Course: 14.8 答复 -- -- 第8章:二次曲线和高要求多元函数

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回答 - Ch 8: 二次曲线和高要求多面函数
Section 8.2 Solving Quadratic Equations by Factoring
::第8.2节通过保理办法解决赤道等量

Review
::回顾

$\begin{align*}x=1,x=\text{-}9\end{align*}$
::x=1,x=9 x=9

$\begin{align*}x=0,x=\text{-}6\end{align*}$
::x=0,x=6xx=6

$\begin{align*}x=\text{-}\frac{3}{2},x=4\end{align*}$
::x=-32,x=4 x=3,x=4

$\begin{align*}x=\frac{2}{3},x=\text{-}\frac{5}{4}\end{align*}$
::x=23x=-54xx=23x=-54

$\begin{align*}x=7,x=\text{-}8\end{align*}$
::x=7,x=8 x=7,x=8

$\begin{align*}x=0,x=\frac{1}{2}\end{align*}$
::x=0,x=12 x=10,x=12

$\begin{align*}x=\text{-}\frac{4}{3},x=\text{-}3\end{align*}$
::x=-43,x=-3

$\begin{align*}x=\frac{7}{6},x=\text{-}\frac{7}{6}\end{align*}$
::x=76x=-76xx=-76

$\begin{align*}x=\text{-}\frac{2}{5},x=\text{-}2\end{align*}$
::x=-25,x=-2

$\begin{align*}x=\text{-}\frac{1}{8},x=9\end{align*}$
::x=18,x=9

Explore More
::探索更多

2 seconds
::2秒2秒

164 feet
::164英尺

$10.50

22 people
::22人 22人

54 people
::54人,54人

14 and 30
::14和30

Section 8.3 Solving Quadratic Equations Using Square Roots
::第8.3节利用平根解决赤道赤道

Review
::回顾

$\begin{align*}x=\pm 12\end{align*}$
::x 12

$\begin{align*}x=\pm 2\end{align*}$
::x% 2

$\begin{align*}x=\pm \sqrt{3}\end{align*}$
::x% 3

$\begin{align*}x=\text{-}2\pm 7\end{align*}$
::x=-2+7x=-2+7

$\begin{align*}x=5\pm \sqrt{3}\end{align*}$
::x=5=3x3

$\begin{align*}x=\pm 2\sqrt{15}\end{align*}$
::x215

$\begin{align*}x=\pm 4\end{align*}$
::x4

$\begin{align*}x=\pm 3\sqrt{14}\end{align*}$
::x314

$\begin{align*}x=\text{-}7\pm \sqrt{13}\end{align*}$
::x=-713

$\begin{align*}x=\frac{\text{-}4\pm 5}{3}\end{align*}$
::x=-453

$\begin{align*}x=10\pm 6\sqrt{2}\end{align*}$
::x=1062

$\begin{align*}x=1\pm \sqrt{37}\end{align*}$
::x=1+37x=1+37

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36.61 by 20.59
::36.61 到20.59岁

The orange hits the ground 1st by 1.62 seconds.
::橙色的击落地面1 乘以1.62秒

60 feet
::60英尺 60英尺

14 in
::14 英寸 14 英寸

$\begin{align*}x=\pm 3\end{align*}$ . Preference to method will vary.
::x3. 对方法的优先程度各有不同。

Both methods give you $\begin{align*}x=2\end{align*}$ and $\begin{align*}x=\text{-}\frac{2}{3}\end{align*}$ . Conclusion: Both methods will help you find the solution(s).
::两种方法都为您提供 x=2 和 x= 23 。结论:两种方法都有助于您找到解决方案。

Section 8.4 Solving Quadratic Equations by Completing the Square
::第8.4节通过完成广场解决赤道等分

Review
::回顾

4

1

64

$\begin{align*}{\left(x+3\right)}^{2}\end{align*}$
:x+3)2

$\begin{align*}{\left(x-\frac{7}{2}\right)}^{2}\end{align*}$
:x-72)2

$\begin{align*}{\left(x-\frac{1}{4}\right)}^{2}\end{align*}$
:x-14)2

$\begin{align*}{\left(x+5\right)}^{2}=\text{-}4; \ x=\text{-}5 \pm 2i\end{align*}$
:x+5)2=-4;x=-52i

$\begin{align*}{\left(x-7\right)}^{2}=\text{-}20; \ x=7 \pm 2i\sqrt{5}\end{align*}$
:-7)2=20; x=7=7°2i5

$\begin{align*}{\left(x+\tfrac{3}{2}\right)}^{2}=\text{-}\tfrac{71}{4}; \ x=\text{-}\tfrac{1}{2} i (\sqrt{71}-3 i), \ x=\tfrac{1}{2} i (\sqrt{71}+3 i)\end{align*}$
:x+32)2=-714; x=12i(71-3i), x=12i(71+3i)

