课程： 5.6 儿科功能的阶段转移

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子类函数的阶段移位

A periodic function that does not start at the sinusoidal axis or at a maximum or a minimum has been shifted horizontally. This horizontal movement allows for different starting points since a sine wave does not have a beginning or an end.
::不从正弦轴开始, 或最大或最小的周期函数被水平移动。此水平移动允许不同的起点, 因为正弦波没有开始或结束。

What are five other ways of writing the function $\begin{aligned} f (x) = 2 \cdot \sin x \end{aligned}$ ?
::写函数 f(x) = 2sinx 的另外五种方式是什么?

Phase Shift of Sinusoidal Functions
::子类函数的阶段移位

The general sinusoidal function is:
::一般的正弦值函数是:

$\begin{aligned} f (x) = \pm a \cdot \sin (b (x + c)) + d \end{aligned}$
:xxx)+d)+(b(x+c)+(d)+(xx)+(xx)+

The constant $\begin{aligned} c \end{aligned}$ controls the phase shift . Phase shift is the horizontal shift left or right for periodic functions. If $\begin{aligned} c = \frac{π}{2} \end{aligned}$ then the sine wave is shifted left by $\begin{aligned} \frac{π}{2} \end{aligned}$ . If $\begin{aligned} c = - 3 \end{aligned}$ then the sine wave is shifted right by 3. This is the opposite direction than you might expect, but it is consistent with the rules of transformations for all functions.
::常数 c 控制了相位移。相位移是用于周期函数的水平向左或右移。如果 c2, 则正弦波由 2 转移。如果 c3 则正弦波由 3 转移。这是与您预期相反的方向, 但与所有函数的转换规则是一致的。

To graph a function such as $\begin{aligned} f (x) = 3 \cdot \cos (x - \frac{π}{2}) + 1 \end{aligned}$ , first find the start and end of one period. Then sketch only that portion of the sinusoidal axis. Finally, plot the 5 important points for a cosine graph while keeping the in mind. The graph is shown below:
::要绘制函数, 如 f( x) = 3cos( x2) +1 , 请先找到一个周期的开始和结束。然后只绘制正弦轴的部分。最后, 绘制正弦图的5个重要点, 并记住。图表显示如下 :

Generally $\begin{aligned} b \end{aligned}$ is always written to be positive. If you run into a situation where $\begin{aligned} b \end{aligned}$ is negative, use your knowledge of even and odd functions to rewrite the function.
::b 一般是肯定的。如果遇到b为否定的情况,请使用您对偶数和奇数功能的知识重写该函数。

\begin{aligned} \cos (- x) & = \cos (x) \\ \sin (- x) & = - \sin (x) \end{aligned}

::cos( - x) =cos ( x) sin ( - x) sin ( x)

Examples
::实例

Example 1
::例1

Earlier, you were asked to write $\begin{aligned} f (x) = 2 \cdot \sin x \end{aligned}$ in five different ways. The function $\begin{aligned} f (x) = 2 \cdot \sin x \end{aligned}$ can be rewritten an infinite number of ways.
::早些时候, 您被要求以五种不同的方式写入 f( x) = 2sinx 。函数 f( x) = 2sinx 可以重写无数种方式。

\begin{aligned} 2 \cdot \sin x = - 2 \cdot \cos (x + \frac{π}{2}) = 2 \cdot \cos (x - \frac{π}{2}) = - 2 \cdot \sin (x - π) = 2 \cdot \sin (x - 8 π) \end{aligned}

::22222222222222222222222222222222(x_88)22828

It all depends on where you choose start and whether you see a positive or negative sine or cosine graph.
::这一切都取决于您选择的起始位置,以及您是否看到正正正正弦或负正弦或正弦图。

Example 2
::例2

Given the following graph, identify equivalent algebraic models.
::根据下图,确定等效代数模型。

Either this is a sine function shifted right by $\begin{aligned} \frac{π}{4} \end{aligned}$ or a cosine graph shifted left $\begin{aligned} \frac{5 π}{4} \end{aligned}$ .
::要么是正弦函数由 +++ 4 向右移动, 要么是正弦图向左移动 5+4 。

$\begin{aligned} f (x) = \sin (x - \frac{π}{4}) = \cos (x + \frac{5 π}{4}) \end{aligned}$
:xx) =sin(x4) =cos(x+54) =cos(x+54)

