Section: 函数和非功能图 | 3.2 职能和非职能图-interactive | The Way To Learn

The Way To Learn

Section outline

The Purpose of this Lesson
::本课程的目的

In this lesson, you will further explore functions and non-functions by examining their equations and graphs in greater detail. You'll also consider the constraints on domain and range that are imposed by the equations and graphs of relations.
::在此教训中, 您将会通过更详细地检查它们的方程式和图表来进一步探索函数和非函数。您还会考虑关系方程式和图表对域和范围的限制。

Introduction : Horizontal and Vertical Lines
::导言:水平线和垂直线

Horizontal and vertical lines provide a useful window into the nature of relations as well as domain and range.
::水平线和垂直线为关系的性质以及域和范围提供了一个有用的窗口。

Activity 1: Creating Linear Models

::活动1:建立线性模型

Work it Out
::工作出来

Dana and Travis are each driving to Salt Lake City, and they are leaving at the same time. Travis lives along the same highway Dana is taking to SLC, but he lives 55 miles closer to SLC than Dana does. Dana starts driving at 70 miles per hour, while Travis drives at 65 miles per hour. Create functions that model the distance each has traveled over time. Graph each. Write and solve an equation to determine when Dana passes Travis. Confirm your result with the graph.
::Dana 和 Travis 都开往盐湖城,他们同时离开。Travis 住在同一条高速公路上 Dana 正在前往 SLC, 但是他住在离SLC更近55英里的地方。 Dana 开始以每小时70英里的速度驾驶, 而Travis 的驾驶则以每小时65英里的速度驾驶。创建每条路程的模型功能。分别绘制图。撰写和解析一个方程式以确定Dana 何时通过Travis 。用图表确认您的结果。

The next week, Dana and Travis are again both planning to drive to SLC, leaving at the same time. But at the moment Dana leaves, Travis' car won't start. Create functions that model the distance each has traveled over time. Graph each. Write and solve an equation to determine when Dana passes Travis. Confirm your result with the graph. What is the range of Travis's function ? Why? Write the range using set notation .
::下周, Dana 和 Travis 将再次计划开车前往 SLC, 并同时离开。但是当Dana 离开时, Travis 的车将无法启动。创建函数来模拟每个时间段的距离。分别绘制图表。写入和解析一个方程以确定 Dana 何时通过Travis 。用图表来确认您的结果。 Travis 的功能范围是什么 ? 为什么? 使用设置的标记来写入范围。

Progress

0 / 3

1.

Recall Dana and Travis's trip to Salt Lake City from the Work it Out above. What is the equation for Dana's distance traveled as a function of time?
::回顾Dana和Travis从上面工作到盐湖城的旅行。Dana距离的方程式是多少?

a
@$\begin{align*}d=70t\end{align*}@$
::@ $\ begin{ ALIGN} d=70t\ end{ legign} $ @ $\ begin{ 等同= d=70t\ end{ 等同$

b
@$\begin{align*}d=55t+70\end{align*}@$
::@ $\ begin{ ALIGN} d=55t+70\ end{ aliign} $

c
@$\begin{align*}d=70t+55\end{align*}@$
::@ $\ begin{ ALIGN} d=70t+55\ end{ aliign} $

Example 1-1
::例1-1

The following are equations for several . Graph each and d iscuss their differences and similarities. One of the equations features an expression that can be further simplified. Simplify it. For each of the functions , are there any $\begin{align*}y\end{align*}$ -values the function does not return? How can you describe the range for each of these functions? What is the range for the last function in the list? Explain.

