Course: 10.3 参数和参数的去除

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参数和参数去除

In a parametric equation , the variables $\begin{align*}x\end{align*}$ and $\begin{align*}y\end{align*}$ are not dependent on one another. Instead, both variables are dependent on a third variable, $\begin{align*}t\end{align*}$ . This is the parameter or a number that affects the behavior of the equation. Usually $\begin{align*}t\end{align*}$ will stand for time. A real world example of the relationship between $\begin{align*}x, y\end{align*}$ and $\begin{align*}t\end{align*}$ is the height, weight and age of a baby.
::在参数方程中,变量 x 和 yare 并不互相依赖。相反, 两种变量都取决于第三个变量。 t。这是影响该方程行为的参数或数字。通常会代表时间。一个真实的世界性实例, 说明 x, yand tis 与婴儿的身高、体重和年龄之间的关系。

Both the height and the weight of a baby depend on time, but there is also clearly a positive relationship between just the height and weight of the baby. By focusing on the relationship between the height and the weight and letting time hide in the background, you create a parametric relationship between the three variables.
::婴儿的身高和体重都取决于时间,但是在婴儿的身高和体重之间显然也有正关系。通过关注身高和体重之间的关系,并让时间隐藏在背景中,你就在三个变量之间建立了参数关系。

What other types of real world situations are modeled with parametric equations?
::以参数方程为模型的另一种真实世界情况是什么?

Eliminating the Parameter
::消除参数

In your graphing calculator there is a parametric mode. Once you put your calculator into parametric mode, on the graphing screen you will no longer see $\begin{align*}y= \underline{\;\;\;\;\;},\end{align*}$ instead, you will see:
::在您的图形计算计算器中有一个参数模式。一旦您将计算器放入参数模式, 在图形显示屏幕上您将不再看到 y, 相反, 您将会看到 :

Notice how for plot one, the calculator is asking for two equations based on variable $\begin{align*}T\end{align*}$ :
::对于图一, 计算器要求两个方程式, 以变量 T为基础 :

\begin{aligned} x_{1 T} = f (t) \\ y_{1 T} = g (t) \end{aligned}

$\begin{align*}& x_{1T}=f(t) \\ & y_{1T}=g(t)\end{align*}$
::x1T=f( t)y1T=g( t)

This is called parametric form . Parametric form refers to a relationship that includes $\begin{align*}x=f(t)\end{align*}$ and $\begin{align*}y=g(t)\end{align*}$ . In order to transform a parametric equation into a normal one, you need to do a process called “eliminating the parameter.” “Eliminating the parameter” is a phrase that means to turn a parametric equation that has $\begin{align*}x=f(t)\end{align*}$ and $\begin{align*}y=g(t)\end{align*}$ into just a relationship between $\begin{align*}y\end{align*}$ and $\begin{align*}x\end{align*}$ . You are eliminating $\begin{align*}t\end{align*}$ . To do this, you must solve the $\begin{align*}x=f(t)\end{align*}$ equation for $\begin{align*}t=f^{-1} (x)\end{align*}$ and substitute this value of $\begin{align*}t\end{align*}$ into the $\begin{align*}y\end{align*}$ equation. This will produce a normal function of $\begin{align*}y\end{align*}$ based on $\begin{align*}x\end{align*}$ .
::这被称为参数形式。参数形式表示包括 x=f( t) andy=g( t) 的关系。要将参数方程式转换成正常方程式, 您需要做一个名为“ 消除参数” 的过程。 “ 缩小参数” 是将x=f( t) 和 y=g( t) 的参数方程式转换成仅仅是 yandx 之间的关系的参数方程式。您正在消除 t。要做到这一点, 您必须解决 t=f( t) 的 x=f( t) 方程式, 并将 t 的数值替换为 Y 方程式。这将产生基于 x 的 y 正常函数。