$\begin{align*}{\left(x-1\right)}^{2}=\tfrac{13}{6}; \ x= 1 \pm \sqrt{\tfrac{13}{6}}\end{align*}$
:x-1)2=136;x=1136

$\begin{align*}{\left(x-3\right)}^{2}= \text{-}16; \ x=3\pm4i\end{align*}$
:x-33)2=16; x=34i

$\begin{align*}{\left(x-\frac{1}{4}\right)}^{2}=\frac{49}{16}; \ x= \text{-}\tfrac{3}{2}, x=2\end{align*}$
:x-144)2=4916; x=-32,x=2

Explore More
::探索更多

In line 3, the right side would be $\begin{align*}\text{-}11+144,\end{align*}$ not $\begin{align*}\text{-}11+36,\end{align*}$ which would make the answer $\begin{align*}x=6\pm \frac{\sqrt{133}}{2}\end{align*}$ .
::在行3中,右侧为 -11+144, 而不是 -11+36, 这样回答 x=61332。

$\begin{align*}x=0.58\end{align*}$
::x=0.58x=0.58

width = 18 m; length = 24 m.
::宽度=18米;长度=24米。

Section 8.5 Solving Quadratic Equations Using the Quadratic Formula
::第8.5节使用二次曲线公式解决二次赤道等量

Review
::回顾

$\begin{align*}x=\text{-}4\pm \sqrt{7}\end{align*}$
::x=-4+7x=-4+7

$\begin{align*}x=\frac{\text{-}1\pm \sqrt{41}}{\text{-}4}\end{align*}$
::x=-141-4

$\begin{align*}x=\frac{\text{-}2\pm \sqrt{19}}{3}\end{align*}$
::x=-2193

$\begin{align*}x=\text{-}15,x=10\end{align*}$
::x=-15x=10xx=10

$\begin{align*}x=\frac{3}{4},x=\text{-}\frac{1}{2}\end{align*}$
::x=34,x=12,x=34,x=12

No real solution
::没有真正的解决办法

$\begin{align*}x=\text{-}\frac{1}{2},x=\frac{2}{5}\end{align*}$
::x=12,x=25 x=25

$\begin{align*}x=8\pm 2\sqrt{15}\end{align*}$
::x=8215

$\begin{align*}x=\text{-}\frac{5}{2}\end{align*}$
::x=-52

$\begin{align*}x=\text{-}3,x=21\end{align*}$
::x=3,x=21xx=21

Explore More
::探索更多

$\begin{align*}p=$22.79\end{align*}$ or $\begin{align*}p=$43.87\end{align*}$
::p=22.79美元或p=43.87美元。

43.2 mph
::43.2平方米

length = 58.73 ft; width = 65.73 ft
::长度=58.73英尺;宽度=65.73英尺

4.45 seconds later
::4.45秒后

Sum of the roots = $\begin{align*}\text{-}\frac{b}{a}\end{align*}$ ; product of the roots = $\begin{align*}\frac{c}{a}\end{align*}$
::根根总和= -ba;根产品= ca

Section 8.6 Complex Numbers
::第8.6节复杂数字

Review
::回顾

$\begin{align*}11i\sqrt{2}\end{align*}$
::1111i2

$\begin{align*}18i\sqrt{5}\end{align*}$
::18i5

$\begin{align*}\text{-}84i\sqrt{2}\end{align*}$
::-84i2

$\begin{align*}\text{-}12\sqrt{6}\end{align*}$

$\begin{align*}\frac{i}{\sqrt{5}}\end{align*}$ or $\begin{align*}\frac{i\sqrt{5}}{5}\end{align*}$
::i5 或 i55

a. -16. b. $\begin{align*}\text{-}9i\end{align*}$ . c. $\begin{align*}\text{-}2i\end{align*}$ .
::a. -16 b. -9i. c. -2i.

$\begin{align*}5+10i\end{align*}$
::5+10i 5+10i

$\begin{align*}0.8+0.5i\end{align*}$
::0.8+0.5i

$\begin{align*}22-13i\end{align*}$
::22-13i

$\begin{align*}\text{-}22i-8\end{align*}$
::-22i-8

$\begin{align*}79i-68\end{align*}$
::79-68

$\begin{align*}92i+60\end{align*}$
::92i+60

$\begin{align*}\text{-}\frac{4i}{3}+3\end{align*}$
::-4i3+3 -4i3+3

$\begin{align*}\frac{4}{5}-\frac{4}{5}i\end{align*}$
::45-45i

$\begin{align*}\frac{\text{-}82+343i}{449}\end{align*}$
::-82+343i449

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::探索更多

Section 7.3; difference of squares
::第7.3节;平方差

$\begin{align*}\sqrt{\text{-}6}\times \sqrt{\text{-}6}=i\sqrt{6}\times i\sqrt{6}=\text{-}6\end{align*}$ ; $\begin{align*}\sqrt{x^2} = \left|x\right|\end{align*}$
::-6x-6=i6xi6=-6; x2x

The slope from the origin to $\begin{align*}2+3i\end{align*}$ is $\begin{align*}\frac{3}{2},\end{align*}$ and the slope from the origin to $\begin{align*}\text{-}3+2i\end{align*}$ is $\begin{align*}\text{-}\frac{2}{3}\end{align*}$ . They are negative reciprocals.