$\begin{aligned} \begin{array}{l} f (x) = 5 x + 7 \\ g (x) = x + 7 \\ p (x) = \frac{1}{4} x + 7 \\ h (x) = 0 x + 7 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{l} f(x)=5x+7\\ g(x)=x+7\\ p(x)=\frac14 x +7\\ h(x)=0x+7\\ \end{array} \end{align*}$
::下面是多个函数的方程。每个图形, 并讨论其差异和相似性。其中的方程之一含有一个可以进一步简化的表达式。简化它。对于每个函数, 是否有函数不返回的 Y 值? 您如何描述这些函数中的每个函数的范围? 列表中最后一个函数的值范围是多少? 解释。 f( x)=5x+7g( x)=x+7p( x)=14x+7h( x)=0x+7

Solution: These functions all have the same $\begin{align*}y\end{align*}$ - intercept . The slopes of the first three are positive. The last function has a of 0. This equation can be simplified to $\begin{align*}y=7.\end{align*}$ The range for the first 3 functions is "all real numbers." This means that every possible real number is produced as a $\begin{align*}y\end{align*}$ -value for each of the first 3 functions. This is evident from the graphs; the constant rate of change gives confidence in a statement of the range. Observe that the range for the last function is one value, 7. All $\begin{align*}x\end{align*}$ -values return the same $\begin{align*}y\end{align*}$ -value.
::解析 : 这些函数都具有相同的 y 界面。前三个函数的斜度是正的。最后一个函数为 0 。这个方程式可以简化为 y= 7 。前三个函数的宽度是 “ 所有真实数字 ” 。这意味着每个可能的真数都是前三个函数的y值。从图表中可以看出, 恒定的变动率能给您对区域语句的信心。观察最后一个函数的宽度是 1 值 , 7. 所有 x 值返回相同的 y 值。

Four linear functions with the same y-intercept

Activity 2: Constant Functions
::活动2:常数函数

Interactive
::交互式互动

Use the interactive below to change the parameters for a linear function and explore the resulting domain and range. Be sure and zoom out to explore beyond the confines of your graphing window. Is the domain always "all real numbers"? Is the range always "all real numbers"? Explain the conditions under which the range is restricted to a single number.
::使用下面的交互效果来改变线性函数的参数, 并探索生成的域和范围。请确定并缩放以在您的图形窗口范围以外进行探索。域是否总是“ 所有真实数字 ” ? 范围是否总是“ 所有真实数字 ” ? 解释范围限于一个数字的条件。

Constant functions of the form $\begin{align*}y=b\end{align*}$ have a slope of 0, and a range of $\begin{align*}\left\{b\right\}\end{align*}$ . All other linear functions have a domain of "all real numbers" and a range of "all real numbers."
::y=b 窗体的常数函数的斜度为 0, 范围为 {b} 。所有其他线性函数都有一个“所有实际数字”和“所有实际数字”的域。

Constant Functions
::常常常函数函数

A constant function is a linear function with a slope of 0.
The graph of a constant function is a horizontal line.
A horizontal line has a domain of "all real numbers", and its range is its $\begin{align*}y\end{align*}$ -intercept.
::恒定函数是斜度为 0 的线性函数。恒定函数的图形是水平线。水平线有“所有实际数字”的域,其范围是 Y 界面。

Work it Out
::工作出来

Create a constant function and graph it. State the domain and range, and explain your answers.
::创建一个恒定函数并将其图形化。指定域和范围, 并解释您的答复。

Is a constant function actually a function, consistent with the definition of a function? Explain. Draw a function map to support your argument.
::解释。绘制一个函数映射以支持您的参数。

Graph a vertical line that's not the $\begin{align*}y\end{align*}$ -axis. What is the $\begin{align*}y\end{align*}$ -intercept, if any? Explain. Can you draw a slope triangle to visualize the slope? Why or why not? What is the slope of this line, if any? Explain. Can you think of a scenario that this equation models? What is the domain and range for this relation ? Is this relation actually a function? Why or why not? Draw a map to support your argument.
::绘制不是 Y 轴的垂直线。 y 界面是什么? 解释一下。您可以绘制一个斜度三角形来直观斜度吗? 为什么或为什么? 该斜度的斜度是什么? 解释一下。你能想象出这个方程模型的情景吗 ? 此关联的域域和范围是什么? 此关系实际上是一个函数吗? 为什么或为什么不是? 绘制一张地图来支持您的论点。

Imagine a wizard. The wizard accepts the number of blueberries you give her and transforms them according to the function $\begin{align*}w(x)=5x+8.\end{align*}$ Is the magic the wizard transforms a function? Why or why not? Draw a function map to support your argument. Is it possible for the wizard to return two different values of blueberries for one value of $\begin{align*}x?\end{align*}$ Why or why not?
::想象一个向导。向导接受您给她的蓝莓数量, 并根据函数 w( x)=5x+8 进行转换。向导变了一个函数吗? 为什么不? 绘制一个函数映射来支持您的论点。向导是否有可能返回两个不同的蓝莓值, 一个值是 x? 为什么或为什么不是 ?

