Course: 9.1 二次曲线的一般形式

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二次曲线的一般形态

Conics are a family of graphs that include , circles, and . All of these graphs come from the same general equation and by looking and manipulating a specific equation you can learn to tell which conic it is and how it can be graphed.
::二次曲线是由包含、圆圈和的图表组成的一组。所有这些图表都来自相同的普通方程式, 通过查找和操控一个特定的方程式, 您可以学会辨别它是哪种二次曲线, 以及如何绘制图形。

What is the one essential skill that enables you to manipulate the equation of a conic in order to sketch its graph?
::能够操纵二次曲线的方程式以绘制其图表的一种基本技能是什么?

Introduction to Conics
::二次曲线导言

The word conic comes from the word cone which is where the shapes of parabolas, circles, ellipses and hyperbolas originate. Consider two cones that open up in opposite directions and a plane that intersects it horizontally. A flat intersection would produce a perfect circle .
::二次曲线来自锥形的词, 锥形、圆圈、椭圆和超双螺旋的形状来自锥体。想象一下两个向相反方向打开的锥体和一个水平交叉的平面。一个平坦的十字路口将产生一个完美的圆圈。

To produce an ellipse, tilt the plane so that the circle becomes elongated and oval shaped. Notice that the angle that the plane is tilted is still less steep than the slope of the side of the cone.
::为了生产椭圆,请倾斜平面,使圆圈变长并形成奥瓦尔形。注意飞机倾斜的角度比锥形侧的斜坡还要小。

As you tilt the plane even further and the slope of the plane equals the slope of the cone edge you produce a parabola . Since the slopes are equal, a parabola only intersects one of the cones.
::当你进一步倾斜平面时,平面的斜坡等于锥形边缘的斜坡时,就会产生抛物线。由于斜坡是相等的,抛物线只交叉一个锥形。

Lastly, if you make the plane steeper still, the plane ends up intersecting both the lower cone and the upper cone creating the two parts of a hyperbola.
::最后,如果你把飞机的陡峭度固定下来, 飞机最终会把下角锥和上角锥交叉, 形成双曲线的两部分。

The intersection of three dimensional objects in three dimensional space to produce two dimensional graphs is quite challenging. In practice, the knowledge of where conics come from is not widely used. It will be more important for you to be able to manipulate an equation into standard form and graph it in a regular coordinate plane. The regular form of a conic is:
::三维空间中三维天体的交叉点以生成两个维图,这相当具有挑战性。实际上,对二次曲线来自何方的知识没有被广泛使用。更重要的是, 您必须能够将方程式操作成标准形式, 并在常规坐标平面中绘制图形。二次曲线的常规形式是 :

$\begin{aligned} A x^{2} + B x y + C y^{2} + D x + E y + F = 0 \end{aligned}$
::Ax2+Bxy+Cy2+Dx+Ey+F=0 Ax2+Bxy+Cy2+Dx+Ey+F=0

Before you start manipulating the general form of a conic equation you should be able to recognize whether it is a circle, ellipse, parabola or hyperbola. In standard form, the two coefficients to examine are $\begin{aligned} A \end{aligned}$ and $\begin{aligned} C \end{aligned}$ .
::在您开始操控二次方程的一般形式之前, 您应该能够识别它是圆、椭圆、 parbola 或 biperbola 。在标准形式上, 需要检查的两个系数是 A 和 C 。

For circles, the coefficients of $\begin{aligned} x^{2} \end{aligned}$ and $\begin{aligned} y^{2} \end{aligned}$ are the same sign and the same value: $\begin{aligned} A = C \end{aligned}$
::对于圆圆, x2 和 y2 的系数是相同的符号和相同值: A= C

For ellipses, the coefficients of $\begin{aligned} x^{2} \end{aligned}$ and $\begin{aligned} y^{2} \end{aligned}$ are the same sign and different values: $\begin{aligned} A, C > 0, A \neq C \end{aligned}$
::对于椭圆, x2 和 y2 的系数是相同的符号和不同的值: A, C>0, AC

For hyperbolas, the coefficients of $\begin{aligned} x^{2} \end{aligned}$ and $\begin{aligned} y^{2} \end{aligned}$ are opposite signs: $\begin{aligned} C < 0 < A \end{aligned}$ or $\begin{aligned} A < 0 < C \end{aligned}$
::对于双光蜡,x2和y2的系数是相反的符号:C<0<A或A<0<C

For parabolas, either the coefficient of $\begin{aligned} x^{2} \end{aligned}$ or $\begin{aligned} y^{2} \end{aligned}$ must be zero: $\begin{aligned} A = 0 \end{aligned}$ or $\begin{aligned} C = 0 \end{aligned}$
::对于parabolas, x2 或 y2 的系数必须为零: A=0 或 C=0

