课程： 5.2 儿科功能家庭

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Sinusoidal功能家庭

The cosine function is the $\begin{aligned} x \end{aligned}$ coordinates of the unit circle and the sine function is the $\begin{aligned} y \end{aligned}$ coordinates . Since the unit circle has radius one and is centered at the origin, both oscillate between positive and negative one.
::余弦函数是单位圆的x坐标,正弦函数是横坐标。由于单位圆为半径一,以原为中心,正向和负向的均在正向和负向之间。

What happens when the circle is not centered at the origin and does not have a radius of 1?
::当圆不以原点为中心,且半径不为1时,会发生什么情况?

Graphs of Sinusoidal Functions
::子函数图

The sinusoidal function family refers to either sine or cosine waves since they are the same except for a horizontal shift. This function family is also called the periodic function family because the function repeats after a given period of time.
::等离子函数族是指正弦波或共弦波,因为它们是相同的,但水平变化除外。这个函数族也称为周期函数族,因为函数在特定时间之后重复。

Consider a Ferris wheel that spins evenly with a radius of 1 unit. It starts at (1, 0) or an angle of 0 radians and spins counterclockwise at a rate of one cycle per $\begin{aligned} 2 π \end{aligned}$ minutes (so you can use time is equal to radians).
::考虑一个以 1 单位半径平均旋转的 Ferris 轮。它从 1 、 0 或 0 弧度角度开始, 逆时速旋转, 每 23 分钟一个周期( 这样您就可以使用时间等于弧度 ) 。

The 16 points around the circle are chosen because they correspond to the key points of the unit circle. Their heights ( $\begin{aligned} y \end{aligned}$ -values) and widths ( $\begin{aligned} x \end{aligned}$ -values) are already known and can be filled in.
::选择圆周的 16 个点是因为它们与单位圆的要点相对应。它们的高度( y- values) 和宽度( x- values) 已经为人所知, 可以填充。

First consider the height at each of the points as you travel around half of the circle from the starting location. Keep track of your work in a table.
::首先考虑每个点的高度, 从起始位置开始, 您在大约半个圆上行走。请在表格中记录您的工作。

Angle (radians)	Height (units)
0	0
$\begin{aligned} \frac{π}{6} \end{aligned}$	$\begin{aligned} \frac{1}{2} \end{aligned}$
$\begin{aligned} \frac{π}{4} \end{aligned}$	$\begin{aligned} \frac{\sqrt{2}}{2} \approx 0.707 \end{aligned}$
$\begin{aligned} \frac{π}{3} \end{aligned}$	$\begin{aligned} \frac{\sqrt{3}}{2} \approx 0.866 \end{aligned}$
$\begin{aligned} \frac{π}{2} \end{aligned}$	1
$\begin{aligned} \frac{2 π}{3} \end{aligned}$	$\begin{aligned} \frac{\sqrt{3}}{2} \approx 0.866 \end{aligned}$
$\begin{aligned} \frac{3 π}{4} \end{aligned}$	$\begin{aligned} \frac{\sqrt{2}}{2} \approx 0.707 \end{aligned}$
$\begin{aligned} \frac{5 π}{6} \end{aligned}$	$\begin{aligned} \frac{1}{2} \end{aligned}$
$\begin{aligned} π \end{aligned}$	0

Notice the of the height around $\begin{aligned} \frac{π}{2} \end{aligned}$ and see the rest of the table in the examples. Once the table is finished, you can plot these points on a regular coordinate plane where the $\begin{aligned} x \end{aligned}$ axis is the angle and the $\begin{aligned} y \end{aligned}$ axis is the height. This is the first part of the graph of the sine function.
::注意% 2 周围的高度, 并在示例中查看表格的其余部分。表格完成后, 您可以在正则坐标平面上绘制这些点, 此处的 x 轴是角度, Y 轴是高度。这是正弦函数图形的第一部分。

To finish the graph of the sine function, finish the table for heights of the points in quadrants III and IV and draw an entire cycle (known as a period) of the sine function.
::要完成正弦函数的图形,完成三和四中点高度的表格,并绘制正弦函数的整个周期(称为时段)。

