parabola with focus at (2, 3) and directrix at $\begin{aligned} y = 5 \end{aligned}$ ?
::以 2 、 3 点为焦点的抛物线和以 y= 5 点为焦点的指向等式是什么?

Solution:
::解决方案 :

The vertex must lie directly between the focus and the directrix, so it must be at (2, 4). The focal length is therefore equal to 1. The parabola opens downwards.
::顶点必须直接位于焦点和准点之间,所以必须位于(2,4),因此,焦距等于1。抛物线向下打开。

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

:x-2)2,2 41(y-4)(x-2)2,44(y-4)

Example 6
::例6

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?

Solution:
::解决方案 :

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 4 times the length from the focus to the vertex. T he l ength from the focus to the vertex 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。焦点为 (6, 52) 。焦点宽度为 19, 是焦点到顶点的四倍。从焦点到顶点的长度必须是194。顶点必须是焦点左侧的焦点长度, 因此顶点在( 6-194, 52) 。这足够信息来写入 parbola 的方程。

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

:y-52)2=4194(x-6+194)(y-52)2=19(x-54)

Example 7
::例7

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,以图解方式绘制以下二次曲线:

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

::y2-4y+12x-32=0。

Solution:
::解决方案 :

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

::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
::摘要

A parabola is the collection of points that are equidistant from a fixed focus and directrix.
::抛物线是收集固定焦点和准点距离相等的点数。

The focus of a parabola is the point that the parabola seems to curve around.
::抛物线的焦点是,抛物线似乎在周围旋转。

The directrix of a parabola is the line that the parabola seems to curve away from.
::抛物线的准点是抛物线似乎向外倾斜的线条。

The general equation for a parabola opening vertically is $\begin{aligned} (x - h)^{2} = \pm 4 p (y - k) \end{aligned}$ .
::垂直开口的抛物线的一般方程为(x-h)2+4p(y-k)。

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

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:
::以图示形式绘制并查明重要信息,以绘制下列图示并找出重要信息:

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)

Please see the Appendix.
::请参看附录。

章节大纲

Introduction ::导言

The Parabola ::帕拉波拉

Standard Equation of Parabola ::帕拉波拉标准方程式

Examples ::实例

Summary ::摘要

Review ::回顾

Review (Answers ) ::回顾(答复)

Introduction
::导言

The Parabola
::帕拉波拉

Standard Equation of Parabola
::帕拉波拉标准方程式

Examples
::实例

Summary
::摘要

Review
::回顾

Review (Answers )
::回顾(答复)