Course: 7.5 图形构造根函数

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图形构造根函数

Mrs. Garcia assigns her student the cube root function $\begin{aligned} y = - \sqrt[3]{(x + 1)} \end{aligned}$ to graph for homework. The following day, she asks her students which quadrant(s) their graph is in.
::Garcia夫人指派她的学生用立方根函数 y(x+1) 3 来图表做家庭作业。第二天,她询问她的学生他们的象限在哪个(s) 。

Alendro says that because of the negative sign, all y values are negative. Therefore his graph is only in the third and fourth quadrants quadrant .
::阿伦德罗说,由于负符号,所有y值都是负值。因此他的图表只在第三和第四四象限中。

Dako says that his graph is in the third and fourth quadrants as well but it is also in the second quadrant.
::Dako说,他的图表在第三和第四四象限中,但也在第二象限中。

Marisha says they are both wrong and that her graph of the function is in all four quadrants.
::Marisha说两者都是错的她的函数图是四个四分位数

Which one of them is correct?
::哪一个是正确的?

Graphing Cubed Root Functions
::图形构造根函数

A cubed root function is different from that of a square root . Their general forms look very similar, $\begin{aligned} y = a \sqrt[3]{x - h} + k \end{aligned}$ and the parent graph is $\begin{aligned} y = \sqrt[3]{x} \end{aligned}$ . However, we can take the cubed root of a negative number, therefore, it will be defined for all values of $\begin{aligned} x \end{aligned}$ . Graphing the parent graph, we have:
::立方根函数与平方根函数不同。其一般形式看起来非常相似, y=ax- h3+k, 母图是 y=x3。然而, 我们可以选取负数的立方根, 因此, 它将被定义为 x 的所有值。绘制母图时, 我们有 :

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

For $\begin{aligned} y = \sqrt[3]{x} \end{aligned}$ , the output is the same as the input of $\begin{aligned} y = x^{3} \end{aligned}$ . The domain and range of $\begin{aligned} y = \sqrt[3]{x} \end{aligned}$ are all real numbers. Notice there is no “starting point” 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 .
::对于 y=x3, 输出与 y=x3 输入相同。 y=x3 的域和范围都是真实数字。注意没有像平方根函数那样的“ 起始点 ” , (h, k) 现在指函数弯曲的点, 称为偏移点。

Let's describe how to obtain the graph of $\begin{aligned} y = \sqrt[3]{x} + 5 \end{aligned}$ from $\begin{aligned} y = \sqrt[3]{x} \end{aligned}$ .
::让我们描述如何从 y=x3+5 从 y=x3 获取 y=x3+5 的图形。

W e know that the +5 indicates a vertical shift of 5 units up. Therefore, this graph will look exactly the same as the parent graph, shifted up five units.
::我们知道, +5 表示向上垂直移动 5 个单位。因此, 此图将看起来和父图完全一样, 向上移动了 5 个单位。

Now, let's graph $\begin{aligned} f (x) = - \sqrt[3]{x + 2} - 3 \end{aligned}$ and find the domain and range.
::现在,让我们来图f(x)x+23-3 并找到域和范围。

From the previous problem , we know that from the parent graph, this function is going to shift to the left two units and down three units. The negative sign will result in a reflection .
::从上一个问题中,我们知道,从父图中,此函数将移到左两个单位,向下三个单位。负信号将导致反射。

Alternate Method: If you want to use a table, that will also work. Here is a table, then plot the points. $\begin{aligned} (h, k) \end{aligned}$ should always be the middle point in your table.
::替代方法 : 如果您想要使用表格, 这也会有效。这是一张表格, 然后绘制点数。 (h, k) 应该始终是表格的中点。

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

Finally, let's graph $\begin{aligned} f (x) = \frac{1}{2} \sqrt[3]{x - 4} \end{aligned}$ .
::最后,让我们用图表f(x)=12x-43。

The -4 tells us that, from the parent graph, the function will shift to the right four units. The $\begin{aligned} \frac{1}{2} \end{aligned}$ effects how quickly the function will “grow”. Because it is less than one, it will grow slower than the parent graph.
::-4 告诉我们,从父图中,函数将转移到右四个单位。12 效果是函数“增长”的速度有多快。由于它小于一个,其增长速度将比父图慢。

