Section: 参数反函数 | 10.4 参数反差

Section outline

You have learned that a graph and its inverse are reflections of each other across the line

\begin{aligned} y = x . \end{aligned}

You have also learned that in order to find an inverse algebraically, you can switch the

\begin{aligned} x \end{aligned}

and

\begin{aligned} y \end{aligned}

variables and solve for

\begin{aligned} y . \end{aligned}

actually make finding inverses easier because both the

\begin{aligned} x \end{aligned}

and

\begin{aligned} y \end{aligned}

variables are based on a third variable

\begin{aligned} t . \end{aligned}

All you need to do to find the inverse of a set of parametric equations and switch the functions for

\begin{aligned} x \end{aligned}

and

\begin{aligned} y . \end{aligned}

Is the inverse of a function always a function?
::函数的反向总是函数吗 ?

Inverses of Parametric Equations
::参数等量的逆数

To find the inverse of a parametric equation you must switch the function of $\begin{aligned} x \end{aligned}$ with the function of $\begin{aligned} y . \end{aligned}$ This will switch all the points from $\begin{aligned} (x, y) \end{aligned}$ to $\begin{aligned} (y, x) \end{aligned}$ and also has the effect of visually reflecting the graph over the line $\begin{aligned} y = x . \end{aligned}$
::要找到参数方程的反向, 您必须用 y 函数切换 x 的函数。这将将所有点从 (x,y) 切换到 (y, x) , 并具有在 y =x 线上直观反映图形的效果。

Similar to the inverses regular functions, the inverses of parametric equations are often restricted so that they are also functions. Take the following parametric equations:
::与正常函数的反向相似,参数方程的反向往往受到限制,因此它们也是函数。

\begin{aligned} x & = 2 t \\ y & = t^{2} - 4 \end{aligned}

::x=2ty=t2-4

To find and graph the inverse of the parametric function on the domain $\begin{aligned} - 2 < t < 2 \end{aligned}$ , first switch the $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ functions and graph.
::要查找和绘制域-2<t<2, 域上的参数函数的反向, 请先切换 x 和 y 函数和图形。

\begin{aligned} x & = t^{2} - 4 \\ y & = 2 t \end{aligned}

::x=t2 - 4y=2t

The original function is shown in blue and the inverse is shown in red.
::原函数以蓝色显示,反面以红色显示。

Examples
::实例

Example 1
::例1

Earlier, you were asked if the inverse of a function is always a function. The inverse of a function is not always a function. In order to see whether the inverse of a function will be a function, you must perform the horizontal line test on the original function. If the function passes the horizontal line test then the inverse will be a function. If the function does not pass the horizontal line test then the inverse produces a relation rather than a function.
::早些时候,有人询问函数的反向是否总是函数。函数的反向并不总是函数。为了查看函数的反向是否是一个函数,您必须在原始函数上进行水平线测试。如果函数通过水平线测试,则反向将是一个函数。如果函数没有通过水平线测试,则反向产生关系而不是函数。

Example 2
::例2

Is the point (4, 8) in the following function or its inverse?
::以下函数中的点(4,8)是反数还是反数?

\begin{aligned} x & = 2 t^{2} - 2 \\ y & = t^{2} - 1 \end{aligned}

::x=2t2--2y=t2-2-1

Try to solve for a matching $\begin{aligned} t \end{aligned}$ in the original function.
::尝试为原始函数中的匹配 t 解析。

$\begin{aligned} x = 2 t^{2} - 2 \end{aligned}$
::x=2t2-2 -2

$\begin{aligned} 4 = 2 t^{2} - 2 \end{aligned}$
::4=2吨2-2-2

$\begin{aligned} 6 = 2 t^{2} \end{aligned}$
::6=2吨2

$\begin{aligned} 3 = t^{2} \end{aligned}$
::3=t2 3=t2

$\begin{aligned} \pm \sqrt{3} = t \end{aligned}$
::3=t

$\begin{aligned} y = t^{2} - 1 \end{aligned}$
::y=t2-1

$\begin{aligned} 8 = t^{2} - 1 \end{aligned}$
::8=t2-1

$\begin{aligned} 9 = t^{2} \end{aligned}$
::9=吨2

$\begin{aligned} \pm 3 = t \end{aligned}$
::3=t

The point does not satisfy the original function. Check to see if it satisfies the inverse.
::点不能满足原始函数。检查是否满足反向功能。

