Course: 8.13 多元函数的拼图图

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多边函数的切入图

A company needs to double the volume of its shipping box. If the box is 1.5 feet, by 1 foot, by 8 inches, and we want to expand the box in the same amount in each direction, how much would we need to add to each dimension to double the volume of the box? (Not for the purposes of shipping the cat seen below!)
::公司需要将其运输箱的容量翻倍。如果箱子是1.5英尺,1英尺,8英寸,我们想要将每个方向的盒子扩大到相同数量,那么每个尺寸需要增加多少,使盒子的容量翻倍? (不是为了运输下面的猫! )

A graph may help us consider this situation, and with a few tools we can draw a sketch of a polynomial graph. We discuss these tools in this section.
::图表也许有助于我们考虑这种情况,用一些工具,我们可以绘制一个多元图的草图。我们在本节中讨论这些工具。

lesson content

Smooth and Continuous
::平滑和连续

One feature to keep in mind when sketching polynomial graphs is that they are smooth and continuous . Smooth means there are no corners or cusps. Continuity means, loosely, that you can draw the entire graph without picking up your pencil. Graphs that are continuous do not have any breaks, like holes or gaps. You can see examples of these in the image below.
::绘制多面图时要记住的一个特征是它们平滑和连续。平滑意味着没有角或角。连续意味着您可以在不捡起铅笔的情况下绘制整张图。连续的图表没有断裂, 比如孔或空隙。您可以在下面的图像中看到这些例子。

Power Functions
::权力功能

To understand the property of end behavior below, it is helpful to consider how one- term polynomial functions behave. We call these functions power functions .
::为了了解以下最终行为的属性,有必要考虑一个时期的多功能功能如何运作。我们称之为这些函数功率功能。

Power Functions
::权力功能

A power function is a function of the form

\begin{aligned} f (x) = a x^{n} \end{aligned}

If the degree of the power function is even, then it will essentially have a U-shape, like $\begin{aligned} y = x^{2} . \end{aligned}$ Notice that as the degree increases, the U-shape becomes more steep for $\begin{aligned} | x | > 1, \end{aligned}$ and flatter for $\begin{aligned} 0 < | x | < 1 \end{aligned}$ . We only call this shape a parabola when it is $\begin{aligned} y = x^{2} . \end{aligned}$
::如果功率函数的幅度是平的,那么它基本上将有一个U形状,如y=x2。请注意,随着度的增加,U形状在 x1 和 x1 之间会变得更陡峭。我们只称它为 y=x2 时的抛物线形状。

In general, power functions of odd degree have the shape below. The same properties hold as the degree increases —the graph becomes more steep for $\begin{aligned} | x | > 1, \end{aligned}$ and flatter for $\begin{aligned} 0 < | x | < 1 \end{aligned}$ .
::一般来说, 奇数的功率函数在下面有形状。度增长的属性相同—— 图表在 x1 中变得更陡, 平滑 0x1 。

End Behavior
::结束行为行为

The end behavior of a graph is how the function behaves when x has a large absolute value— that is, x -values that are not close to 0 on the number line.
::图形的结尾行为是当 x 具有巨大的绝对值时函数的行为方式, 即数字线上不接近 0 的 X 值。

End Behavior
::结束行为行为

For large

\begin{aligned} | x | \end{aligned}

, the polynomial function

\begin{aligned} f (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + . . . + a_{1} x + a_{0} \end{aligned}

behaves like the power function

\begin{aligned} f (x) = a_{n} x^{n} \end{aligned}

This creates four possibilities, depending on the leading coefficient .
::这创造了四种可能性,取决于主要系数。

This video by Mathispower4u describes the end behavior of a polynomial function.
::Mathispower4u的这段影片描述多元函数的结束行为。

Zeros and Their Multiplicity
::零及其多样性

The other property we can use to sketch polynomial function graphs is the zeros—that is, the x- intercepts and their multiplicities. We have described many ways in this chapter to find the zeros of a polynomial function. The multiplicity of a zero is the number of times the factor occurs to create that zero.
::用于绘制多边函数图形的其他属性是零-即 X 界面及其多重性。我们在本章中描述了许多方法来找到多边函数的零。0的多重性是生成零的系数发生次数。

