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Your mission, should you choose to accept it, as Agent Trigonometry is to the find the domain and the range of the function $\begin{aligned} y = \frac{1}{2} \sin (x + 2) - 3 \end{aligned}$ .
::您的任务, 如果您选择接受它, 因为Trigono量度代理是找到域和函数 y= 12sin( x+2) - 3 的范围。

Graphing Functions
::图图函数

Here you'll change the , the horizontal shifts, vertical shifts , and reflections.
::在这里,你将改变, 横向的转变,垂直的转变, 和反射。

Let's graph $\begin{aligned} y = 4 \sin (x - \frac{π}{4}) \end{aligned}$ and find the domain and range.
::让我们绘制 y= 4sin(x4) 并找到域和范围。

First, stretch the curve so that the amplitude is 4, making the maximums and minimums 4 and -4. Then, shift the curve $\begin{aligned} \frac{π}{4} \end{aligned}$ units to the right.
::首先,拉伸曲线,使振幅为 4 , 使最大和最小值为 4 和 - 4 。然后, 将曲线 + 4 单位向右移动。

As for the domain, it is all real numbers because the sine curve is periodic and infinite. The range will be from the maximum to the minimum; $\begin{aligned} y \in [- 4, 4] \end{aligned}$ .
::至于域,它都是真实的数字, 因为正弦曲线是周期性的, 无限的。范围从最大到最小; y y [- 4, 4] 。

Now, let's graph $\begin{aligned} y = - 2 \cos (x - 1) + 1 \end{aligned}$ and find the domain and range.
::现在,让我们来绘制 y2cos (x- 1) +1, 并找到域名和范围。

The -2 indicates the cosine curve is flipped and stretched so that the amplitude is 2. Then, move the curve up one unit and to the right one unit.
::-2表示余弦曲线被翻转和拉伸,使振幅为2,然后将曲线向上移动一个单位,向右移动一个单位。

The domain is all real numbers and the range is $\begin{aligned} y \in [- 1, 3] \end{aligned}$ .
::域名是所有真实数字,范围是y[-1,3]。

Finally, let's find the equation of the sine curve below.
::最后,让我们在下面找到正弦曲线的方程。

First, let’s find the amplitude. The range is from 1 to -5, which is a total distance of 6. Divided by 2, we find that the amplitude is 3. Halfway between 1 and -5 is $\begin{aligned} \frac{1 + (- 5)}{5} = - 2 \end{aligned}$ , so that is our vertical shift. Lastly, we need to find the horizontal shift. The easiest way to do this is to superimpose the curve $\begin{aligned} y = 3 \sin (x) - 2 \end{aligned}$ over this curve and determine the movement from one maximum to the closest maximum of this curve.
::首先,让我们找到振幅。范围从1到5, 总距离是6, 除以2, 我们发现振幅是3, 1到5之间的半径是1+( - 5-5)52, 这就是我们垂直的转变。最后, 我们需要找到横向的转变。最简单的办法是将曲线 y=3sin( x)-2 叠加到这个曲线上, 并确定从这个曲线的最大一个向最接近的最大移动。

Subtracting $\begin{aligned} \frac{π}{2} \end{aligned}$ and $\begin{aligned} \frac{π}{6} \end{aligned}$ , we have:
::以2和6为减号,我们已:

$\begin{aligned} \frac{π}{2} - \frac{π}{6} = \frac{3 π}{6} - \frac{π}{6} = \frac{2 π}{6} = \frac{π}{3} \end{aligned}$

Making the equation $\begin{aligned} y = 3 \sin (x + \frac{π}{3}) - 2 \end{aligned}$ .
::使公式y=3sin(x3)-2。

Examples
::实例

Example 1
::例1

Earlier, you were asked to find the domain and range of the function $\begin{aligned} y = \frac{1}{2} \sin (x + 2) - 3 \end{aligned}$ .
::早些时候, 您被要求查找 y= 12sin {( x+2) - 3 函数的域名和范围。

The $\begin{aligned} \frac{1}{2} \end{aligned}$ indicates the sine curve is smooshed so that the amplitude is $\begin{aligned} \frac{1}{2} \end{aligned}$ . Then, move the curve down two units and to the left three units.
::12 表示正弦曲线被打乱, 使振幅为12。然后, 将曲线向下移动两个单位, 向左移动三个单位。

The domain is all real numbers and the range is $\begin{aligned} y \in [- \frac{5}{2}, - \frac{3}{2}] \end{aligned}$ .
::域名是所有实际数字,范围是y[-52,-32]。

For Examples 2 & 3, graph the functions. State the domain and range. Show two full periods.
::对于例2和例3,请绘制函数图。说明域和范围。显示两个完整的周期。

Example 2
::例2

$\begin{aligned} y = - 2 \sin (x - \frac{π}{2}) \end{aligned}$
::y2sin(x%2)