There are two major benefits of graphing in parametric form. First, it is straightforward to graph a portion of a regular function using the $\begin{align*}T_{min}, T_{max}\end{align*}$ and $\begin{align*}T_{step}\end{align*}$ in the window setting. Second, parametric form enables you to graph projectiles in motion and see the effects of time.
::以参数形式绘制图形有两大好处。首先,使用窗口设置中的 Tmin、Tmax 和 Tstep 来绘制正常函数的一部分。其次, 参数形式可以使您在移动中绘制射弹图, 并查看时间效果。

A tortoise and a hare start 202 feet apart and then race to a flag halfway between them. The hare decides to take a nap and give the tortoise a 21 second head start. The hare runs at 9.8 feet per second and the tortoise hustles along at 3.2 feet per second. This situation can be represented by parametric equations and we can use the equations to determine who wins this epic race and by how much.
::一只乌龟和一只野兔开始相距202英尺,然后在两只野兔之间的半边跑到旗子上。野兔决定午睡,给乌龟一个21秒的起点。野兔以每秒9.8英尺的速度运行,乌龟以每秒3.2英尺的速度运行。这一情况可以用参数方程式来表示,我们可以用方程式来决定谁赢得这场史诗比赛,谁赢得多少。

First draw a picture and then represent each character with a set of parametric equations.
::首先绘制图片,然后用一组参数方程代表每个字符。

The tortoise’s position is (-101, 0) at $\begin{align*}t=0\end{align*}$ and (-97.8, 0) at $\begin{align*}t=1\end{align*}$ . You can deduce that the equation modeling the tortoise’s position is:
::乌龟的位置是(-101,0) t=0和(-97.8,0) t=1。您可以推断,乌龟的方程模式是:

\begin{aligned} x_{1} = - 101 + 3.2 \cdot t \\ y_{1} = 0 \end{aligned}

$\begin{align*}& x_1=-101+3.2 \cdot t \\ & y_1=0\end{align*}$
::x1101+3.21=0

The hare’s position is (101, 0) at $\begin{align*}t=21\end{align*}$ and (91.2, 0) at $\begin{align*}t=22\end{align*}$ . Note that it does not make sense to make equations modeling the hare’s position before 21 seconds have elapsed because the Hare is napping and not moving. You can set up an equation to solve for the hare’s theoretical starting position had he been running the whole time.
::兔子的位置是(101,0) t=21和(91.2,0) t=22。请注意,在兔子已经过21秒之前,用方程式模拟兔子的位置是没有道理的,因为兔子正在午睡,没有移动。如果兔子一直运行着,你可以设置一个方程式来解决兔子的理论起点。

\begin{aligned} x_{2} = b - 9.8 t \\ 101 = b - 9.8 \cdot 21 \\ 305.8 = b \end{aligned}

$\begin{align*}& x_2=b-9.8t \\ & 101=b-9.8 \cdot 21 \\ & 305.8=b\end{align*}$
::x2=b-9.8t101=b-9.821305.8=b

The hare’s position equation after $\begin{align*}t=21\end{align*}$ can be modeled by:
::T=21之后的兔子位置方程式可模拟如下:

\begin{aligned} x_{2} = 305.8 - 9.8 \cdot t \\ y_{2} = 0 \end{aligned}

$\begin{align*}& x_2=305.8 - 9.8 \cdot t \\ & y_2=0\end{align*}$
::x2=305.8-9.8.82=0

The tortoise crosses $\begin{align*}x=0\end{align*}$ when $\begin{align*}t \approx 31.5\end{align*}$ . The hare crosses $\begin{align*}x=0\end{align*}$ when $\begin{align*}t \approx 31.2\end{align*}$ . The hare wins by about 1.15 feet.
::乌龟十字x=0,t31.5。兔子十字xx0,t31.2。野兔以约1.15英尺的距离赢得胜利。

Now, use your calculator to display these parametric equations. T
::现在,使用您的计算器来显示这些参数方程。 T

here are many settings you should know for parametric equations that bring questions like this to life. The TI-84 has features that allow you to see the race happen.
::这里有许多您应该知道的参数方程设置, 给这样的问题带来生命。 TI- 84 具有能让您看到比赛发生的特性。