Example 1
::例1

Describe the zeros of $\begin{aligned} f (x) = x (x + 7) (x - 1)^{2} (3 x - 4)^{5} \end{aligned}$ and their multiplicities.
::描述 f( x) =x( x+7) (x- 1) 2 (3x-4) 5 及其倍数的零。

Solution: To find the zeros, we can set each factor equal to 0 and solve.
::解决方案: 要找到零, 我们可以设定每个系数等于 0 并解答。

\begin{aligned} f (x) = x (x + 7) (x - 1)^{2} (3 x - 4)^{5} \\ x = 0 x + 7 = 0 x - 1 = 0 3 x - 4 = 0 \\ x = 0 x = - 7 x = 1 x = \frac{4}{3} \end{aligned}

::f( x) =xx( x+7)( x- 1) 2( 3x- 4) 5x=0x+7=0x=0x1=03x=4=0x=0x=0x=7x=1x=43

The factor that led to the zero at 0 occurs one time, so zero at 0 has multiplicity 1.
::导致0时为0的系数发生一次,因此0时为0时为多重系数1。

The factor that led to the zero at -7 occurs one time, so the zero at -7 has multiplicity 1.
::导致在 - 7 时为零的系数发生一次,因此在 - 7 时为零的系数有多重 1。

The factor that led to the zero at 1 occurs twice, so the zero at 1 has multiplicity 2.
::导致1时为零的系数发生两次,因此1时为零的系数有多重系数2。

The factor that led to the zero at $\begin{aligned} \frac{4}{3} \end{aligned}$ occurs five times, so the zero at $\begin{aligned} \frac{4}{3} \end{aligned}$ has multiplicity 5.
::导致43时零位的系数为5倍,43时零位的系数为5倍。

This video by aggieneer02 shows how to find the zeros of a polynomial function and their multiplicities.
::Aggieneer02的这段影片展示了如何找到多元函数的零及其多重性。

The multiplicity of a zero plays a role in how the graph behaves around that zero.
::一个零的多重性对图表在零周围的行为方式起着一定作用。

Even Multiplicity	Odd Multiplicity

touch	cross

Example 2
::例2

Sketch the graph of $\begin{aligned} f (x) = 2 x^{2} (x - 3) (x + 4) \end{aligned}$ .
::F(x) = 2x2(x-3)(x+4) 的图解。

Solution: First we identify the zeros and their multiplicities.
::解决方案:首先,我们确定零及其多重性。

Zeros	Multiplicities	Cross/Touch
0	2	touch
3	1	cross
-4	1	cross

Next we need to determine the end behavior. We do not need to multiply out all of the factors. We just need to multiply the highest power from each term.
::接下来我们需要决定最终行为。我们不需要把所有因素都乘以。我们只需要从每个任期中乘以最高的权力。

\begin{aligned} f (x) = 2 x^{2} (x - 3) (x + 4) \\ 2 x^{2} \cdot x \cdot x = 2 x^{4} \end{aligned}

::f( x) = 2x2( x-3) (x+4) 2x2x2x=2x4

This means that the end behavior is like the power function $\begin{aligned} y = 2 x^{4}, \end{aligned}$ so both ends face upward.
::这意味着最终行为就像 y=2x4 的功率函数, 所以两端都向上。

To sketch the graph, we represent what we know. These parts are in red below. Then we start at the left and go to the right. We connect the arrow on the left to the zero at -4. Since the multiplicity is odd, we cross the x -axis. We then go down, but since the origin is a point, we have to connect the graph back up to that point. At 0, the graph touches the x -axis. This part is in green below.
::要绘制图形, 我们代表我们所知道的。这些部分在下面的红色。然后从左边开始, 到右边。我们把左边的箭头连接到 - 4 点的零点。由于多重性是奇特的, 我们跨过 X 轴。然后我们往下走, 但是由于起源是一个点, 我们必须将图表连接到那个点。在 0 点, 图形会碰到 x 轴。这部分是在下面的绿色。