Using the graphing calculator: If you wanted to graph this function using the TI-83 or 84, press $\begin{aligned} Y = \end{aligned}$ and clear out any functions. Then, press $\begin{aligned} (1 \div 2) \end{aligned}$ , MATH and scroll down to 4 : $\begin{aligned} \sqrt[3]{} \end{aligned}$ and press ENTER . Then, type in the rest of the function, so that $\begin{aligned} Y = (\frac{1}{2}) \sqrt[3]{} (X - 4) \end{aligned}$ . Press GRAPH and adjust the window.
::使用图形计算器 : 如果您想要使用 TI- 83 或 84 来显示此函数, 请按 Y = 并清除任何函数。然后按 (1\\ 2) 、 MATH 并滚动到 4: 3 并按 ENTER 键。然后键入此函数的其余部分, 以便 Y= (123) (X- 4) 按 GRAPH 并调整窗口。

Important Note: The domain and range of all cubed root functions are both all real numbers.
::重要注意: 所有立方根函数的域和范围都是真实数字。

Examples
::实例

Example 1
::例1

Earlier, you were asked to determine which student was correct.
::早些时候,有人要求你确定哪个学生是正确的。

If you graph the function $\begin{aligned} y = - \sqrt[3]{(x + 1)} \end{aligned}$ , you see that the domain is all real numbers, which makes all quadrants possible. However, for all positive values of x , y is negative because of the negative sign in front of the cube root. That rules out the first quadrant. Therefore, Dako is correct.
::如果您图形显示 y( x+1) 3 函数, 您就会看到域是所有真实数字, 这使得所有四分位数成为可能。但是, 对于 x 的所有正值来说, y 是负的, 因为立方根前面有负的符号。这排除了第一个象限。因此, Dako 是正确的。

Example 2
::例2

Evaluate $\begin{aligned} y = \sqrt[3]{x + 4} - 11 \end{aligned}$ when $\begin{aligned} x = - 12 \end{aligned}$ .
::在 x12 时评价 y=x+43- 11。

Plug in $\begin{aligned} x = - 12 \end{aligned}$ and solve for $\begin{aligned} y \end{aligned}$ .
::插入 x12 并解决 y 。

$\begin{aligned} y = \sqrt[3]{- 12 + 4} - 11 = \sqrt[3]{- 8} + 4 = - 2 + 4 = 2 \end{aligned}$
::y12+43-1183+42+4=2

Example 3
::例3

Describe how to obtain the graph of $\begin{aligned} y = \sqrt[3]{x + 4} - 11 \end{aligned}$ from $\begin{aligned} y = \sqrt[3]{x} \end{aligned}$ .
::描述如何从 y=x+43- 11 从 y=x3 获取 y=x+43- 11 的图形。

Starting with $\begin{aligned} y = \sqrt[3]{x} \end{aligned}$ , you would obtain $\begin{aligned} y = \sqrt[3]{x + 4} - 11 \end{aligned}$ by shifting the function to the left four units and down 11 units.
::从 y=x3 开始,您可以通过将函数向左四个单位和向下11 个单位移动来获取 Y=x+43- 11 。

Graph the following cubed root functions. Check your graphs on the graphing calculator.
::绘制下面的立方根函数。请检查图形计算器上的图表。

Example 4
::例4

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

This function is a horizontal shift to the right two units and down four units.
::此函数是向右两个单位和向下四个单位的横向移动。

Example 5
::例5

$\begin{aligned} f (x) = - 3 \sqrt{x} - 1 \end{aligned}$
:xx)%%3x- 1)

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 one unit.
::此函数是 y=x3 的反射, 伸展为 y=x3 的三倍。最后, 它会向下移动一个单位。

Review
::回顾

Evaluate $\begin{aligned} f (x) = \sqrt[3]{2 x - 1} \end{aligned}$ for the following values of x .
::下列x的数值评价 f(x)=2x-13。

$\begin{aligned} f (14) \end{aligned}$
::f(14) f(14)

$\begin{aligned} f (- 62) \end{aligned}$
::f( - 62)

$\begin{aligned} f (20) \end{aligned}$
::f( 20)

Graph the following cubed root functions. Use your calculator to check your answers.
::绘制下面的立方根函数。使用您的计算器检查答案。

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

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

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

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

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

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

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

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

$\begin{aligned} y = - \frac{1}{3} \sqrt[3]{x - 5} + 2 \end{aligned}$
::y13x-53+2

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

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

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

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

图形构造根函数

Graphing Cubed Root Functions ::图形构造根函数

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Graph the following cubed root functions. Check your graphs on the graphing calculator. ::绘制下面的立方根函数。请检查图形计算器上的图表。

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)