First, set the mode to simultaneous graphing. This will show both the tortoise and hare’s position at the same time.
::首先, 设置同步图形绘制模式。这将同时显示龟和兔子的位置。

Next, change the graphing window so that $\begin{align*}t\end{align*}$ varies between 0 and 32 seconds. The $\begin{align*}T_{step}\end{align*}$ determines how often the calculator will calculate points. The larger the $\begin{align*}T_{step}\end{align*}$ , the faster and less accurately the graph will plot. Also change the $\begin{align*}x\end{align*}$ to vary between -110 and 110 so you can see the positions of both characters.
::接下来,修改图形窗口,使 t 在 0 至 32 秒之间变化。 Tstep 将决定计算点的频率。图的绘制越大, 速度越快, 精确度就越小。还要将 x 更改为 - 110 至 110 秒, 以便您看到两个字符的位置。

Input the parametric equations. Toggle to the left of the $\begin{align*}x\end{align*}$ and change the cursor from a line to a line with a bubble at the end. This shows their position more clearly.
::输入参数方程式。切换到 x 的左侧, 并将光标从一行修改为一行, 在结尾处插入一个气泡的直线。这样可以更清楚地显示它们的位置。

Now when you graph you should watch the race unfold as the two position graphs race towards each other.
::当你的图表中,你应该看到比赛的展开当两个位置的图表相互对冲时。

Examples
::实例

Example 1
::例1

Earlier, you were asked what types of real world situations can be modeled by parametric equations. Parametric equations are often used when only a portion of a graph is useful. By limiting the domain of $\begin{align*}t\end{align*}$ , you can graph the precise interval of the function you want. Parametric equations are also useful when two different variables jointly depend on a third variable and you wish to look at the relationship between the two dependent variables. This is very common in statistics where an underlying variable may actually be the cause of a problem and the observer can only examine the relationship between the outcomes that they see. In the physical world, parametric equations are exceptional at graphing position over time because the horizontal and vertical vectors of objects in free motion are each dependent on time, yet independent of one another.
::早些时候,有人问您可以通过参数方程式模拟真实世界状况的哪一类。参数方程式通常只在某个图形的一部分有用时才使用。通过限制 t 的域, 您可以绘制您想要的函数的精确间隔。当两个不同的变量共同依赖第三个变量时, 参数方程式也是有用的, 您想要查看两个依赖变量之间的关系。这在统计中非常常见, 其中潜在的变量可能实际上是一个问题的原因, 观察者只能检查它们所看到的结果之间的关系。在物理世界中, 参数方程式在时间的图形位置上是例外的, 因为自由运动物体的横向和垂直矢量都取决于时间, 但彼此独立。

Example 2
::例2

Eliminate the parameter in the following equations.
::消除以下方程式中的参数。

\begin{aligned} x = 6 t - 2 \\ y = 5 t^{2} - 6 t \end{aligned}

$\begin{align*}& x=6t-2 \\ & y=5t^2-6t\end{align*}$
::x=6t--2y=5t2-6t

$\begin{align*}x=6t-2\end{align*}$ So $\begin{align*}\frac{x+2}{6}=t\end{align*}$ . Now, substitute this value for $\begin{align*}t\end{align*}$ into the second equation:
::x=6t- 2 So x+26=t。现在, 将 t 的这个值替换为第二个方程式 :

$\begin{align*}y=5\left (\frac{x+2}{6}\right )^2-6\left (\frac{x+2}{6}\right)\end{align*}$
::y=5(x+262)-6(x+26)

Example 3
::例3

For the given parametric equation, graph over each interval of $\begin{align*}t\end{align*}$ .
::对于给定的参数方程,用图显示 t 之间的每一间距。