::从原地到2+3i的斜坡为32,从原地到-3+2i的斜坡为23,是负对等的。

When $\begin{align*}c=\text{-}1,\end{align*}$ the five values are 0, -1, 0, -1, 0. When $\begin{align*}c=i,\end{align*}$ the five values are $\begin{align*}0,i,(\text{-}1+i),\text{-}i,(1+i).\end{align*}$
::当 c=-1 时, 5 个值为 0, -1, 0, -1, 0。当 c=i 时, 5 个值为 0, i, (1+i), (1+i), i, (1+i) 。

$\begin{align*}Z=11-2j\end{align*}$
::11-2j

Section 8.7 Solving Quadratic Equations with Complex Number Solutions
::第8.7节具有复杂数字解决办法的二次赤道解决办法

Review
::回顾

One solution
::一种解决办法

Two solutions
::二. 两种解决办法

No solutions
::无解决方案

Two solutions
::二. 两种解决办法

$\begin{align*}x=i \sqrt{5}\end{align*}$
::x=5x=5

$\begin{align*}x=\text{-}3,x=20\end{align*}$
::x=3,x=20 x=20

$\begin{align*}x=\text{-}1\pm 11i\end{align*}$
::x=-111i

$\begin{align*}x=\pm \sqrt{\frac{10}{3}}\end{align*}$
::x 103

$\begin{align*}x=\pm 3i\end{align*}$
::x3i

$\begin{align*}x=\pm \sqrt{6}i\end{align*}$
::x 6i

$\begin{align*}x=\frac{\text{-}5\pm 3i\sqrt{7}}{4}\end{align*}$
::x=-53i74

$\begin{align*}x=\text{-}6\pm 2i\sqrt{3}\end{align*}$
::x=-62i3

$\begin{align*}x=\frac{1\pm 2i\sqrt{7}}{2}\end{align*}$
::x=12i72

$\begin{align*}x=\frac{\text{-}5\pm 6i\sqrt{2}}{6}\end{align*}$
::x=-56i26

$\begin{align*}x=\frac{3\pm i\sqrt{\frac{53}{7}}}{4}\end{align*}$
::x=35374

Explore More
::探索更多

$\begin{align*}4+i\end{align*}$ ; $\begin{align*}6-2i\end{align*}$
::4+一;6-2i

a. $\begin{align*}x^2 - 6x +13\end{align*}$ ; b. $\begin{align*}x^2 - 2x +2\end{align*}$
::a. x2 - 6x+13; b. x2 - 2x+2

a. $\begin{align*}c=\text{-}3\end{align*}$ ; $\begin{align*}c=1\end{align*}$ ; $\begin{align*}c=5\end{align*}$    b. $\begin{align*}c=\text{-}3\end{align*}$ ; $\begin{align*}c=9\end{align*}$ ; $\begin{align*}c=11\end{align*}$
::a. c=3; c=1; c=5 b. c=3; c=9; c=11; c=11; c=5 b. c=3; c=9; c=9; c=11

Yes
::是是

No
::否无

Mistake is in Step 2. The solution is $\begin{align*}x=6\pm i\sqrt{2}\end{align*}$ .
::错误在第二步,答案是 x=6i2。

Not with real number coefficients.
::没有实际数字系数。

Section 8.8 Graphing Quadratic Functions
::第8.8节构造二次函数

Review
::回顾

Explore More

a. Slingshot A: 26.7 m; Slingshot B: 14.8 m; Slingshot C: 21.1 m. b. Slingshot A: 36.7 m; Slingshot B: 44.7 m; Slingshot C: 62 m. c. Answers will vary.
::a. 弹片A:26.7米;弹片B:14.8米;弹片C:21.1米。 b. 弹片A:36.7米;弹片B:44.7米;弹片C:62米。 c. 回答将有所不同。

a. $\begin{align*}y=\text{-}0.0215({\left(x-17.47\right)}^{2}+6.56,\end{align*}$ b. 4.60 feet from the left and 30.3 feet from the right.
::a.y=-0.0215((x-17.47)2+6.56,b. 4.60英尺从左侧,30.3英尺从右侧。

a. 11.25 feet. b. 0.625 seconds. c. 1.46 seconds.
::a 11.25英尺. b. 0.625秒. c. 1.46秒.