Each specific type of conic has its own graphing form, but in all cases the technique of completing the square is essential.
::每种具体类型的二次曲线都有自己的图形形式,但在所有情况下,完成方形的技术都是必不可少的。

For review, let's complete the square in the expression $\begin{aligned} x^{2} + 6 x \end{aligned}$ . and demonstrate graphically what completing the square represents.
::用于审查,让我们完成表达式 x2+6x. 中的方块, 并用图形显示完成方块代表的方块。

Algebraically, completing the square just requires you to divide the coefficient of $\begin{aligned} x \end{aligned}$ by 2 and square the result. In this case $\begin{aligned} {(\frac{6}{2})}^{2} = 3^{2} = 9 \end{aligned}$ . Since you cannot add nine to an expression without changing its value, you must simultaneously add nine and subtract nine so the net change will be zero.
::代数, 完成正方块只需要您将x系数除以 2 和结果平方即可。在这种情况下( 622= 32= 9) 。由于您无法在不改变表达式值的情况下添加 9, 因此您必须同时添加 9 和减去 9, 这样净变化将是零。

$\begin{aligned} x^{2} + 6 x + 9 - 9 \end{aligned}$
::x2+6x+9-9

Now you can factor by recognizing a perfect square.
::现在您可以通过识别一个完美的正方块来考虑因素。

$\begin{aligned} (x + 3)^{2} - 9 \end{aligned}$
:x+3)2-9

Graphically the original expression $\begin{aligned} x^{2} + 6 x \end{aligned}$ can be represented by the area of a rectangle with sides $\begin{aligned} x \end{aligned}$ and $\begin{aligned} (x + 6) \end{aligned}$ .
::图形化的原始表达式 x2+6x 可以用侧形 x 和 (x+6) 的矩形区域表示。

The term “complete the square” has visual meaning as well algebraic meaning. The rectangle can be rearranged to be more square-like so that instead of small rectangle of area $\begin{aligned} 6 x \end{aligned}$ at the bottom, there is a rectangle of area $\begin{aligned} 3 x \end{aligned}$ on two sides of the $\begin{aligned} x^{2} \end{aligned}$ square.
::“ 完成正方” 一词具有视觉意义和代数含义。矩形可以重新排列, 使其更像正方形, 这样, x2 方的两侧就有一个区域 3x 的矩形, 而不是下方的区域 6x 的小矩形。

Notice what is missing to make this shape a perfect square? A little corner square of 9 is missing which is why the 9 should be added to make the perfect square of $\begin{aligned} (x + 3) (x + 3) \end{aligned}$ .
::要让这个形状成为完美的正方形, 缺少什么? 缺少一个小角方块 9 的正方块。这就是为什么应该添加 9 来使( x+3 ( x+3) +3 ) 的正方块成为完美的正方块的原因。

Examples
::实例

Example 1
::例1

Earlier, you were asked what skills you need for conics. The one essential skill that you need for conics is completing the square. If you can complete the square with two variables then you will be able to graph every type of conic.
::早些时候,有人问您需要什么样的二次曲线技能。二次曲线所需的一种基本技能就是完成方形。如果您能用两个变量完成方形, 那么您就可以绘制每一种二次曲线的图表。

Example 2
::例2

What type of conic is each of the following relations?
::下列关系中,每种是哪种二次曲线?

$\begin{aligned} 5 y^{2} - 2 x^{2} = - 25 \end{aligned}$
1. Hyperbola because the $\begin{aligned} x^{2} \end{aligned}$ and $\begin{aligned} y^{2} \end{aligned}$ coefficients are different signs.
  ::Hyperbola 因为 x2 和 y2 系数是不同的符号。
::5y2-2x225 超博拉,因为x2和y2系数不同。

$\begin{aligned} x = - \frac{1}{2} y^{2} - 3 \end{aligned}$
1. Parabola (sideways) because the $\begin{aligned} x^{2} \end{aligned}$ term is missing.
  ::抛物线( 边缘) , 因为 X2term 缺失。
::x12y2-3 parabola( 边道) , 因为 X2term 缺失。

$\begin{aligned} 4 x^{2} + 6 y^{2} = 36 \end{aligned}$
1. Ellipse because the $\begin{aligned} x^{2} \end{aligned}$ and $\begin{aligned} y^{2} \end{aligned}$ coefficients are different values but the same sign.
  ::椭圆, 因为 x2 和 y2 系数是不同的值, 但相同符号。
::4x2+6y2=36 椭圆, 因为 x2 和 y2 系数是不同的值, 但相同符号。

$\begin{aligned} x^{2} - \frac{1}{4} y = 1 \end{aligned}$
1. Parabola (upright) because the $\begin{aligned} y^{2} \end{aligned}$ term is missing.
  ::parabola(直截了当),因为缺少 y2 term 。
::x2 - 14y=1 帕拉波拉(直立),因为缺少 y2term 。