$\begin{align*}x=t^2-4 \end{align*}$
::x=t2 - 4xx=t2 - 4

$\begin{align*}y=2t\end{align*}$
::y=2吨

$\begin{align*}-2 \le t \le 0\end{align*}$
::- -20

$\begin{align*}0 \le t \le 5\end{align*}$
::05

$\begin{align*}-3 \le t \le 2\end{align*}$
::- 32

a. A good place to start is to find the coordinates where $\begin{align*}t\end{align*}$ indicates the graph will start and end. For $\begin{align*}-2 \le t \le 0, \ t=-2\end{align*}$ and $\begin{align*}t=0\end{align*}$ indicate that the points (0, -4) and (-4, 0) are the endpoints of the graph.
::a. 一个良好的起点是找到坐标,其中 t 表示图的起点和终点。对于- 2t0, t2和 t=0, 表示点(0, 4)和(4, 0)是图的终点。

b. $\begin{align*}0 \le t \le 5\end{align*}$
::b. 05

c. $\begin{align*}-3 \le t \le 2\end{align*}$
::c. - 32

Example 4
::例4

Eliminate the parameter and graph the following parametric curve.
::删除参数,并绘制以下参数曲线图。

$\begin{align*}x=3 \cdot \sin t \end{align*}$
::x=3=sint

$\begin{align*}y=3 \cdot \cos t\end{align*}$
::y y = 3cost

When parametric equations involve trigonometric functions you can use the Pythagorean Identity, $\begin{align*}\sin^2t + \cos^2t=1\end{align*}$ . In this problem, $\begin{align*}\sin t=\frac{x}{3}\end{align*}$ (from the first equation) and $\begin{align*}\cos t=\frac{y}{3}\end{align*}$ (from the second equation). Substitute these values into the Pythagorean Identity and you have:
::当参数方程包含三角函数时, 您可以使用 Pythagorean 身份, sin2t+cos2t=1. 在此问题上, sint=x3( 从第一个方程) 和 cost=y3( 从第二个方程) 。将这些值替换为 Pythagorean 身份, 您有 :

\begin{aligned} {(\frac{x}{3})}^{2} + {(\frac{y}{3})}^{2} = 1 \\ x^{2} + y^{2} = 9 \end{aligned}

$\begin{align*}& \left (\frac{x}{3}\right)^2 + \left (\frac{y}{3}\right)^2=1 \\ & x^2+y^2=9\end{align*}$
:

x3)2+(y3)2=1x2+y2=9)

This is a circle centered at the origin with radius 3.
::这是一个圆, 以原点为中心, 半径为 3 。

Example 5
::例5

Find the parameterization for the line segment connecting the points (1, 3) and (4, 8).
::查找连接点(1、3)和点(4、8)的线段的参数化。

Use the fact that a point plus a vector yields another point. A vector between these points is $\begin{align*}<4 -1, 8 - 3> = <3, 5>\end{align*}$
::使用一个点加上矢量产生另一个点这一事实。这些点之间的矢量为 < 4 - 1,8 - 33-5>

Thus the point (1, 3) plus $\begin{align*}t\end{align*}$ times the vector $\begin{align*}<3, 5>\end{align*}$ will produce the point (4, 8) when $\begin{align*}t=1\end{align*}$ and the point (1, 3) when $\begin{align*}t=0\end{align*}$ .
::因此,当 t=1 和 t=0 的点(1, 3) 和点(1, 3) 时,矢量 < 3,5 > 将产生点(4, 8) 和点(1, 3) 时, t=0 。

$\begin{align*}(x, y)=(1, 3) + t \cdot <3, 5>\end{align*}$ , for $\begin{align*}0 \le t \le 1\end{align*}$
:x,y) =(1,3)+t3,5>, 0t1

You than then break up this vector equation into parametric form.
::而不是将矢量方程式分解成参数形式。

\begin{aligned} x = 1 + 3 t \\ y = 3 + 5 t \\ 0 \leq t \leq 1 \end{aligned}

$\begin{align*}& x=1+3t \\ & y=3+5t \\ & 0 \le t \le 1\end{align*}$
::x=1+3ty=3+5t0%t1