No
::否无

$\begin{align*}y=\frac{\text{-}{b}^{2}+4ac}{4a}\end{align*}$
::y=-b2+4ac4a y=-b2+4ac4a

10.06 feet at 0.56 seconds
::10.06英尺,0.56秒

The value of a makes the graph wider or narrower. If a is positive, then the graph opens up. If a is negative, the graph opens down. The value of h is the horizontal translation, and therefore this value will indicate where the curve moves left or right on the x-axis. The value of k is the vertical translation, and therefore this value will indicate where the curve moves up or down on the y-axis.

::使图表更宽或更窄。如果为正值, 则图形向上打开。如果为负值, 图形向下打开。 h 值为水平翻译, 因此这个数值将显示曲线在 x 轴上左或右移动的位置。 k 值为垂直翻译, 因此这个数值将显示曲线在 Y 轴上上上下移动的位置。

Section 8.9 Quadratic Inequalities
::第8.9节严重不平等

Review
::回顾

$\begin{align*}x\le 7\;\;\text{or}\;\;x\ge \text{-}2\end{align*}$
::x7orx-2

$\begin{align*}\text{-}5\le x\le 10\end{align*}$
::- 5x10 - 5x10

$\begin{align*}\text{-}6\le x\le 8\end{align*}$
::- 6x_8

$\begin{align*}\text{-}2\le x\le 10\end{align*}$
::- 25x10 - 10

$\begin{align*}x\le \text{-}7;x\ge \text{-}3\end{align*}$
::x-7;x-3

$\begin{align*}x< \text{-}6 \;\;\text{or}\;\;x> 3\end{align*}$
::x- 6orx>3

$\begin{align*}x< \text{-}1 \;\;\text{or}\;\;x> 8\end{align*}$
::x-1orx > 8

$\begin{align*}x\le \text{-}9 \;\;\text{or}\;\;x\ge 10\end{align*}$
::x-- 9orx- 10

$\begin{align*}\text{-}\frac{1}{3}\le x\le 8\end{align*}$
::-13x_8

$\begin{align*}x\le \text{-}3 \;\;\text{or}\;\;x\ge 2\end{align*}$
::x- 3orx% 2

Explore More
::探索更多

$\begin{align*}0.25s\le x\le 0.50s\end{align*}$
::0.25sx0.50s

$\begin{align*}1s\le x\le 1.41s\end{align*}$ { Note: Equation $\begin{align*}h=20-5{t}^{2}\end{align*}$ will be added to the question.}
::{注: 等量 h=20-5t2将添加到问题中。 }

The width would be between 30 and 58 feet. The length would be between 40 and 68.
::宽度在30至58英尺之间,长度在40至68英尺之间。

$\begin{align*}x=5\end{align*}$ and $\begin{align*}x=27\end{align*}$ are the break-even points. The profit is found when $\begin{align*}5< x < 27\end{align*}$ .
::x=5 和 x=27 是平衡点。利润在 5 < x < 27 " 时找到。

$\begin{align*}x=11\end{align*}$ and $\begin{align*}x=23\end{align*}$ are the break-even points. The profit is found when $\begin{align*}11< x < 23\end{align*}$ .
::x=11 和 x=23 是平衡点。利润在 11 < x < 23 时找到。

79.06 feet. No, you will not need more gas.
::79.06英尺不,你不需要更多汽油了

$\begin{align*}0 < w \le 5\end{align*}$
::0<w=5>

Section 8.10 Quadratic Models
::第8.10节赤道模型

Review
::回顾

Width = 125 feet. Length = 125 feet.
::宽度=125英尺长度=125英尺

Width = 30 feet. Length = 20 feet. Yes, it will fit.
::宽度 = 30英尺长度 = 20英尺

Revenue = $5,625.
::收入=5 625美元。

Height = 144.8 feet. Total distance = 436.39 feet.
::高度 =144.8英尺总距离 = 436.39英尺

Leg 1 = 32.23 feet. Leg 2 = 38.23 feet.
::腿1 = 32.23英尺腿2 = 38.23英尺

Explore More

Radius = 0.56 feet. Height = 0.57 feet.
::半径=0.56英尺高度=0.57英尺

a. $\begin{align*}h(t)=\text{-}\frac{1}{80}{\left(t-80\right)}^{2}+80\end{align*}$ . b. $\begin{align*}h(t)=\text{-}\frac{1}{40}{\left(t-80\right)}^{2}+80.\end{align*}$
::a. h(t)=-180(t-80)2+80。 b. h(t)=-140(t-80)2+80。

6 price increases
::6 涨价

Price = $175 and make 35,000 smart phones.
::价格=175美元,制作35 000部智能电话。

$\begin{align*}y=\text{-}\frac{5}{72}\left({x}^{2}-18x\right)\end{align*}$
::y=-572(x2-18x)

$\begin{align*}y=\text{-}\frac{71}{6084}{\left(x-39\right)}^{2}+23\end{align*}$
::y=-716084(x-39)2+23

a. $\begin{align*}y=\text{-}0.18{x}^{2}+8.52x-3.95\end{align*}$ . b. Answers will vary.
::a.y=-0.18x2+8.52x-3.95.b.答复将有所不同。