Angle (radians)	Height (units)
$\begin{aligned} π \end{aligned}$	0
$\begin{aligned} \frac{7 π}{6} \end{aligned}$	$\begin{aligned} - \frac{1}{2} \end{aligned}$
$\begin{aligned} \frac{5 π}{4} \end{aligned}$	$\begin{aligned} - \frac{\sqrt{2}}{2} \approx - 0.707 \end{aligned}$
$\begin{aligned} \frac{4 π}{3} \end{aligned}$	$\begin{aligned} - \frac{\sqrt{3}}{2} \approx - 0.866 \end{aligned}$
$\begin{aligned} \frac{3 π}{2} \end{aligned}$	-1
$\begin{aligned} \frac{5 π}{3} \end{aligned}$	$\begin{aligned} - \frac{\sqrt{3}}{2} \approx - 0.866 \end{aligned}$
$\begin{aligned} \frac{7 π}{4} \end{aligned}$	$\begin{aligned} - \frac{\sqrt{2}}{2} \approx - 0.707 \end{aligned}$
$\begin{aligned} \frac{11 π}{6} \end{aligned}$	$\begin{aligned} - \frac{1}{2} \end{aligned}$
$\begin{aligned} 2 π \end{aligned}$	0

Similar to sine, you can use your knowledge of the angles $\begin{aligned} 0, \frac{π}{2}, π, \frac{3 π}{2}, 2 π \end{aligned}$ on the unit circle to get a complete cycle of the cosine graph. While the sine function uses the $\begin{aligned} y - \end{aligned}$ coordinates, the cosine function is the $\begin{aligned} x - \end{aligned}$ coordinates of the unit circle and measures width. By referring to a unit circle or your memory, you can fill out a much shorter table than before.
::类似正弦, 您可以使用您对单位圆上角度 0, 2, 3, 2, 2 的知识来获取正弦图的完整周期。当正弦函数使用 Y 坐标时, 余弦函数是单位圆的 x 坐标和测量宽度。通过引用单位圆或内存, 您可以填写比以前短得多的表格。

Angle (radians)	Width (units)
0	1
$\begin{aligned} \frac{π}{2} \end{aligned}$	0
$\begin{aligned} π \end{aligned}$	-1
$\begin{aligned} \frac{3 π}{2} \end{aligned}$	0
$\begin{aligned} 2 π \end{aligned}$	1

First plot these five points and then connect them with a smooth curve. This will produce the cosine graph.
::首先绘制这五个点, 然后用一个平滑的曲线连接它们。这将生成余弦图。

Determining these five main points is the key to graphing sine or cosine graphs even when the graph is shifted or stretched.
::确定这五个主要点是绘制正弦或余弦图形的关键,即使图形被移动或拉伸。

Examples
::实例

Example 1
::例1

Earlier, you were asked what happens when the circle is not centered at the origin and does not have a radius of 1. The unit circle produces the parent function sine and cosine graphs. When the unit circle is shifted up or down, made wider or narrower, or spun faster or slower in either direction, then the graphs of the sine and cosine functions will be transformed using basic function transformation rules.
::早些时候,有人询问当圆不以原点为中心且没有半径为 1 时会发生什么情况。单位圆产生父函数正弦和正弦图形。当单位圆向上或向下移动,变宽或缩小,或向任一方向快速或放慢旋转时,正弦和正弦函数的图形将使用基本函数转换规则转换。

Example 2
::例2

What happens on either side of the sine and cosine graphs? Can you explain why?
::弦和弦图的两侧发生了什么?你能解释一下为什么吗?

The graphs of the sine (blue) and cosine (red) functions repeat forever in both directions.
::正弦(蓝色)和正弦( 红) 的图形函数在双向中永远重复。

If you think about the example with the Ferris wheel, the ride will keep on spinning and has been spinning forever. This is why the same cycle of the graph repeats over and over.
::如果您想一想Ferris轮子的例子, 骑车将保持旋转, 并且一直旋转。这就是为什么相同的图形循环会反复重复。

Example 3
::例3

How are the sine and cosine graphs the same and how are they different?
::弦形和弦形图形如何相同,它们有何不同?

The sine graph is the same as the cosine graph offset by $\begin{aligned} \frac{π}{2} \end{aligned}$ . Besides the shift, both curves are identical due to the perfect symmetry of circles.
::正弦图与正弦图相同,共弦图由 2 抵消,除此变化外,由于圆的完美对称,两个曲线是相同的。

Example 4
::例4

Where are two maximums and two minimums of the sine graph?
::正弦图的两个最大值和两个最低值在哪里?