$\begin{aligned} x = t^{2} - 1 \end{aligned}$
::x=t2-2-1

$\begin{aligned} 4 = t^{2} - 1 \end{aligned}$
::4=t2-1

$\begin{aligned} \pm \sqrt{5} = t \end{aligned}$
::5=t

$\begin{aligned} y = 2 t^{2} - 2 \end{aligned}$
::y= y2t2-2-2

$\begin{aligned} 8 = 2 t^{2} - 2 \end{aligned}$
::8=2吨2-2-2

$\begin{aligned} 10 = 2 t^{2} \end{aligned}$
::10=2吨2

$\begin{aligned} \pm \sqrt{5} = t \end{aligned}$
::5=t

The point does satisfy the inverse of the function.
::点能满足函数的反向。

Example 3
::例3

Parameterize the following function and then graph the function and its inverse.
::参数化以下函数,然后图形化函数及其反向。

$\begin{aligned} f (x) = x^{2} + x - 4 \end{aligned}$
:xx)=x2+x-4)

For the original function, the parameterization is:
::对于原始函数,参数化是:

\begin{aligned} x & = t \\ y & = t^{2} + t - 4 \end{aligned}

::x=ty=t2+t-4xxy=t2+t-4

The inverse is:
::反之:

\begin{aligned} x & = t^{2} + t - 4 \\ y & = t \end{aligned}

::x=t2+t-4y=t

Example 4
::例4

An intersection for two sets of parametric equations happens when the points exist at the same $\begin{aligned} x, y \end{aligned}$ and $\begin{aligned} t . \end{aligned}$ Find the points of intersection of the function and its inverse from Example 2.
::两组参数方程式的交叉路段发生于点位于相同的 x、y 和 t。从例2中查找函数的交叉点及其反向。

The parameterized function is:
::参数化函数为:

\begin{aligned} x_{1} & = t \\ y_{1} & = t^{2} + t - 4 \end{aligned}

::x1=ty1=t2+t-4

The inverse is:
::反之:

\begin{aligned} x_{2} & = t^{2} + t - 4 \\ y_{2} & = t \end{aligned}

::x2=t2+t-4y2=t

To find where these intersect, set $\begin{aligned} x_{1} = x_{2} \end{aligned}$ and $\begin{aligned} y_{1} = y_{2} \end{aligned}$ and solve.
::要找到这些交叉点, 请设置 x1=x2 和 y1=y2 并解析。

\begin{aligned} t & = t^{2} + t - 4 \\ t^{2} & = 4 \\ t & = \pm 2 \end{aligned}

::t = t2+t- 4t2=4t2

You still need to actually calculate the points of intersection on the graph. You can tell from the graph in Example C that there seem to be four points of intersection. Since $\begin{aligned} t \end{aligned}$ can mean time, the question of intersection is more complicated than simply overlapping. It means that the points are at the same $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ coordinate at the same time. Note what the graphs look like when $\begin{aligned} - 1.8 < t < 1.8. \end{aligned}$
::您仍需要实际计算图形上的交叉点。您可以从例C中的图表中看到, 似乎有四个交叉点。由于 t 代表时间, 交叉点的问题比简单的重叠复杂。这意味着点在相同的 x 和 y 坐标的同时。注意当 - 1. 8 < 1. 8 时这些图形的外观。

Note what the graphs look like $\begin{aligned} t > 2.2 \end{aligned}$ or $\begin{aligned} t < - 2.2 \end{aligned}$
::注意图表看起来像 t>2.2 或 t2.2 。

Notice how when these partial graphs are examined there is no intersection at anything besides $\begin{aligned} t = \pm 2 \end{aligned}$ and the points (2, 2) and (-2, -2) While the paths of the graphs intersect in four places, they intersect at the same time only twice.
::注意检查这些部分图表时,除了 t2 和点(2, 2) 和点(2, 2) 和点(2, 2) 外,没有任何地方没有交叉点。虽然这些图形的路径在四个地方交叉,但同时交叉只两次。

Example 5
::例5

Identify where the following parametric function intersects with its inverse.
::确定下列参数函数与其反向相交之处。