Example 3
::例3

At $\begin{aligned} t = 5 \end{aligned}$ minutes William steps up 2 feet to sit at the lowest point of the Ferris wheel that has a diameter of 80 feet. A full hour later he finally is let off the wheel after making only a single revolution. During that hour he wondered how to model his height over time in a graph and equation.
::T=5分钟后,威廉升起2英尺,坐在直径为80英尺的Ferris轮的最低点。整整一小时后,他终于在一次革命后被从方向盘上解脱出来。在那个时候,他想知道如何用图表和方程来模拟他的身高。

Since the period is 60 which works extremely well with the $\begin{aligned} 360^{\circ} \end{aligned}$ in a circle, this problem will be shown in degrees.
::由于60岁这一时期在一个圆圈里与360极为有效,这一问题将以度表示。

Time (minutes)	Height (feet)
5	2
20	42
35	82
50	42
65	2

William chooses to see a negative cosine in the graph. He identifies the amplitude to be 40 feet. The vertical shift of the sinusoidal axis is 42 feet. The horizontal shift is 5 minutes to the right.
::威廉选择在图形中看到负余弦。他将振幅确定为40英尺。正弦轴的垂直移动为42英尺。水平移动向右为5分钟。

The period is 60 (not 65) minutes which implies $\begin{aligned} b = 6 \end{aligned}$ when graphed in degrees.
::时间段为60分钟(不是65分钟),这意味着以度表示时 b=6。

$\begin{aligned} 60 = \frac{360}{b} \end{aligned}$
::60=360b

Thus one equation would be:
::因此,一个等式是:

$\begin{aligned} f (x) = - 40 \cdot \cos (6 (x - 5)) + 42 \end{aligned}$
::f(x) +42 (6(x)-5)

Example 4
::例4

Tide tables report the times and depths of low and high tides. Here is part of tide report from Salem, Massachusetts dated September 19, 2006.
::潮汐表报告了低潮和高潮的时间和深度,这是2006年9月19日马萨诸塞州萨利姆的潮汐报告的一部分。

10:15 AM ::上午10时15分	9 ft. ::9英尺	High Tide ::高潮下
4:15 PM ::下午4:15	1 ft. ::1英尺	Low Tide ::低潮下
10:15 PM ::上午10时15分	9 ft. ::9英尺	High Tide ::高潮下

Find an equation that predicts the height based on the time. Choose when $\begin{aligned} t = 0 \end{aligned}$ carefully.
::查找一个根据时间预测高度的方程式。仔细选择 t=0 的时间。

There are two logical places to set $\begin{aligned} t = 0 \end{aligned}$ . The first is at midnight the night before and the second is at 10:15 AM. The first option illustrates a phase shift that is the focus of this concept, but the second option produces a simpler equation. Set $\begin{aligned} t = 0 \end{aligned}$ to be at midnight and choose units to be in minutes.
::有两个逻辑位置可以设定 t=0 。第一个是前一天午夜, 第二个是上午10: 15。第一个选项显示的是这个概念的焦点, 但第二个选项产生一个简单的方程式。设定 t=0 在午夜, 并在分钟内选择单位。

Time (hours : minutes)	Time (minutes)	Tide (feet)
10:15	615	9
16:15	975	1
22:15	1335	9
	$\begin{aligned} \frac{615 + 975}{2} = 795 \end{aligned}$	5
	$\begin{aligned} \frac{1335 + 975}{2} = 1155 \end{aligned}$	5

These numbers seem to indicate a positive cosine curve. The amplitude is 4 and the vertical shift is 5. The horizontal shift is 615 and the period is 720.
::这些数字似乎表示正余弦曲线。振幅为 4, 垂直变化为 5。水平变化为 615, 周期为 720。

$\begin{aligned} 720 = \frac{2 π}{b} \to b = \frac{π}{360} \end{aligned}$
::720=2bb360

Thus one equation is:
::因此,一个方程式是:

$\begin{aligned} f (x) = 4 \cdot \cos (\frac{π}{360} (x - 615)) + 5 \end{aligned}$
::f(x) = 4cos( 360(x-615) +5

Example 5
::例5

Use the equation from Example 4 to find out when the tide will be at exactly 8 ft on September $\begin{aligned} 19^{t h} \end{aligned}$ .
::利用例4的方程式来找出9月19日潮水将到达8英尺