One maximum of the sine graph occurs at $\begin{aligned} (\frac{π}{2}, 1) \end{aligned}$ . One minimum occurs at $\begin{aligned} (\frac{3 π}{2}, - 1) \end{aligned}$ . This is one cycle of the sine graph. Since it completes a cycle every $\begin{aligned} 2 π \end{aligned}$ , when you add $\begin{aligned} 2 π \end{aligned}$ to an $\begin{aligned} x \end{aligned}$ -coordinate you will be on the same point of the cycle giving you another maximum or minimum.
::正弦图的最大值在(%2, 1) 时发生, 最低值在( 3°2, - 1) 时发生。这是正弦图的一个周期。因为正弦图每 2 + 等于等于等于等于等于等于等于等于 2 等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于

$\begin{aligned} (\frac{5 π}{2}, 1) \end{aligned}$ is another maximum. $\begin{aligned} (\frac{7 π}{2}, - 1) \end{aligned}$ is another minimum.
:52,1)是另一个最高值。 (72,1)是另一个最低值。

Example 5
::例5

In the interval $\begin{aligned} [- 2 π, 4 π) \end{aligned}$ where does cosine have zeroes?
::间隔 [-2,4] 余弦在哪里有零?

Observe where the cosine curve has $\begin{aligned} x \end{aligned}$ -coordinates equal to zero. Note that $\begin{aligned} 4 π \end{aligned}$ is excluded from the interval. The values are $\begin{aligned} - \frac{3 π}{2}, - \frac{π}{2}, \frac{π}{2}, \frac{3 π}{2}, \frac{5 π}{2}, \frac{7 π}{2} \end{aligned}$ .
::观察余弦曲线的 X 坐标等于 0 的地方。请注意, 4 被从间隔中排除。值为 - 32, 22, 22, 32, 52, 72 。

Summary

In the unit circle, the cosine function corresponds to the x-coordinates, and the sine function to the y-coordinates.
::在单位圆中,余弦函数对应于x坐标,正弦函数对应于y坐标。

Sinusoidal functions refer to sine or cosine waves, which are periodic and repeat after a given period of time.
::Sinusoid 函数指正弦波或余弦波,它们周期性波或余弦波在特定时间之后重复波。

When graphing the sine or cosine functions, remember the five main points at angles $\begin{aligned} 0, \frac{π}{2}, π, \frac{3 π}{2}, 2 π . \end{aligned}$
::当绘制正弦或余弦函数时,请记住角度 0,%2,%,%2,%2,%2,%2的五个主要点。

Review
::回顾

1. Sketch $\begin{aligned} p (x) = \sin x \end{aligned}$ from memory.
::1. 内存的 Slich p(x) =sinx 。

2. Sketch $\begin{aligned} j (x) = \cos x \end{aligned}$ from memory.
::2. 内存的 Sletch j(x) =cosx 。

3. Where do the maximums of the cosine graph occur?
::3. 余弦图的最大值在哪里出现?

4. Where do the minimums of the cosine graph occur?
::4. 余弦图的最小值在哪里出现?

5. Find all the zeroes of the sine function on the interval $\begin{aligned} [- π, \frac{5 π}{2}] \end{aligned}$ .
::5. 在间隔处查找正弦函数的所有零[,52]。

6. Find all the zeroes of the cosine function on the interval $\begin{aligned} (- \frac{π}{2}, \frac{7 π}{2}] \end{aligned}$ .
::6. 在间隔处查找余弦函数的所有零(2,7+2) 。

7. Preview: Using your knowledge of function transformations and the cosine graph, predict what the graph of $\begin{aligned} f (x) = 2 \cos x \end{aligned}$ will look like.
::7. 预览:利用您对函数转换和余弦图的了解,预测 f( x) =2cosx 的图形看起来像什么。

8. Preview: Using your knowledge of function transformations and the cosine graph, predict what the graph $\begin{aligned} g (x) = \cos x + 2 \end{aligned}$ of will look like.
::8. 预览:利用您对函数转换和余弦图的了解,预测图形g(x)=cosx+2的形状。

9. Preview: Using your knowledge of function transformations and the cosine graph, predict what the graph of $\begin{aligned} h (x) = \cos (x - π) \end{aligned}$ will look like.
::9. 预览:利用您对函数转换和余弦图的了解,预测h(x)=cos(x))的图形将看起来像什么。

10. Preview: Using your knowledge of function transformations and the cosine graph, predict what the graph of $\begin{aligned} k (x) = - \cos x \end{aligned}$ will look like.
::10. 预览:利用您对函数转换和余弦图的了解,预测 k(x) cosx 的图形看起来像什么。

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

Sinusoidal功能家庭

Graphs of Sinusoidal Functions ::子函数图

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)