Section: 构造立方根函数 | 10.8 构造立方根函数

Section outline

The power needed for an airplane to overcome the drag on an aircraft is approximately $\begin{aligned} v = 4 \sqrt[3]{P} \end{aligned}$ , where P is power and v is the velocity at which the plane is traveling ¹ . We can determine what velocities we can overcome given different amounts of power.
::飞机克服飞机拖力所需的动力大约为 v=4P3, P是动力, V是飞机旅行的速度1。我们可以确定,由于电力量不同,我们能克服什么速度。

With a graph, we can see all the solutions at one time. We discuss how to graph cube root functions in this section.
::使用图表,我们可以同时看到所有解决方案。我们讨论如何在本节中绘制立方根函数的图形。

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Graphing Cube Root Functions
::构造立方根函数

A cube root function is different from that of a square root even though their standard forms look quite similar.
::立方体根函数与平方根函数不同,尽管其标准表格看起来相当相似。

Standard Form of a Cube Root Function
::立方根函数的标准格式

\begin{aligned} y = a \sqrt[3]{x - h} + k, \end{aligned}

::y=ax-h3+k, y=ax-h3+k, y=ax-h3+k, y=ax-h3+k, y=ax-h3+k, y=ax-h3+k,

where $\begin{aligned} a, h, k \end{aligned}$ are real numbers.
::a,h,k是真实数字。

$\begin{aligned} a, h, \end{aligned}$ and $\begin{aligned} k \end{aligned}$ play the same roles as they did for square root functions.
::a, h, k 发挥与平方根函数相同的作用。

However, we can take the cube root of any real number , including negative numbers; therefore , cube root functions will be defined for all values of $\begin{aligned} x \end{aligned}$ .
::然而,我们可以选择任何实际数字的立方根, 包括负数; 因此, 立方根函数将被定义为 x 的所有值。

Once again, knowing perfect cubes will be helpful.
::再一次,知道完美的立方体将会有帮助。

$\begin{aligned} 1^{3} = 1 \end{aligned}$	$\begin{aligned} 6^{3} = 216 \end{aligned}$
$\begin{aligned} 2^{3} = 8 \end{aligned}$	$\begin{aligned} 7^{3} = 343 \end{aligned}$
$\begin{aligned} 3^{3} = 27 \end{aligned}$	$\begin{aligned} 8^{3} = 512 \end{aligned}$
$\begin{aligned} 4^{3} = 64 \end{aligned}$	$\begin{aligned} 9^{3} = 729 \end{aligned}$
$\begin{aligned} 5^{3} = 125 \end{aligned}$	$\begin{aligned} 10^{3} = 1, 000 \end{aligned}$

We see the graph of the parent function below.
::下面是父函数图。

Example 1
::例1

Graph $\begin{aligned} y = \sqrt[3]{x} \end{aligned}$ .
::图y=x3。

Solution: First we make a table of values.
::解决办法:首先,我们编制一个数值表。

x	y
-27	-3
-8	-2
-1	-1
0	0
1	1
8	2
27	3

Next, plot these points and draw the graph.
::接下来,绘制这些点,绘制图表。

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The domain and range of $\begin{aligned} y = \sqrt[3]{x} \end{aligned}$ are all real numbers. Notice there is no endpoint like the square root functions; the $\begin{aligned} (h, k) \end{aligned}$ now refers to the point where the function bends, called a point of inflection .
::y=x3 的域和范围都是真实数字。注意没有像平方根函数这样的端点; (h, k) 现在指函数弯曲的点, 称为折移点。

Example 2
::例2

Graph $\begin{aligned} y = \sqrt[3]{x} + 5 \end{aligned}$ .
::图y=x3+5。

Solution: From the concept on Transformations , we know that the +5 indicates a vertical shift of 5 units up. Therefore, this graph will look exactly the same as the parent graph above, shifted up 5 units.
::解决方案: 从“ 变换” 的概念中, 我们知道 +5 表示向上垂直移动 5 个单位。因此, 此图将和上面的父图完全一样, 向上移动了 5 个单位。

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Example 3
::例3

Graph $\begin{aligned} y = \sqrt[3]{x - 2} - 4 \end{aligned}$ .
::图y=x-23-4。

Solution: This function is a horizontal shift to the right 2 units and down 4 units. To get a new table of values, add 2 to each of the x -values from the parent graph.
::解析 : 此函数是向右 2 单位向下向下 4 个单位的横向移动。要获得新的值表, 请在父图中的 x 值中各增加 2 。

$\begin{aligned} x \end{aligned}$	$\begin{aligned} y \end{aligned}$
$\begin{aligned} - 8 + 2 = - 6 \end{aligned}$	-6
$\begin{aligned} - 1 + 2 = 1 \end{aligned}$	-5
$\begin{aligned} 0 + 2 = 2 \end{aligned}$	-4
$\begin{aligned} 1 + 2 = 3 \end{aligned}$	-3
$\begin{aligned} 8 + 2 = 10 \end{aligned}$	-2

Next, we plot the points and draw the graph.
::接下来,我们绘制点和图。

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by Mr. Yasuda's Math Videos demonstrates how to graph a cube root function without a calculator.
::由Yasuda先生的数学视频展示如何用图解来显示一个没有计算器的立方根函数。