$\begin{aligned} - \frac{x^{2}}{8} + \frac{y^{2}}{4} = 1 \end{aligned}$
1. Hyperbola because the $\begin{aligned} x^{2} \end{aligned}$ and $\begin{aligned} y^{2} \end{aligned}$ coefficients are different signs.
  ::Hyperbola 因为 x2 和 y2 系数是不同的符号。
::-x28+y24=1 超博拉,因为x2和y2系数是不同的符号。

$\begin{aligned} - x^{2} + 99 y^{2} = 12 \end{aligned}$
1. Hyperbola because the $\begin{aligned} x^{2} \end{aligned}$ and $\begin{aligned} y^{2} \end{aligned}$ coefficients are different signs.
  ::Hyperbola 因为 x2 和 y2 系数是不同的符号。
::-x2+99y2=12 Heperbola,因为x2和y2系数是不同的符号。

Example 3
::例3

Complete the square for both the $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ terms in the following equation.
::以下方程式中的 x 和 y 条件的 x 和 y 条件填全方形。

$\begin{aligned} x^{2} + 6 x + 2 y^{2} + 16 y = 0 \end{aligned}$
::x2+6x+2y2+16y=0

First write out the equation with space so that there is room for the terms to be added to both sides. Since this is an equation, it is appropriate to add the values to both sides instead of adding and subtracting the same value simultaneously. As you rewrite with spaces, factor out any coefficient of the $\begin{aligned} x^{2} \end{aligned}$ or $\begin{aligned} y^{2} \end{aligned}$ terms since your algorithm for completing the square only works when this coefficient is one.
::首先用空格写出方程式, 以便给两边添加条件。由于这是一个方程式, 将数值加到两边是合适的, 而不是同时增减相同值。在您用空格重写时, 请将 X2 或 y2 条件中的任何系数除掉, 因为您在完成正方形时的算法只有在此系数为一时才有效。

$\begin{aligned} x^{2} + 6 x + \underline{} + 2 (y^{2} + 8 y + \underline{}) = 0 \end{aligned}$
::x2+6x 2(y2+8y)=0

Next complete the square by adding a nine and what looks like a 16 on the left (it is actually a 32 since it is inside the parentheses).
::下一个通过在左侧加一个9和16的形状(实际上是32,因为它在括号内)来填充方形。

$\begin{aligned} x^{2} + 6 x + 9 + 2 (y^{2} + 8 y + 16) = 9 + 32 \end{aligned}$
::x2+6x+9+9+2(y2+8y+16)=9+32

Factor.
::因素。

$\begin{aligned} (x + 3)^{2} + 2 (y + 4)^{2} = 41 \end{aligned}$
:x+3)2+2(y+4)2=41

Example 4
::例4

Identify the type of conic in each of the following relations.
::确定下列关系中每种关系中的二次曲线类型。

$\begin{aligned} 3 x^{2} = 3 y^{2} + 18 \end{aligned}$
1. The relation is a hyperbola because when you move the $\begin{aligned} 3 y^{2} \end{aligned}$ to the left hand side of the equation, it becomes negative and then the coefficients of $\begin{aligned} x^{2} \end{aligned}$ and $\begin{aligned} y^{2} \end{aligned}$ have opposite signs.
  ::关系是双曲线,因为当您将3y2移动到方程式的左侧时,它变成负值,然后x2和y2的系数有相反的符号。
::3x2=3y2+18 关系是一个双曲线,因为当将3y2移动到方程式的左侧时,它变成负值,然后x2和y2的系数有相反的符号。

$\begin{aligned} y = 4 (x - 3)^{2} + 2 \end{aligned}$
1. Parabola
  ::帕拉波拉
::y=4(x-3)2+2 百兆波拉

$\begin{aligned} x^{2} + y^{2} = 4 \end{aligned}$
1. Circle
  ::圆圆
::x2+y2=4 圆

$\begin{aligned} y^{2} + 2 y + x^{2} - 6 x = 12 \end{aligned}$
1. Circle
  ::圆圆
::y2y+x2 - 6x=12 圆

$\begin{aligned} \frac{x^{2}}{6} + \frac{y^{2}}{12} = 1 \end{aligned}$
1. Ellipse
  ::椭圆
::x26+y212=1 椭圆

$\begin{aligned} x^{2} - y^{2} + 4 = 0 \end{aligned}$
1. Hyperbola
  ::超重波
::x2-y2+4=0 双波拉

Example 5
::例5

Complete the square for both $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ in the following equation.
::以下方程式中 x 和 y 的 x 和 y , 填全正方形。