The domain is all real numbers and the range is $\begin{aligned} y \in [- 2, 2] \end{aligned}$ .
::域名是所有真实数字,范围是y[-2,2]。

Example 3
::例3

$\begin{aligned} y = \frac{1}{3} \cos (x + 1) - 2 \end{aligned}$
::y=13cos( x+1)-2

The domain is all real numbers and the range is $\begin{aligned} y \in [- 2 \frac{1}{3}, - 1 \frac{2}{3}] \end{aligned}$ .
::域名是所有实际数字,范围是y[-213,-123]。

Example 4
::例4

Write one sine equation and one cosine equation for the curve below.
::为下面的曲线写一个正弦方程和一个正弦方程。

The amplitude and vertical shift is the same, whether the equation is a sine or cosine curve. The vertical shift is -2 because that is the number that is halfway between the maximum and minimum. The difference between the maximum and minimum is 1, so the amplitude is half of that, or $\begin{aligned} \frac{1}{2} \end{aligned}$ . As a sine curve, the function is $\begin{aligned} y = - 2 + \frac{1}{2} \sin x \end{aligned}$ . As a cosine curve, there will be a shift of $\begin{aligned} \frac{π}{2}, y = \frac{1}{2} \cos (x - \frac{π}{2}) - 2 \end{aligned}$ .
::振幅和垂直变化是相同的, 不论方程式是正弦曲线还是正弦曲线。垂直变化是 - 2, 因为数字是最大和最小值之间的一半。最大和最小值之间的差值是 1, 因此振幅是该值的一半, 或12。作为正弦曲线, 函数是 y2+12sinx。作为正弦曲线, 将会有 2, y=12cos( x2)-2 的移动。

Review
::回顾

Determine if the following statements are true or false.
::确定以下声明是真实的还是虚假的。

To change a cosine curve into a sine curve, shift the curve $\begin{aligned} \frac{π}{2} \end{aligned}$ units.
::将余弦曲线改变为正弦曲线,将曲线 2 单位移动。

For any given sine or cosine graph, there are infinitely many possible equations that can be written to represent the curve.
::对于任何给定的正弦或正弦图,可以写出无数可能的方程式来表示曲线。

The amplitude is the same as the maximum value of the sine or cosine curve.
::振幅与正弦或余弦曲线的最大值相同。

The horizontal shift is always in terms of $\begin{aligned} π \end{aligned}$ .
::横向变化总是以为单位。

The domain of any sine or cosine function is always all real numbers.
::任何正弦或余弦函数的域始终是所有实际数字。

Graph the following sine or cosine functions such that $\begin{aligned} x \in [- 2 π, 2 π] \end{aligned}$ . State the domain and range.
::下图的正弦函数或余弦函数为 x[-222]。请标明域和范围。

$\begin{aligned} y = \sin (x + \frac{π}{4}) + 1 \end{aligned}$
::y=sin(x4)+1

$\begin{aligned} y = 2 - 3 \cos x \end{aligned}$
::y=2 - 3cosx

$\begin{aligned} y = \frac{3}{4} \sin (x - \frac{2 π}{3}) \end{aligned}$
::y=34sin(x-23)

$\begin{aligned} y = - 5 \sin (x - 3) - 2 \end{aligned}$
::y5sin(x-3)-2

$\begin{aligned} y = 2 \cos (x + \frac{5 π}{6}) - 1.5 \end{aligned}$
::y=2cos(x+56)-1.5

$\begin{aligned} y = - 2.8 \cos (x - 8) + 4 \end{aligned}$
::y2.8cos(x-8)+4

Use the graph below to answer questions 12-15.
::用下图回答问题12-15。

Write a sine equation for the function where the amplitude is positive.
::为振幅为正数的函数写入正数等式。

Write a cosine equation for the function where the amplitude is positive.
::为振幅为正数的函数写入余弦方程。

How often does a sine or cosine curve repeat itself? How can you use this to help you write different equations for the same graph?
::正弦或余弦曲线的重复频率是多少? 您如何使用它来帮助您为同一图表写出不同的方程式 ?

Write a second sine and cosine equation with different horizontal shifts.
::以不同的水平移动写入二次正弦和连弦方程。

Use the graph below to answer questions 16-20.
::用下图回答问题16-20。

Write a sine equation for the function where the amplitude is positive.
::为振幅为正数的函数写入正数等式。

Write a cosine equation for the function where the amplitude is positive.
::为振幅为正数的函数写入余弦方程。

Write a sine equation for the function where the amplitude is negative .
::在振幅为负值的函数中写入正弦方程。

Write a cosine equation for the function where the amplitude is negative .
::为振幅为负的函数写入余弦方程。

Describe the similarities and differences between the four equations from questions 16-19.
::说明问题16-19的四个方程之间的相似和不同之处。

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。

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