You are driving a car at 50 miles per hour for two hours. Then you stop for an hour. Then you continue on at 60 miles per hour. Create a graph of a piecewise function that models your distance as a function of time. Create a function map with some values showing that this is actually a function. Is it possible for your function to return two different values for a single $\begin{align*}x\end{align*}$ -value? Why or why not?
::您正在驾驶一辆每小时50英里的汽车,时间为两个小时。然后停了一个小时。然后继续以每小时60英里的速度运行。然后,您会继续以每小时60英里的速度运行。创建一个以时间函数来模拟距离的片段函数图。创建一个功能映射图,其中含有一些值,显示这是一个函数。您是否可能返回一个 x 值的两个不同的值?为什么或为什么没有?

You encounter a magical phone booth. When you step inside and close the door, suddenly you are nowhere! You sit in the phone booth in nowhere for several hours, and nothing happens. Suddenly, at the 4-hour mark, the phone booth transports you 5 miles away, and, at the same time, 7 miles away. That is to say, at the 4-hour mark ( $\begin{align*}x=4\end{align*}$ ), you are in two places at once! I t only lasts for an instant, and then you are back to boring nowhere again. Create a graph of this bizarre scenario. Is this a function? Why or why not? Create a map to support your argument.
::遇到一个神奇的电话亭。当你走出门关上门时, 你突然无处可去。你无处可去地坐在电话亭里数小时, 没有发生任何事。突然, 在4小时关头, 电话亭将你传送到5英里之外, 同时, 7英里以外。这就是说, 在4小时关口( x=4) , 你同时在两个地方! 它只停留一瞬间, 然后你又回到一个无趣的无处。创建这个怪异的场景的图。这是功能吗? 为什么或为什么不? 绘制一张地图来支持你的论点。

Write the equation for a vertical line that's not the $\begin{align*}y\end{align*}$ -axis. What is its $\begin{align*}x\end{align*}$ -intercept? Why? Write the equation for the $\begin{align*}y\end{align*}$ -axis. What is its $\begin{align*}x\end{align*}$ -intercept? Describe its $\begin{align*}y\end{align*}$ - intercepts .
::写入一个不是 Y 轴的垂直线的方程式。什么是它的 X 界面? 为什么? 为 y 轴写这个方程式。什么是它的 x 界面? 描述它的 y 界面。

Activity 3 : Vertical Lines
::活动3:垂直线

Example 3-1

::例3-1

Graph a vertical line that's not the $\begin{align*}y\end{align*}$ -axis. A vertical line violates the definition of a function in an infinite number of points. Explain. Use a function map to support your argument.
::图形为不是 Y 轴的垂直线。垂直线在无限的点数中违反了函数的定义。解释。使用函数映射支持您的参数。

Solution: The definition of a function is a relation between the domain and range such that for each value in the domain there is only one corresponding value in the range. For a vertical line, there is only one value in the domain. This value maps to an infinite number of values in the range. The "first" value that it maps to doesn't violate the definition of a function, but every value after that does!

::解析 : 函数的定义是域与范围之间的关系, 使域中每个值在范围中只有一个相应的值。对于垂直线, 域中只有一个值。此值映射为范围中数不尽的值。它绘制的“ 第一” 值并不违反函数的定义, 但随后的每一个值都违反函数的定义 !