Lastly, we go through the point at (3,0) and connect up to the arrow on the right. This part is in blue above.
::最后,我们从(3,0)处走过点,然后连接到右边的箭头,这部分是上面蓝色的。

Note that we cannot determine how low the graph goes without some calculus concepts. Our sketch is just an approximation.
::请注意,我们无法确定图表的下方没有某些微积分概念。我们的草图只是一个近似。

This video by ValenciaMathJoel shows examples of sketching polynomial graphs.
::这段影片由巴伦西亚马特乔尔拍摄,

Example 3
::例3

Sketch the graph of $\begin{aligned} y = - \frac{1}{2} (x - 2) (x + 5)^{2} . \end{aligned}$
::绘制 y= 12(x- 2)(x+5) 的图形 2。

Solution: There is a zero at 2 of multiplicity 1. The graph will cross the x -axis there. At (-5,0), the graph will touch the x -axis because the the multiplicity is 2.
::解析度: 多重度为 2 时为 0 。图形将穿过 X 轴。在 ( 5, 0) 时, 图形将触摸 x 轴, 因为多重度为 2 。

The end behavior of the graph is $\begin{aligned} y = - \frac{1}{2} x^{3} \end{aligned}$ , so the arrow on the left will face up and the arrow on the left will face down.
::图形的末端行为为 Y= 12x3, 左侧的箭头会向上, 左侧的箭头会向下。

How to Graph a Polynomial Function With a Graphing Utility
::如何用图形化工具绘制多面函数

To graph a polynomial function with a graphing utility, enter the function in with $\begin{aligned} Y = \end{aligned}$ and graph.
::要用图形工具绘制多边函数,请在 Y = 和图中输入该函数。

Note:
::注:

This will produce a more accurate graph than the sketches in this section.
::这将产生比本节的草图更准确的图表。

If you see horizontal line segments on your graph, those are an approximation. Polynomial function graphs are smooth.
::如果您在图形上看到水平线段, 这些是近似值。多面函数图形是平滑的。

Summary
::摘要

Polynomial function graphs are smooth and continuous.
::多面函数图形是光滑和连续的。

Power functions are one-term polynomial functions and help us recognize the end behavior of our function , or what happens for large $\begin{aligned} | x | \end{aligned}$ .
::权力功能是一线多功能, 帮助我们识别我们功能的最终行为, 或大型的 x会发生什么。

The multiplicity of the zero is the power of the factor that yielded that zero. This determines whether the graph will cross the x -axis or touch the x- axis at that zero.
::零的倍数是产生零的系数的功率。这决定图形是跨过x轴,还是触碰零的X轴。

Using the end behavior and the zeros and their multiplicities, we can sketch a graph of a polynomial function.
::利用结束行为和零及其多重特性, 我们可以绘制一个多面函数的图形。

Review
::回顾

Sketch the graph of the functions below. Then find the domain and range.
::绘制以下函数的图形。然后找到域和范围。

1. $\begin{aligned} y = 2 (x - 3) (x + 4) (x + 7) \end{aligned}$
::1. y=2(x-3)(x+4)(x+7)

2. $\begin{aligned} f (x) = - \frac{2}{3} (x + 2) (x - 2)^{4} \end{aligned}$
::2. f(x)=23(x+2)(x-2)4

3. $\begin{aligned} y = (x + 1) (x - 2) (x + 3) (x - 6) \end{aligned}$
::3. y=(x+1)(x-2)(x+3)(x-6)

4. $\begin{aligned} g (x) = - 2 x^{2} (x + 3)^{2} (x - 5) \end{aligned}$
::4. g(x)=-2x2(x+3)(x-5)

5. $\begin{aligned} h (x) = - 4 x^{2} (x + 2)^{2} (x - 3) (x - 7) \end{aligned}$
::5. h(x)=-4x2(x+2)(x-3)(x-7)

6. $\begin{aligned} y = x^{3} + 2 x^{2} - 5 x - 6 \end{aligned}$
::6. y=x3+2x2 - 5x-6