Example 4
::例4

The power needed for an airplane to overcome the drag on an aircraft is approximately $\begin{aligned} v = 4 \sqrt[3]{P} \end{aligned}$ , where P is power and v is the velocity at which the plane is traveling ¹ . Graph this function.
::飞机克服飞机拖力所需的动力约为 v=4P3, P是动力, v是飞机旅行的速度。请绘制此函数。

Solution: This is a transformation of the parent graph, where the outputs are all multiplied by a factor of 4, so we can use the values in Example 1 and multiply them by 4 to get the points on this graph. The graph is below.
::解决方案 : 这是父图的转换, 输出全部乘以 4 系数, 这样我们就可以使用例1 中的值乘以 4 来获得此图上的点。图表如下。

Example 5
::例5

Graph $\begin{aligned} f (x) = - 3 \sqrt{x} - 1 \end{aligned}$ .
::图f(x)%3x-1。

Solution: This function is a reflection of $\begin{aligned} y = \sqrt[3]{x} \end{aligned}$ and stretched to be three times as large. Lastly, it is shifted down 1 unit.
::解析度: 此函数是 y=x3 的反射值, 伸展为 y=x3 的三倍。最后, 它会向下移动 1 个单位。

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Example 6
::例6

Graph $\begin{aligned} f (x) = - \sqrt[3]{x + 2} - 3 \end{aligned}$ .
::图f(x)x+23-3。

Solution: This function is going to shift to the left 2 units and down 3 units. Multiplying by -1 will result in a reflection.
::解决方案 : 此函数将移到左2个单位和下3个单位。乘以 - 1 将导致反射。

Here is a table; next, plot the points. To get the x -values, add 2 to the typical x -values of the parent graph.
::这是一张表格; 下一张, 绘制点。要获取 x 值, 请在父图的典型 x 值中添加 2 。

x	y
6	-5
-1	-4
-2	-3
-3	-2
-10	-1

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by Mathispower4u demonstrates how to graph transformations of the cube root function.
::由 Mathispower4u 演示如何图形化立方体根函数的转换。

Summary
::摘要

To graph cube root functions, make a table of values, plot the points, and graph the function.
::要绘制立方体根函数,请绘制一个数值表,绘制点图,并绘制函数图。

Review
::回顾

Graph the following cube root functions:
::绘制以下立方根函数图 :

1. $\begin{aligned} y = \sqrt[3]{x} + 4 \end{aligned}$
::1. y=x3+4

2. $\begin{aligned} y = \sqrt[3]{x - 3} \end{aligned}$
::2.y=x-33 y=x-33

3. $\begin{aligned} f (x) = \sqrt[3]{x + 2} - 1 \end{aligned}$
::3. f(x)=x+23-1

4. $\begin{aligned} g (x) = - \sqrt[3]{x} - 6 \end{aligned}$
::4. g(x)=-x3-6

5. $\begin{aligned} f (x) = 2 \sqrt[3]{x + 1} \end{aligned}$
::5. f(x)=2x+13

6. $\begin{aligned} h (x) = - 3 \sqrt[3]{x} + 5 \end{aligned}$
::6. h(x)=-3x3+5

7. $\begin{aligned} y = \frac{1}{2} \sqrt[3]{1 - x} \end{aligned}$
::7.y=121-x3

8. $\begin{aligned} y = 2 \sqrt[3]{x + 4} - 3 \end{aligned}$
::8.y=2x+43-3

9. $\begin{aligned} y = - \frac{1}{3} \sqrt[3]{x - 5} + 2 \end{aligned}$
::9.y=-13x-53+2

10. $\begin{aligned} g (x) = \sqrt[3]{6 - x} + 7 \end{aligned}$
::10. g(x)=6-x3+7

11. $\begin{aligned} f (x) = - 5 \sqrt[3]{x - 1} + 3 \end{aligned}$
::11. f(x)=-5x-13+3

12. $\begin{aligned} y = 4 \sqrt[3]{7 - x} - 8 \end{aligned}$
::y=47-x3-8

Explore More
::探索更多

1. Write the equation for each function graphed below.
::1. 写下下列各函数的方程。

a.
::a 。

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b.
::b. b. 数据

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2. Write the equation of a cube root function with $\begin{aligned} (h, k) = (2, 7) \end{aligned}$ passing through $\begin{aligned} (10, 11) \end{aligned}$ .
::2. 写入立方根函数的方程式(h,k)=(2,7)通过(10,11)。

Answers for Review and Explore More Problems
::回顾和探讨更多问题的答复

Please see the Appendix.
::请参看附录。

PLIX
::PLIX

Try this interactive that reinforces the concepts explored in this section:
::尝试这一互动,强化本节所探讨的概念:

References
::参考参考资料

1. "Drag (physics)," last edited May 18, 2017,
::1. 2017年5月18日上次编辑的"碎片(物理)"

Section outline

Graphing Cube Root Functions ::构造立方根函数

Standard Form of a Cube Root Function ::立方根函数的标准格式

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Example 6 ::例6

Summary ::摘要

Review ::回顾

Explore More ::探索更多

Answers for Review and Explore More Problems ::回顾和探讨更多问题的答复

PLIX ::PLIX

References ::参考参考资料

Graphing Cube Root Functions
::构造立方根函数

Standard Form of a Cube Root Function
::立方根函数的标准格式

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Example 6
::例6

Summary
::摘要

Review
::回顾

Explore More
::探索更多

Answers for Review and Explore More Problems
::回顾和探讨更多问题的答复

PLIX
::PLIX

References
::参考参考资料