Vertical Lines
::垂直直线

The equation for a vertical line has the form $\begin{align*}x=a\end{align*}$ , where $\begin{align*}a\end{align*}$ is the $\begin{align*}x\end{align*}$ -intercept.
Vertical lines have no slope.
A vertical line that's not the $\begin{align*}y\end{align*}$ -axis has no $\begin{align*}y\end{align*}$ -intercept.
A vertical line that is the $\begin{align*}y\end{align*}$ -axis has an infinite number of $\begin{align*}y\end{align*}$ -intercepts.
A vertical line is not a function; its domain is the $\begin{align*}x\end{align*}$ -intercept, and its range is "all real numbers."
::垂直线的方程式为 x=a, 其中一个是 X 界面。垂直线没有斜度。不是 y 轴的垂直线没有 Y 界面。 Y 轴的垂直线有无限数量的 y 界面。垂直线不是一个函数; 其域是 x 界面, 其范围是“ 所有实际数字 ” 。

Activity 4 : Relations That Are Not Functions
::活动4:非职能关系

Work it Out
::工作出来

Solve the equation $\begin{align*}y^2=36\end{align*}$ for $\begin{align*}y.\end{align*}$ How many solutions are there? Explain.
::解决y的 Y2=36等式。有多少解决方案? 请解释。

Complete the table and graph the following equation: $\begin{align*}y=\pm\sqrt{x}.\end{align*}$ Are there any values that are not in the domain? Explain. For an $\begin{align*}x\end{align*}$ -value of 0, how many $\begin{align*}y\end{align*}$ -values are returned? For every other $\begin{align*}x\end{align*}$ -value in the domain, how many $\begin{align*}y\end{align*}$ -values are returned? Explain. How are the observations you just made evident in the graph? Explain. Is this equation representative of a function? Why or why not? How many times does this relation violate the definition of a function? Explain.

$\begin{aligned} \begin{array}{cc} x & y \\ 0 \\ 1 \\ 1 \\ 2 \\ 2 \\ 3 \\ 3 \\ 4 \\ 4 \\ 5 \\ 5 \\ 9 \\ 9 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{c|c} x & y\\ \hline 0\\ 1\\ 1\\ 2\\ 2\\ 3\\ 3\\ 4\\ 4\\ 5\\ 5\\ 9\\ 9\\ \end{array}\end{align*}$
::完成表格并绘制以下方程式 : yx 。是否有不在域内的值 ? 解释。对于 0 的 x 值, 返回多少 Y 值? 对于域内的其他每个 x 值, 返回多少 Y 值? 解释。您刚才在图表中显示的观察如何? 解释。此方程式是否代表一个函数? 为什么或为什么不是? 这个关系多少次违反函数的定义? 解释。 xy0112233445599

Relations That Are Not Functions
::非职能关系

A function is a relation between domain and range such that each value in the domain corresponds to only one value in the range.
Relations that are not functions violate this definition. They feature at least one value in the domain that corresponds to two or more values in the range.
::函数是域与范围之间的一种关系,使域中的每个值仅对应区域中的一个值。非函数的关系违反此定义。它们至少含有一个值,与区域中两个或两个以上的值相对应。

Example 4-1
::例4-1

The Pythagorean Theorem presents another context for exploring relations that are not functions. The Pythagorean Triple, 3, 4, 5 represents a right triangle as shown below. Sketch your own version. Notice that the triangle has been placed so that the vertex of one acute angle is at the origin.
::Pytagorean Theorem 为探索非功能关系提供了另一个背景。 Pytagoren Triple, 3, 4, 5 代表右三角, 如下文所示。涂抹您自己的版本。请注意, 三角形已放置, 以使一个急性角的顶端位于源头。

What are the coordinates of point P?
::P点的坐标是多少?

If this triangle is reflected across the $\begin{align*}x\end{align*}$ -axis, what are the coordinates of P'? Is triangle OCP' also a right triangle? How do you know?
::如果这个三角形反射到 X 轴上, P 的坐标是什么? OCP 三角形是否也是右三角形? 你怎么知道 ?

If both triangles are then reflected across the $\begin{align*}y\end{align*}$ -axis, what are the resulting coordinates of the images of P and P'? Are these triangles also right? How do you know?
::如果两个三角在 Y 轴反射, P 和 P 图像的坐标是什么? 这些三角是否也正确? 你怎么知道?