$\begin{aligned} x = 4 t \end{aligned}$
::x=4tx=4吨

$\begin{aligned} y = t^{2} - 16 \end{aligned}$
::y=t2-16

$\begin{aligned} x_{1} = 4 t; y_{1} = t^{2} - 16 \end{aligned}$ The inverse is:
::x1=4t; y1=t2- 16 逆数为:

\begin{aligned} x_{2} & = t^{2} - 16 \\ y_{2} & = 4 t \end{aligned}

::x2=t2 - 16y2=4t

Solve for $\begin{aligned} t \end{aligned}$ when $\begin{aligned} x_{1} = x_{2} \end{aligned}$ and $\begin{aligned} y_{1} = y_{2} . \end{aligned}$
::当 x1=x2 和 y1=y2 时解决 t 。

\begin{aligned} 4 t & = t^{2} - 16 \\ 0 & = t^{2} - 4 t - 16 \\ t & = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot (- 16)}}{2} = \frac{4 \pm 4 \sqrt{5}}{2} = 2 \pm 2 \sqrt{5} \end{aligned}

::4t=t2-160=t2-4t-16t=416-41(-16)2=4452=225

The points that correspond to these two times are:
::与这两次相对应的要点是:

\begin{aligned} x & = 4 (2 + 2 \sqrt{5}), y = (2 + 2 \sqrt{5})^{2} - 16 \\ x & = 4 (2 - 2 \sqrt{5}), y = (2 - 2 \sqrt{5})^{2} - 16 \end{aligned}

::x=4(2+25)、y=(2+25)、y=(2+25)、y=16x=4(2 - 25)、y=(2 - 25)、y=16

Summary

To find the inverse of a parametric equation, simply switch the functions for $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y . \end{aligned}$
::要找到参数方程的反向,只需切换 x 和 y 的函数。

Inverses of parametric equations are often restricted so that they are also functions.
::参数方程式的逆向往往受到限制,因此它们也是功能。

Review
::回顾

Use the function $\begin{aligned} x = t - 4; y = t^{2} + 2 \end{aligned}$ for #1 - #3.
::在 # 1 - # 3 中使用函数 x=t- 4; y=t2+2。

1. Find the inverse of the function.
::1. 查找函数的反向。

2. Does the point (-2, 6) live on the function or its inverse?
::2. 点(-2, 6)是否活在函数上或其反向上?

3. Does the point (0, 1) live on the function or its inverse?
::3. 点(0,1)是否在函数上存在或反向存在?

Use the relation $\begin{aligned} x = t^{2}; y = 4 - t \end{aligned}$ for #4 - #6.
::使用关系 x=t2; y=4 - t 用于 # 4 - # 6 。

4. Find the inverse of the relation.
::4. 找出这种关系的反面。

5. Does the point (4, 0) live on the relation or its inverse?
::5. 点(4,0)是否以相关关系或相反关系为依据?

6. Does the point (0, 4) live on the relation or its inverse?
::6. 点(0, 4)是否维持在关系或反向关系上?

Use the function $\begin{aligned} x = 2 t + 1; y = t^{2} - 3 \end{aligned}$ for #7 - #9.
::使用函数 x=2t+1; y=t2- 3为 # 7 - # 9 使用函数 x=2t+1; y=t2- 3 为 # 7 - # 9 使用函数 x=2t+1; y=t2- 3 for # 7 - # 9 。

7. Find the inverse of the function.
::7. 查找函数的反向。

8. Does the point (1, 5) live on the function or its inverse?
::8. 点(1,5)是否在函数上存在,或其反向存在?

9. Does the point (9, 13) live on the function or its inverse?
::9. 点(9,13)是否在函数上存在,还是反之?

Use the function $\begin{aligned} x = 3 t + 14; y = t^{2} - 2 t \end{aligned}$ for #10 - #11.
::为 # 10 - # 11 使用函数 x=3t+14; y=t2- 2t。

10. Find the inverse of the function.
::10. 查找函数的反向。

11. Identify where the parametric function intersects with its inverse.
::11. 查明参数函数与其反相交错之处。

Use the relation $\begin{aligned} x = t^{2}; y = 4 t - 4 \end{aligned}$ for #12 - #13.
::使用关系 x=t2; y=4- 4用于 # 12 - # 13 。

12. Find the inverse of the relation.
::12. 找出相互关系的反面。

13. Identify where the relation intersects with its inverse.
::13. 查明关系与其反向交错之处。

14. Parameterize $\begin{aligned} f (x) = x^{2} + x - 6 \end{aligned}$ and then graph the function and its inverse.
::14. 参数f(x)=x2+x-6,然后绘制函数及其反向图。

15. Parameterize $\begin{aligned} f (x) = x^{2} + 3 x + 2 \end{aligned}$ and then graph the function and its inverse.
::15. f(x)=x2+3x+2的参数,然后绘制函数及其反向图。

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

Inverses of Parametric Equations ::参数等量的逆数

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)