Course: 11.2 平根函数的转移

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平根函数移位
Shifts of Square Root Functions
::平根函数移位

We will now look at how graphs are shifted up and down in the Cartesian plane .
::我们现在将研究如何在笛卡尔飞机上向上和向下移动图表。

Graph the functions $\begin{aligned} y = \sqrt{x}, y = \sqrt{x} + 2 \end{aligned}$ and $\begin{aligned} y = \sqrt{x} - 2 \end{aligned}$ .
::函数=x,y=x+2andy=x-2图。

When we add a constant to the right-hand side of the equation , the graph keeps the same shape, but shifts up for a positive constant or down for a negative one.
::当我们在方程式的右侧添加一个常数时, 图形保持相同的形状, 但向正常数移动或向下移动到负常数。

Graphing Multiple Functions
::绘制多个函数

Graph the functions $\begin{aligned} y = \sqrt{x}, y = \sqrt{x - 2}, \end{aligned}$ and $\begin{aligned} y = \sqrt{x + 2} \end{aligned}$ .
::函数y=x,y=x-2和y=x+2的图形。

When we add a constant to the argument of the function (the part under the radical sign), the function shifts to the left for a positive constant and to the right for a negative constant.
::当我们在函数参数(激进符号下的部分)中加上一个常数时,函数向左移,向右转,向右转,向右转,向右转,向右转,向右转,向右转,向右转,向右转,向右转,向右转,向右转,向右转,向右转。

Now let’s see how to combine all of the above types of transformations .
::现在让我们来看看如何将上述所有类型的转变结合起来。

Combining Transformations
::组合转换

Graph the function $\begin{aligned} y = 2 \sqrt{3 x - 1} + 2 \end{aligned}$ .
::函数 y= 23x- 1+2 图形。

We can think of this function as a combination of shifts and stretches of the basic square root function $\begin{aligned} y = \sqrt{x} \end{aligned}$ . We know that the graph of that function looks like this:
::我们可以将这个函数视为基本平方根函数 y=x 的移动和拉长的组合。我们知道该函数的图表是这样的:

If we multiply the argument by 3 to obtain $\begin{aligned} y = \sqrt{3 x} \end{aligned}$ , this stretches the curve vertically because the value of $\begin{aligned} y \end{aligned}$ increases faster by a factor of $\begin{aligned} \sqrt{3} \end{aligned}$ .
::如果我们将参数乘以 3 以获得 y= 3x, 则会垂直延伸曲线, 因为 y 的值增加更快, 增加系数为 3 。

Next, when we subtract 1 from the argument to obtain $\begin{aligned} y = \sqrt{3 x - 1} \end{aligned}$ this shifts the entire graph to the left by one unit.
::接下来,当我们从参数中减去 1 以获取 y= 3x- 1 时, 将整张图移到左侧一个单元。

Multiplying the function by a factor of 2 to obtain $\begin{aligned} y = 2 \sqrt{3 x - 1} \end{aligned}$ stretches the curve vertically again, because $\begin{aligned} y \end{aligned}$ increases faster by a factor of 2.
::将函数乘以 2 乘以 2 以获得 y= 23x- 1 的 y= 23x- 1 ,将曲线垂直再拉长一次,因为 y 增速快于 2 。

Finally we add 2 to the function to obtain $\begin{aligned} y = 2 \sqrt{3 x - 1} + 2 \end{aligned}$ . This shifts the entire function vertically by 2 units.
::最后,在获取y=23x-1+2的函数中增加2个。这将整个函数垂直改变为 2 个单位。

Each step of this process is shown in the graph below. The purple line shows the final result.
::此过程的每个步骤都显示在下图中。紫线显示最终结果。

Now we know how to graph square root functions without making a table of values. If we know what the basic function looks like, we can use shifts and stretches to transform the function and get to the desired result.
::现在我们知道如何绘制平方根函数, 而不绘制一个数值表。如果我们知道基本函数的外观, 我们可以使用移动和伸展来转换函数, 并达到预期的结果。

Example
::示例示例示例示例

Example 1
::例1

Graph the function $\begin{aligned} y = - \sqrt{x + 3} - 5 \end{aligned}$ .
::函数 yx+3 - 5 的图形。

We can think of this function as a combination of shifts and stretches of the basic square root function $\begin{aligned} y = \sqrt{x} \end{aligned}$ . We know that the graph of that function looks like this:
::我们可以将这个函数视为基本平方根函数 y=x 的移动和拉长的组合。我们知道该函数的图表是这样的:

Next, when we add 3 to the argument to obtain $\begin{aligned} y = \sqrt{x + 3} \end{aligned}$ this shifts the entire graph to the right by 3 units.
::接下来,当我们为获取 y=x+3 在参数中添加 3 时, 将整张图向右移动 3 个单位。

Multiplying the function by -1 to obtain $\begin{aligned} y = - \sqrt{x + 3} \end{aligned}$ which reflects the function across the $\begin{aligned} x \end{aligned}$ -axis.
::将函数乘以 -1 以获取 yx+3, 反映 X 轴的函数。

Finally we subtract 5 from the function to obtain $\begin{aligned} y = - \sqrt{x + 3} - 5 \end{aligned}$ . This shifts the entire function down vertically by 5 units.
::最后,我们从函数中减去5,以获取 yx+3-5。这样整个函数垂直下移5个单位。

Review
::回顾

Graph the following functions.
::如下图所示函数。

$\begin{aligned} y = \sqrt{2 x - 1} \end{aligned}$
::y=2x-1

$\begin{aligned} y = \sqrt{x - 100} \end{aligned}$
::y=x-100 y=x-100

$\begin{aligned} y = \sqrt{4 x + 4} \end{aligned}$
::y=4x+4 y=4x+4

$\begin{aligned} y = \sqrt{5 - x} \end{aligned}$
::y=5-x y=5-x

$\begin{aligned} y = 2 \sqrt{x} + 5 \end{aligned}$
::y=2x+5 y=2x+5

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

$\begin{aligned} y = 4 + 2 \sqrt{x} \end{aligned}$
::y=4+2x y=4+2x

$\begin{aligned} y = 2 \sqrt{2 x + 3} + 1 \end{aligned}$
::y=22x+3+1

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

$\begin{aligned} y = \sqrt{x + 1} - \sqrt{4 x - 5} \end{aligned}$
::y=x+1 - 4x - 5

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

平根函数移位

Shifts of Square Root Functions ::平根函数移位

Graphing Multiple Functions ::绘制多个函数

Combining Transformations ::组合转换

Example ::示例示例示例示例

Example 1 ::例1

Review ::回顾

Review (Answers) ::回顾(答复)

Shifts of Square Root Functions
::平根函数移位

Graphing Multiple Functions
::绘制多个函数

Combining Transformations
::组合转换

Example
::示例示例示例示例

Example 1
::例1

Review
::回顾

Review (Answers)
::回顾(答复)