What equation relates the $\begin{align*}x\end{align*}$ and $\begin{align*}y\end{align*}$ coordinates of these points? Explain.
::哪些方程与这些点的 x 和 Y 坐标相关? 请解释。

The Pythagorean Triple 3, 4, 5 represents a right triangle.

Solution: The coordinates of point P are $\begin{align*}(3,4).\end{align*}$ The graph below shows the results of the described transformations . The triangles are congruent because reflection is a rigid motion transformation. The image points are $\begin{align*}P'(3,-4), ~P''(-3,-4), ~\text{and}~ P'''(-3,4).\end{align*}$ The equation relating $\begin{align*}x\end{align*}$ and $\begin{align*}y\end{align*}$ is the Pythagorean Theorem: $\begin{align*}x^2+y^2=5^2.\end{align*}$ This equation holds true for the negative values of $\begin{align*}x\end{align*}$ and $\begin{align*}y\end{align*}$ because squaring them makes them positive.
::解析度: P 点的坐标 (3, 4) 。下图显示描述的变换结果。三角形是相同的, 因为反射是一个硬运动变换。图像点是 P {( 3, 4) 、 P {( 3, 4) 和 P {( 3, 4) 。有关 x 和 y 的方程式是 Pythagorean Theorem : x2+y2=52 。此方程式对 x 和 y 的负值来说是正确的, 因为它们是正数。

A 3, 4, 5 triangle and reflections across the x-axis, y-axis, and both axes.

Activity 5: More Relations That Are Not Functions
::活动5:更多非职能关系

Interactive
::交互式互动

The points P, P', P'', and P''' above form four right triangles with a hypotenuse of 5. Those specific points were chosen because they form triangles with sides that are a Pythagorean Triple: 3, 4, and 5.
::P、P'、P'、P'和P'以上各点组成四个右三角形,次数为5。这些特定点之所以被选中,是因为它们形成三角形,侧面是Pythagoren Triple:3、4和5。

Are there other points that would result in right triangles with a hypotenuse of 5 that are not ? Use the interactive below to explore the possibilities. If you graph all the points that make the equation $\begin{align*}x^2+y^2=5^2\end{align*}$ true, what shape results? Why?
::是否还有其它点可以导致右三角形, 下限为 5 不 ? 使用下面的交互来探索可能性。如果您用图表显示使方程式 x2+y2=52 真实的所有点, 那么结果是什么 ? 为什么 ?

There are many triangles with a hypotenuse of 5 and at least one irrational leg . The shape of the triangle changes, but the hypotenuse remains 5. This means every point P created is 5 units from the origin. The set of points 5 units from the origin is a circle with radius 5. (In general, a circle is the set of points equidistant from a given point called the center.) The equation $\begin{align*}x^2+y^2=5^2\end{align*}$ describes the relationship between the coordinates of all the points on the circle. Every point on the circle created by the interactive makes the equation $\begin{align*}x^2+y^2=5^2\end{align*}$ true.
::有许多三角形, 其下限为 5, 至少不合理腿。三角形的形状是 5 。但下限的形状是 5 。这意味着每个点 P 是从原点创建的 5 个单位。 5 个点的集合点是半径 5 的圆形。 (一般而言, 一个圆形是从一个指定点( 中点) 的等距点组。) 方程式 x2+y2=52 描述圆上所有点的坐标之间的关系。互动所创建的圆形的每个点都使方程式 x2+y2=52 成为真实。

The Equation for A Circle
::A圆的方程式

$\begin{align*}x^2+y^2=r^2\end{align*}$ , where $\begin{align*}(x,y)\end{align*}$ is a point on the circle and $\begin{align*}r\end{align*}$ is the radius.

Work it Out
::工作出来

Write the equation for all the points on a circle with radius 13. Confirm that the points $\begin{align*}(5,12), (5,-12), (-5,-12), \text{and } (-5,12)\end{align*}$ all satisfy the equation. (To satisfy an equation means to make it true.) Sketch the graph of a circle with radius 13. Are the equation and graph representative of a function? Why or why not?
::以半径13为圆形写出所有点的方程。确认点(5,12,5,-12,5,5,5,5,12)和(5,5,12)都符合方程。 (要满足方程手段使其成为事实。 ) 将圆形图与半径13相拼贴。方程和图代表函数吗?为什么或为什么不是?