7. $\begin{aligned} y = x^{3} - 8 \end{aligned}$
::7. y=x3-8

8. $\begin{aligned} f (x) = - x^{4} + 3 x^{2} - 2 \end{aligned}$
::8. f(x)=-x4+3x2-2

9. $\begin{aligned} y = x^{4} + 6 x^{3} + 23 x^{2} + 36 x + 36 \end{aligned}$
::9. y=x4+6x3+23x2+36x36

10. $\begin{aligned} g (x) = - x^{5} + 4 x^{3} + x^{2} - 4 \end{aligned}$
::10. g(x)=-x5+4x3+x2-4

Explore More
::探索更多

1. Determine if the following statements are SOMETIMES, ALWAYS, or NEVER true. Explain your reasoning.
::1. 确定以下陈述是否是时事性陈述、ALWAYS声明或从来不属实,解释你的推理。

The range of an even function is $\begin{aligned} (- \infty, m a x] \end{aligned}$ , where max is the maximum of the function.
::一个偶数函数的范围是 (- , max) , 最大值是函数的最大值。

The domain and range of all odd functions are all real numbers.
::所有奇数函数的域和范围都是真实数字。

A function can have exactly three imaginary solutions.
::函数可以完全有三个假想的解决方案。

An $\begin{aligned} n^{t h} \end{aligned}$ degree polynomial has $\begin{aligned} n \end{aligned}$ real solutions.
::Nnth 度多面性具有真正的解决方案。

The parent graph of any polynomial function has one zero.
::任何多边函数的母形图为 0。

2. Explain why the end behavior of an even-degree function faces in the same direction, even though we consider very large positive numbers and very large negative numbers.
::2. 解释为什么一等分函数的结束行为会朝着同一方向面临同样方向,尽管我们认为非常多的正数和非常大负数。

3. Find the two complex solutions in Number 2 of the Review.
::3. 在《审查报告》第2号中找到两个复杂的解决办法。

4. Using the techniques for solving quadratic inequalities in Section 9 of this chapter, solve $\begin{aligned} x^{3} + 2 x^{2} - 5 x - 6 \geq 0 \end{aligned}$ .
::4. 利用本章第9节中的解决二次不平等的技术,解决x3+2x2-5x-6x-0。

5. To make a fair race between a dragster and a funny car, a scientist devised the following polynomial equation:
::5. 为了在拖车和滑稽汽车之间公平竞争,一名科学家设计了以下多元等式:

$\begin{aligned} f (x) = 71.682 x - 60.427 x^{2} + 84.710 x^{3} - 27.769 x^{4} + 4.296 x^{5} - 0.262 x^{6} . \end{aligned}$ What is the maximum point of this function's graph?
::f(x) = 71.682x- 60.427x2+84. 710x3- 27.769x4+4.296x5-0. 262.x6. 本函数图的最大点是什么?

Answers for Review and Explore More Problems
::回顾和探讨更多问题的答复

Please see the Appendix.
::请参看附录。

Section outline

General

多边函数的切入图

Smooth and Continuous ::平滑和连续

Power Functions ::权力功能

Power Functions ::权力功能

End Behavior ::结束行为行为

End Behavior ::结束行为行为

Zeros and Their Multiplicity ::零及其多样性

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

How to Graph a Polynomial Function With a Graphing Utility ::如何用图形化工具绘制多面函数

Summary ::摘要

Review ::回顾

Explore More ::探索更多

Answers for Review and Explore More Problems ::回顾和探讨更多问题的答复

Smooth and Continuous
::平滑和连续

Power Functions
::权力功能

Power Functions
::权力功能

End Behavior
::结束行为行为

End Behavior
::结束行为行为

Zeros and Their Multiplicity
::零及其多样性

Example 1
::例1

Example 2
::例2

Example 3
::例3

How to Graph a Polynomial Function With a Graphing Utility
::如何用图形化工具绘制多面函数

Summary
::摘要

Review
::回顾

Explore More
::探索更多

Answers for Review and Explore More Problems
::回顾和探讨更多问题的答复