Section 8.11 Finding Rational and Real Zeros of Polynomial Functions
::第8.11节确定多功能的合理和实际零

Review
::回顾

$\begin{align*}x=\pm 1,\pm 2,\pm 4,\pm 5,\pm 10,\pm 20\end{align*}$
::x1,2,4,5,5,10,XX20

$\begin{align*}x=\pm 1,\pm \frac{1}{2},\pm \frac{1}{4}, \pm 3,\pm \frac{3}{2},\pm \frac{3}{4},\pm 5,\pm \frac{5}{2},\pm \frac{5}{4}, \pm 15,\pm \frac{15}{2},\pm \frac{15}{4}\end{align*}$
::1, 12, 14, 3, 32, 34, 5, 52, 54, 15, 15, 152, 154

$\begin{align*}x=\pm 1,\pm \frac{1}{2}, \pm 2,\pm 4, \pm 8\end{align*}$
::1, 12, 2, 4, 8

$\begin{align*}x=\pm 1,\pm 3,\pm 9\end{align*}$
::x1,%3,%9

$\begin{align*}x=\pm 1,\pm \frac{1}{2},\pm \frac{1}{4}, \pm \frac{1}{8},\pm 2,\pm \frac{1}{4},\pm 3,\pm \frac{3}{2},\pm \frac{3}{4}, \pm \frac{3}{8},\pm 4,\pm 6,\pm 8,\pm 12,\pm 24\end{align*}$
::1, 12, 14, 18, 2, 14, 3, 32, 34, 38, 4, 6, 8, 12, 24

$\begin{align*}x=2,x=\frac{1}{2},x=\frac{1}{3}\end{align*}$
::x=2,x=12,x=13,x=13

$\begin{align*}x=2,x=\text{-}1, x = \text{-}4\end{align*}$
::x=2,x=1,x=4

$\begin{align*}x=3,x=0.604,x=\text{-}0.104\end{align*}$
::x=3,x=0.604,x=-0.104

$\begin{align*}x=1.5\end{align*}$
::x=1.5x=1.5

$\begin{align*}x=5\end{align*}$
::x=5x=5

$\begin{align*}x=\text{-}1.69,x=1.32\end{align*}$
::x=-1.69,x=1.32

$\begin{align*}x=3.05\end{align*}$
::x=3. 05

$\begin{align*}x = \text{-}4, x = 1.5\end{align*}$ a
::x=4,x=1.5 a

$\begin{align*}x=5,x=\text{-}5,x=\frac{1}{3},x=\text{-}\frac{1}{3}\end{align*}$
::x=5,x=5,x=5,x=5,x=13,x=13

Explore More
::探索更多

a. $\begin{align*}V=(x)(10-2x)(15-2x).\end{align*}$ b. $\begin{align*}x=0,x=5,x=7.5.\end{align*}$ c. None are physically possible, as each causes a zero dimension (height, width, and depth).
::a. V=(x)(10-2x)(15-2x) b. x=0,x=5,x=7.5.c. 不存在实际可能的情况,因为每个情况都有零维(高度、宽度和深度)。

a. $\begin{align*}C=4,\!500x+729,\!000.\end{align*}$ b. 32 cars. c. 45 cars. d. $9,280.
::a. C=4 500x+729 000.b. 32辆汽车.c.45辆.d. 9 280美元。

141.44 weeks
::141.44周

17.92 months
::17.92个月

Section 8.12 Finding Complex Zeros of Polynomial Functions
::第8.12节多元职能的复杂零星查找

Review
::回顾

$\begin{align*}x=3.16,x=\text{-}3.16,x=\text{-}2,x=1\end{align*}$
::x3.16x=-3.16,x=-2,x=1

$\begin{align*}x=5,x=\text{-}0.866,x=0.866\end{align*}$
::x=5,x=-0.866,x=0.866

$\begin{align*}x=2.45,\text{-}2.45,x=1.58i,x=\text{-}1.58i\end{align*}$
::x=2.45,-2.45,x=1.58i,x=1.58i

$\begin{align*}x=\text{-}3\end{align*}$
::x=-3x=3

$\begin{align*}x=\text{-}2,x=1.5+0.5i,x=1.5-0.5i\end{align*}$
::x=-2,x=1.5+0.5i,x=1.5-0.5i

$\begin{align*}x=1.29,x=\text{-}1.29,x=1.73i,x=\text{-}1.73i\end{align*}$
::x=1.29,x=1.29,x=1.73i,x=-1.73i

$\begin{align*}x=3\end{align*}$
::x=3x=3

$\begin{align*}x=8,x=\text{-}5,x=\text{-}2.33,x=0.5\end{align*}$
::x=8,x=5,x=2.33,x=0.5

$\begin{align*}x=\text{-}2.08, x=\text{-}0.59, x = 1\end{align*}$
::x= 2.08,x=-059,x=1