Summary

In a parametric equation , variables x and y are dependent on a third variable, t, which is the parameter, usually representing time.
::在参数方程式中,变量 x 和 y 取决于第三个变量 t,即参数,通常代表时间。

Parametric form refers to a relationship that includes $\begin{align*}x = f(t),~ y = g(t).\end{align*}$ To transform a parametric equation into a normal one, you need to eliminate the parameter.
::参数形式是指包含 x=f(t), y=g(t) 的关系。要将参数方程式转换成正常方程式,需要删除参数。

Eliminating the parameter involves solving the $\begin{align*}x\end{align*}$ equation for $\begin{align*}t\end{align*}$ and substituting this value into the $\begin{align*}y\end{align*}$ equation, producing a normal function of $\begin{align*}y\end{align*}$ based on $\begin{align*}x.\end{align*}$
::去除参数需要解决 t 的 x 方程式,并将此值替换为 y 方程式,产生基于 x 的 y 正常函数 y 。

Review
::回顾

Eliminate the parameter in the following sets of parametric equations.
::消除以下几组参数方程中的参数。

1. $\begin{align*}x=3t-1; \ y=4t^2-2t\end{align*}$
::1. x=3t- 1;y=4t2-2t

2. $\begin{align*}x=3t^2+6t; \ y=2t-1\end{align*}$
::2. x=3t2+6t;y=2t- 1

3. $\begin{align*}x=t+2; \ y=t^2+4t+4\end{align*}$
::3. x=t+2;y=t2+4+4

4. $\begin{align*}x=t-5; \ y=t^3+1\end{align*}$
::4. x=t-5;y=t3+1

5. $\begin{align*}x=t+4; \ y=t^2-5\end{align*}$
::5. x=t+4;y=t2-5

For the parametric equation $\begin{align*}x=t, y=t^2+1\end{align*}$ , graph over each interval of $\begin{align*}t\end{align*}$ .
::参数方程式 x=t,y=t2+1, t之间每一间隔的图形。

6. $\begin{align*}-2 \le t \le -1\end{align*}$
::6-21

7. $\begin{align*}-1 \le t \le 0\end{align*}$
::7.--10

8. $\begin{align*}-1 \le t \le 1\end{align*}$
::8.--11

9. $\begin{align*}-2 \le t \le 2\end{align*}$
::9.-22

10. $\begin{align*}-5 \le t \le 5\end{align*}$
::10. - 55

11. Eliminate the parameter and graph the following parametric curve: $\begin{align*}x=\sin t, \ y=-4+3 \cos t\end{align*}$ .
::11. 删去参数并用下列参数曲线图解:x=sint,y4+3cost。

12. Eliminate the parameter and graph the following parametric curve: $\begin{align*}x=1+2 \cos t, \ y=1+2 \sin t\end{align*}$ .
::12. 删去参数和图解如下参数曲线:x=1+2cost,y=1+2sint。

13. Using the previous problem as a model, find a parameterization for the circle with center (2, 4) and radius 3.
::13. 以前一个问题为模型,为圆圆找到一个参数化,以中间(2、4)和半径3为中心。

14. Find the parameterization for the line segment connecting the points (2, 7) and (1, 4).
::14. 为连接点(2、7)和点(1、4)的线段找出连接点的线段参数化。

15. Find a parameterization for the ellipse $\begin{align*}\frac{x^2}{4} + \frac{y^2}{25}=1\end{align*}$ . Use the fact that $\begin{align*}\cos^2 t+\sin^2 t=1\end{align*}$ . Check your answer with your calculator.
::15. 为椭圆 x24+y225=1. 寻找参数。请使用 cos2\\\ t+sin2\\\t=1. 使用计算器检查您的回答。

16. Find a parameterization for the ellipse $\begin{align*}\frac{(x-4)^2}{9} + \frac{(y+1)^2}{36}=1\end{align*}$ . Check your answer with your calculator.
::16. 查找椭圆(x-4)29+(y+1)236=1的参数。请使用计算器检查您的回答。

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

参数和参数去除

Eliminating the Parameter ::消除参数

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)