$\begin{aligned} - 3 x^{2} - 24 x + 4 y^{2} - 32 y = 8 \end{aligned}$
::-3x2-24x+4y2-32y=8

\begin{aligned} - 3 x^{2} - 24 x + 4 y^{2} - 32 y & = 8 \\ - 3 (x^{2} + 8 x + \underline{}) + 4 (y^{2} - 8 y + \underline{}) & = 8 \\ - 3 (x^{2} + 8 x + 16) + 4 (y^{2} - 8 y + 16) & = 8 - 48 + 64 \\ - 3 (x + 4)^{2} + 4 (y - 4)^{2} & = 24 \end{aligned}

::3x2-24x+4y2-32y=8-3(x2+8x)+4(y2-8y)=8-3(x2+8x+16)+4(y2-8y+16)+4(y2-8y+16)=8-48-64-3(x+4)2+4(y-4)2=24

Summary

Conics are a family of graphs that include parabolas, circles, ellipses, and hyperbolas.
::二次曲线是一个由图表组成的组合,包括parabolas、圆圈、椭圆和超光谱。

The regular form of a conic is: $\begin{aligned} A x^{2} + B x y + C y^{2} + D x + E y + F = 0. \end{aligned}$
::二次曲线的正态形式是:Ax2+Bxy+Cy2+Dx+Ey+F=0。

Based on the regular form, the coefficients A and C signify the type of conic.
- Circles: $\begin{aligned} A = C \end{aligned}$
  ::圆圆: A=C
- Ellipses: $\begin{aligned} A \end{aligned}$ and $\begin{aligned} C \end{aligned}$ are the same sign and $\begin{aligned} A \neq C . \end{aligned}$
  ::缩略语:A和C是相同的标志和AQC。
- Hyperbolas: $\begin{aligned} A \end{aligned}$ and $\begin{aligned} C \end{aligned}$ are opposite signs.
  ::Hyperbolas:A和C是相反的信号。
- Parabolas: $\begin{aligned} A = 0 \end{aligned}$ or $\begin{aligned} C = 0. \end{aligned}$
  ::parabolas:A=0或C=0。
::根据常规形式,系数A和系数C表示二次曲线的类型。圆形:A=C 椭圆:A和C是相同的符号,A=C。Hyperbolas:A和C是相反的符号。Parabolas:A=0或C=0。

Completing the square is an essential technique for manipulating conic equations and graphing them.
::完成广场是操纵二次方程和绘制图象的基本技术。

Review
::回顾

Identify the type of conic in each of the following relations.
::确定下列关系中每种关系中的二次曲线类型。

1. $\begin{aligned} 3 x^{2} + 4 y^{2} = 12 \end{aligned}$
::1. 3x2+4y2=12

2. $\begin{aligned} x^{2} + y^{2} = 9 \end{aligned}$
::2. x2+y2=9

3. $\begin{aligned} \frac{x^{2}}{4} + \frac{y^{2}}{9} = 1 \end{aligned}$
::3. x24+y29=1

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

5. $\begin{aligned} x^{2} + 2 x - y^{2} + 6 y = 15 \end{aligned}$
::5. x2+2x-y2+6y=15

6. $\begin{aligned} x^{2} = y - 1 \end{aligned}$
::6. x2=y- 1

Complete the square for $\begin{aligned} x \end{aligned}$ and/or $\begin{aligned} y \end{aligned}$ in each of the following expressions.
::下列表达式中的 x 和/或 y 的复方。

7. $\begin{aligned} x^{2} + 4 x \end{aligned}$
::7. x2+4x

8. $\begin{aligned} y^{2} - 8 y \end{aligned}$
::8.y2-8 y2-8y

9. $\begin{aligned} 3 x^{2} + 6 x + 4 \end{aligned}$
::9. 9. 3x2+6x+4

10. $\begin{aligned} 3 y^{2} + 9 y + 15 \end{aligned}$
::10. 3y2+9y+15

11. $\begin{aligned} 2 x^{2} - 12 x + 1 \end{aligned}$
::11. 2x2 - 12x+1

Complete the square for $\begin{aligned} x \end{aligned}$ and/or $\begin{aligned} y \end{aligned}$ in each of the following equations.
::下列方程式中的 x 和/或 y , 填全正方形。

12. $\begin{aligned} 4 x^{2} - 16 x + y^{2} + 2 y = - 1 \end{aligned}$
::12. 4x2 - 16x+y2+2y1

13. $\begin{aligned} 9 x^{2} - 54 x + y^{2} - 2 y = - 81 \end{aligned}$
::13. 9x2-54x+y2-2y_2y81

14. $\begin{aligned} 3 x^{2} - 6 x - 4 y^{2} = 9 \end{aligned}$
::14. 3x2-6x-4y2=9

15. $\begin{aligned} y = x^{2} + 4 x + 1 \end{aligned}$
::15.y=x2+4x+1

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

二次曲线的一般形态

Introduction to Conics ::二次曲线导言

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)