$\begin{align*}x=5.54+1.21i,x=5.54-1.21i,x=\text{-}1.96,x=0.178+1.07i,x=0.178-1.07i\end{align*}$
::x=5.54+1.21i,x=5.54-1.21i,x=1.96,x=0.178+1.07i,x=0.178-1.07i

$\begin{align*}x^3-4x^2+x-4\end{align*}$
::x3 - 4x2+x-4

$\begin{align*}x^3 + 3x^2 + 4x + 12\end{align*}$
::x3+3x2+4x+12

$\begin{align*}{x}^{4}+2{x}^{3}-7{x}^{2}-10x+10\end{align*}$
::x4+2x3 - 7x2 - 10x+10

$\begin{align*}{x}^{4}-\frac{31}{3}{x}^{3}+\frac{100}{3}{x}^{2}-38x+\frac{28}{3}\end{align*}$
::x4-313x3+1003x2-38x+283

Explore More
::探索更多

a. Answers will vary. b. Answers will vary. c. An odd-degree polynomial must have at least one real root, since the function approaches negative infinity on one end and positive infinity on the other.
::a. 答案将有所不同。 b. 答案将有所不同。 c. 奇异度多数值必须至少有一个真正的根根,因为函数一端接近负无穷,另一端接近正无穷。

For $\begin{align*}n=3\end{align*}$ , roots of unity are 1. For $\begin{align*}n=4\end{align*}$ , the roots of unity are $\begin{align*}\left\{\pm 1,\pm i\right\}.\end{align*}$ For $\begin{align*}n=5\end{align*}$ , the roots of unity are $\begin{align*}\left\{1,\text{-}0.809\pm 0.588i,0.309\pm 0.951i\right\}.\end{align*}$
::n=3, 团结的根源是1. n=4, 团结的根源是+1, i}。 n=5, 团结的根源是{1,-0.8090.5888i,0.3090.951}。

$\begin{align*}x=3i,x=\text{-}3i,x=2,x=\text{-}2\end{align*}$
::x=3i,x=3i,x=3i,x=2,x=2,x=2

No. The zeros of the function are $\begin{align*}x=1.609i,x=\text{-}1.609i,x=0.5073,x=\text{-}0.5073.\end{align*}$
::否。函数的零为 x=1.609i,x=1.6091,x=0.5073,x=-05073。

$\begin{align*}x=\text{-}6.646,x=\text{-}1.354,x=\pm 2i\end{align*}$
::x=-6.646,x=-1.354,x2i

Section 8.13 Sketching Graphs of Polynomial Functions
::第8.13节多面函数的拼图图

Review
::回顾

Domain: $\begin{align*}(\text{-}\infty ,\infty )\end{align*}$ , Range: $\begin{align*}(\text{-}\infty ,\infty )\end{align*}$

::域 : (-) 域 : (-) 域 : (-) 域 : (-) 域 : (-) 域 : (-) 域 : (-)

Domain: $\begin{align*}(-\infty ,\infty )\end{align*}$ , Range: $\begin{align*}(-\infty ,\infty )\end{align*}$

::域 ,), 范围,)

Domain: $\begin{align*}(\text{-}\infty ,\infty )\end{align*}$ , Range: $\begin{align*}[\text{-}154.96 ,\infty )\end{align*}$

::域 - , ), 范围 : [-154.96, ]

Domain: $\begin{align*}(\text{-}\infty ,\infty )\end{align*}$ , Range: $\begin{align*}(\text{-}\infty ,\infty )\end{align*}$

::域 : (-) 域 : (-) 域 : (-) 域 : (-) 域 : (-) 域 : (-) 域 : (-)

Domain: $\begin{align*}(\text{-}\infty ,\infty )\end{align*}$ , Range: $\begin{align*}(\text{-}\infty,27731.3]\end{align*}$

::域 : (-) , 范围 : (-) , 27731.3 , 范围 : (-) , 27731.3 , 范围 :

Domain: $\begin{align*}(\text{-}\infty ,\infty )\end{align*}$ , Range: $\begin{align*}(\text{-}\infty ,\infty )\end{align*}$

::域 : (-) 域 : (-) 域 : (-) 域 : (-) 域 : (-) 域 : (-) 域 : (-)

Domain: $\begin{align*}(\text{-}\infty ,\infty )\end{align*}$ , Range: $\begin{align*}(\text{-}\infty ,\infty )\end{align*}$

::域 : (-) 域 : (-) 域 : (-) 域 : (-) 域 : (-) 域 : (-) 域 : (-)

Domain: $\begin{align*}(\text{-}\infty ,\infty )\end{align*}$ , Range: $\begin{align*}(\text{-}\infty, 0.25 ]\end{align*}$

::域 : (- , ) , 范围 : (- , 0. 25)]