Determine if each of the following equations represents a function or not. Explain your decision. Describe the type of relation or function, and its graph.
$\begin{align*}\begin{array}{ll} \text{a.} & y=\pm\sqrt{x}\\ \text{b.} & y=\pm5\sqrt{x}\\ \text{c.} & x^2+y^2=36\\ \text{d.} & y=\sqrt{x}\\ \text{e.} & y=6\\ \text{f.} & x=7\\ \text{g.} & y=x^2\\ \text{h.} & y=5x^2\\ \text{i.} & y=100x\\ \text{j.} & y=x-12\\ \end{array}\end{align*}$
::确定以下方程式是否代表函数。请解释您的决定。描述关系或函数的类型。描述其图形。 a. y\\ xb. y\\\\ 5xxc. x2+y2= 36d. y= 36d. y=xe. y= 6f. x= 7g. y= 7g. y=x2h. y= 5x2i. y= 100xj. y=x- 12

(extension) Describe the domain and range of each of the functions above.
:扩展) 描述上述每一项功能的域和范围。

Determine if each of the following graphs represent a function or not. Explain your decision. If possible, describe the type of relation or function.
::确定下图是否代表函数。请解释您的决定。如果可能, 请描述关系或函数的类型。

any quadratic function
::任何二次函数

any linear function that is not horizontal
::非水平的任何线性函数

any circle
::任意圆

any ellipse
::任何椭圆

any square root function
::任何平方根函数

any horizontal line
::任何水平线

any vertical line
::任何垂直线

any set of 10 points
::任何一套10点的点数

any set of 10 points such that $\begin{align*}x\end{align*}$ -values aren't repeated
::任何一组 10 点, 使 x 值不重复

any polygon
::任何多边形

a sine graph
::一个正弦图形

an exponential function
::一个指数函数

a rational function with an identified vertical asymptote at $\begin{align*}x=3\end{align*}$
::a 合理函数,在 x=3 时有被识别的垂直等同点。

From the graph, how can you distinguish a function from a non-function? This technique is sometimes called the vertical line test because you can draw a vertical line to expose a violation of the definition of a function. For each of the non-functions above, sketch them on your own paper and draw a vertical line showing the $\begin{align*}x\end{align*}$ -value for which the relation failed the vertical line test.
::从图形中,您如何区分函数和非功能的函数?这种技术有时被称为垂直线测试,因为您可以绘制垂直线以暴露对函数定义的违反。对于以上每个非功能,您可以在自己的纸上绘制这些功能的图解,并绘制一条垂直线,显示关系未通过垂直线测试的 x 值。

Sketch the graphs of 5 functions. Use the last exercise as an example. You can make these free-hand, you do not need to create the graphs from equations. Then sketch the graph of 5 non-functions. Draw a vertical line showing where (that means at which $\begin{align*}x\end{align*}$ -value) the relation fails the vertical line test.
::拖曳 5 函数的图形。使用最后一个练习作为示例。您可以将这些自由手画出来, 您不需要用方程式创建图形。然后绘制5个非函数的图形。绘制一条垂直线, 显示关系在什么位置( 意味着 x 值) 的垂直线测试失败。

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Summary
::摘要

Horizontal lines are functions that have a range that is a single value.
::水平线是具有单值范围的函数。

Vertical lines are not functions.
::垂直线条不是函数。

The equations $\begin{align*}y=\pm\sqrt{x}\end{align*}$ and $\begin{align*}x^2+y^2=9\end{align*}$ are examples of non-functions because there is at least one $\begin{align*}x\end{align*}$ -value with two or more $\begin{align*}y\end{align*}$ -values.
::yx 和 x2+y2=9等式是非函数的例子,因为至少有一个x值,有两个或两个以上y值。

T he vertical line test is a great way to visualize a violation of the definition of a function.
::垂直线测试是将违反函数定义的行为直观化的一个很好的方法。

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