This problem gives you the $\begin{aligned} y \end{aligned}$ and asks you to find the $\begin{aligned} x \end{aligned}$ . Later you will learn how to solve this algebraically, but for now use the power of the intersect button on your calculator to intersect the function with the line $\begin{aligned} y = 8 \end{aligned}$ . Remember to find all the $\begin{aligned} x \end{aligned}$ values between 0 and 1440 to account for the entire 24 hours.
::此问题给了您 y , 并要求您找到 x 。以后您将学习如何解析这个代数, 但现在您将使用计算器上的交叉按钮的力量将函数与 y= 8 线交叉。记住要找到 0 至 1440 之间的所有 x 值, 以计算整个 24 小时。

There are four times within the 24 hours when the height is exactly 8 feet. You can convert these times to hours and minutes if you prefer.
::在24小时之内,高度为8英尺,有4次。如果您愿意,可以将这些时间转换为小时和分钟。

$\begin{aligned} t \approx \end{aligned}$ 532.18 (8:52), 697.82 (11:34), 1252.18 (20:52), 1417.82 (23:38)
::T 532.18 (8:52)、697.82 (11:34)、1252.18 (20:52)、1417.82 (23:38)

Summary

For the general form of the sinusoidal function $\begin{aligned} f (x) = \pm a \cdot \sin (b (x + c)) + d, \end{aligned}$ the constant $\begin{aligned} c \end{aligned}$ represents the phase shift, or horizontal shift.
- If $\begin{aligned} c \end{aligned}$ is positive, the function is shifted left. If $\begin{aligned} c \end{aligned}$ is negative, the function is shifted right.
  ::如果 c 是正,则函数向左移。如果 c 是负,则函数向右移。
::对于正弦函数f(x)asin(b(x+c)+d)的一般形式,恒定 c 表示相位变化或水平变化。如果c为正,则函数向左移动。如果c为负,则函数向右移动。

Review
::回顾

Graph each of the following functions.
::绘制下列函数的每一个图。

1. $\begin{aligned} f (x) = 2 \cos (x - \frac{π}{2}) - 1 \end{aligned}$
::1. f(x)=2cos(x)2-1

2. $\begin{aligned} g (x) = - \sin (x - π) + 3 \end{aligned}$
::2. g(x)+3 g(x) @sin @(x)+3

3. $\begin{aligned} h (x) = 3 \cos (2 (x - π)) \end{aligned}$
::3. h(x)=3cos(2(x))

4. $\begin{aligned} k (x) = - 2 \sin (2 x - π) + 1 \end{aligned}$
::4. k(x) @ @% 2sin @ (2x)+1

5. $\begin{aligned} j (x) = - \cos (x + \frac{π}{2}) \end{aligned}$
::5. j(x) (x) 2

Give one possible sine equation for each of the graphs below.
::给下方图中的每个图表提供一个可能的正弦方程。

Give one possible cosine function for each of the graphs below.
::给下方每个图表提供一个可能的余弦函数。

10.

11.

The temperature over a certain 24 hour period can be modeled with a sinusoidal function. At 3:00, the temperature for the period reaches a low of $\begin{aligned} 22^{\circ} F \end{aligned}$ . At 15:00, the temperature for the period reaches a high of $\begin{aligned} 40^{\circ} F \end{aligned}$ .
::特定24小时的温度可模拟为正弦函数。 3: 00时, 这一期间的温度低至22°F。下午3:00时, 这一期间的温度高至40°F。

12. Find an equation that predicts the temperature based on the time in minutes. Choose $\begin{aligned} t = 0 \end{aligned}$ to be midnight.
::12. 找到一个根据分钟中的时间预测温度的方程。选择 t=0 为午夜。

13. Use the equation from #12 to predict the temperature at 4:00 PM.
::13. 使用第12号方程式预测下午4点的温度。

14. Use the equation from #12 to predict the temperature at 8:00 AM.
::14. 利用12号的方程来预测早上8点的温度。

15. Use the equation from #12 to predict the time(s) it will be $\begin{aligned} 32^{\circ} F \end{aligned}$ .
::15. 使用第12号方程式预测时间为32-F。

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。

章节大纲

常规

子类函数的阶段移位

Phase Shift of Sinusoidal Functions ::子类函数的阶段移位

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)