Domain: $\begin{align*}(\text{-}\infty ,\infty )\end{align*}$ , Range: $\begin{align*}[17.625 ,\infty )\end{align*}$

::域 - , ), 范围 : [17.625, ]

Domain: $\begin{align*}(\text{-}\infty ,\infty )\end{align*}$ , Range: $\begin{align*}(\text{-}\infty ,\infty )\end{align*}$

::域 : (-) 域 : (-) 域 : (-) 域 : (-) 域 : (-) 域 : (-) 域 : (-)

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::探索更多

Sometimes true. It depends on the leading coefficient.
::有时是真的,这取决于主要系数。

Always true.
::总是真实的。

Never. Complex roots always come in pairs.
::从来没有复杂的根总是成对的

Always true. This is the Fundamental Theorem of Algebra.
::总是对的,这是代数的基本理论

Always true. It goes through the origin.
::永远是真实的,它经过了源头

::有时是真实的。它取决于主要系数。总是真实的。从来没有。复杂的根总是成对的。总是真实的。这是代数的基本理论, 总是真实的。它贯穿了源头。

The end behavior of the graph shows how the function behaves when x has large absolute values. For even functions, it will always be positive.
::图形的结尾行为显示函数在 x 有大绝对值时的行为方式。即使是函数, 也总是正的。

$\begin{align*}x=2+6.396i,x=2-6.396i\end{align*}$
::x=2+6.396i,x=2-6.396i

$\begin{align*}\text{-}3\le x\le \text{-}1\end{align*}$ and $\begin{align*}x\ge 2\end{align*}$
::-x-1 和 x2

$\begin{align*}(6.15,1754.43)\end{align*}$



Section 8.14 Connections: Solar Power
::第8.14节连接:太阳能

The graph opens down (and has a maximum) since the coefficient for $\begin{align*}{x}^{2}\end{align*}$ is -7. The y -intercept is $\begin{align*}(0,64).\end{align*}$ The input is x (the month) and the output is y (the number of kilowatt hours).
::由于 x2 的系数为 -7. y 接口为 0,64。输入为 x (月份),输出为 y (千瓦小时数)。

Realistic values for x would be from 1 to 12, given there are 12 months in a year. A sample table for this would be the following:
$\begin{align*}\begin{array}{|c|c|} \hline x & y \\\hline 1 & 151.5 \\\hline 2 & 224 \\\hline 5 & 351.5 \\\hline 10 & 264 \\\hline \end{array}\end{align*}$
::x 的现实值为 1 至 12, 因为一年中有12个月, 其样本表如下: xy1151.52225351.510264

There is increasing power until x = 6.33, and then it decreases after this point.

::在 x = 6.33 之前,电量在不断增强,然后在此点后下降。

$\begin{align*}(6.33,364.83)\end{align*}$ . The vertex tells which month had used the greatest power.
:6.33 364.83) 顶点显示哪个月使用了最大功率。

Maximum output is 364.83 kilowatt hours in the month 6.33.
::最大输出为6.33月的364.83千瓦小时。

In month 13.39, there was zero output. The answers do not make sense, as the months should be whole numbers. The power reading could have been taken during the month. Also, if month 6 is June, then why does it have such a high output of solar energy?
::在13.39月中,产出为零。答案没有意义,因为月份应该是整月。电力读数可以在当月进行。此外,如果6月是6月,那么为什么太阳能产出如此之高?

Answers will vary. A quadratic function would be appropriate here, because the energy output would increase during winter months and then decrease during summer months.
::答案会各有不同。这里的二次函数是合适的, 因为冬季的能源产出会增加, 夏季的能源产出会减少。

71,332 is the y -intercept.
::71 332是 Y 的接口。

::71 332是 Y 的接口。

71,332 is the initial output of power.
::71 332是电力的初始输出。

::71 332是电力的初始输出。

The vertex is $\begin{align*}(6.54,279,\!469.34)\end{align*}$ .
::顶部为6.54,279,469.34。

::顶部为6.54,279,469.34。

279,469.34 kilowatts
::279 469.34千瓦特

::279 469.34千瓦特

April and September
::4月和9月

::4月和9月

The y -intercept is $\begin{align*}(0,71,\!332),\end{align*}$ so the minimum amount of energy should be here.
::y 界面是( 0,71,332) 所以最小能量的数量应该在这里。

::y 界面是( 0,71,332) 所以最小能量的数量应该在这里。

Mid-march to mid-September

::9月中旬至9月中旬

::9月中旬至9月中旬

Section 8.15 Connections: Designing Suspension Bridges
::第8.15节连接:设计悬吊桥

Vertex: $\begin{align*}(2,\!100,8)\end{align*}$
::顶数2,100,8)

$\begin{align*}y=0.00011{\left(x-2,\!100\right)}^{2}+8\end{align*}$
::y=0.00011(x-2100)2+8

Vertex: $\begin{align*}(2,\!130,4);\end{align*}$ equation: $\begin{align*}y=0.00010{\left(x-2,\!130\right)}^{2}+4\end{align*}$ .

::vertex : (2,130,4); 等式 : y = 0. 0000010(x-2,130) 2+ 4。

Answers will vary, but should note that both graphs have the same shape.
::答案将有所不同,但应指出,两个图表的形状相同。

Section 8.16 Connections: Law of Diminishing Returns
::第8.16节关联:减少回返法

Equation: $\begin{align*}y=\text{-}0.1{x}^{2}+3.237x-1.714\end{align*}$ .

::方程式:y=-0.1x2+3.237x-1.714。

About 25 products with 16 people
::约25种产品,16人

When there are too many people working
::当有太多人工作的时候

The marginal product is the difference in the y -values (9,11,7,-4,-7).
::边际产品是Y值的差别(9,11,7,4,7-7)。

Between 0 and 5 workers, 5 and 10 workers, and 1 0 and 15 workers
::0至5名工人,5至10名工人,10至15名工人

Between 15 and 20 workers, and 2 0 and 25 workers
::15至20名工人,20至25名工人

For between 15 and 20 workers, and 2 0 and 25 workers
::15至20名工人和20至25名工人

Both graphs have the same shape. The marginal products graph has a lower vertex, and is translated to the left.

::两个图形的形状都相同。边际产品图形的顶点较低,并被翻译成左边。

The graph opens down (and has a maximum) since the coefficient for x^2 is -7. The y-intercept is (0,64). The input is x (the month) and the output is y (the number of kilowatt hours).
::图形向下打开( 并有最大值) , 因为 x% 2 的系数是 - 7 。 y 接口是 (0, 64) 。输入是 x (当月) , 输出是 y (千瓦小时数 ) 。

Section outline

General

回答 - Ch 8: 二次曲线和高要求多面函数

Section 8.2 Solving Quadratic Equations by Factoring ::第8.2节 通过保理办法解决赤道等量

Section 8.3 Solving Quadratic Equations Using Square Roots ::第8.3节 利用平根解决赤道赤道

Section 8.4 Solving Quadratic Equations by Completing the Square ::第8.4节 通过完成广场解决赤道等分

Section 8.5 Solving Quadratic Equations Using the Quadratic Formula ::第8.5节 使用二次曲线公式解决二次赤道等量

Section 8.6 Complex Numbers ::第8.6节 复杂数字

Section 8.7 Solving Quadratic Equations with Complex Number Solutions ::第8.7节 具有复杂数字解决办法的二次赤道解决办法

Section 8.8 Graphing Quadratic Functions ::第8.8节 构造二次函数

Section 8.9 Quadratic Inequalities ::第8.9节 严重不平等

Section 8.10 Quadratic Models ::第8.10节 赤道模型

Section 8.11 Finding Rational and Real Zeros of Polynomial Functions ::第8.11节 确定多功能的合理和实际零

Section 8.12 Finding Complex Zeros of Polynomial Functions ::第8.12节 多元职能的复杂零星查找

Section 8.13 Sketching Graphs of Polynomial Functions ::第8.13节 多面函数的拼图图

Section 8.14 Connections: Solar Power ::第8.14节 连接:太阳能

Section 8.15 Connections: Designing Suspension Bridges ::第8.15节连接:设计悬吊桥

Section 8.16 Connections: Law of Diminishing Returns ::第8.16节 关联:减少回返法

Section 8.2 Solving Quadratic Equations by Factoring
::第8.2节通过保理办法解决赤道等量

Section 8.3 Solving Quadratic Equations Using Square Roots
::第8.3节利用平根解决赤道赤道

Section 8.4 Solving Quadratic Equations by Completing the Square
::第8.4节通过完成广场解决赤道等分

Section 8.5 Solving Quadratic Equations Using the Quadratic Formula
::第8.5节使用二次曲线公式解决二次赤道等量

Section 8.6 Complex Numbers
::第8.6节复杂数字

Section 8.7 Solving Quadratic Equations with Complex Number Solutions
::第8.7节具有复杂数字解决办法的二次赤道解决办法

Section 8.8 Graphing Quadratic Functions
::第8.8节构造二次函数

Section 8.9 Quadratic Inequalities
::第8.9节严重不平等

Section 8.10 Quadratic Models
::第8.10节赤道模型

Section 8.11 Finding Rational and Real Zeros of Polynomial Functions
::第8.11节确定多功能的合理和实际零

Section 8.12 Finding Complex Zeros of Polynomial Functions
::第8.12节多元职能的复杂零星查找

Section 8.13 Sketching Graphs of Polynomial Functions
::第8.13节多面函数的拼图图

Section 8.14 Connections: Solar Power
::第8.14节连接:太阳能

Section 8.15 Connections: Designing Suspension Bridges
::第8.15节连接:设计悬吊桥

Section 8.16 Connections: Law of Diminishing Returns
::第8.16节关联:减少回返法