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# 多项式的末端走向

## 什么是 "末端走势"?

#### 检查你是否理解

1) 这是 y, equals, g, left parenthesis, x, right parenthesis的图像.
g 的末端走势如何?

## 代数方法确定末端走势

• x, right arrow, plus, infinity,f, left parenthesis, x, right parenthesis趋近于?
• x, right arrow, minus, infinity,f, left parenthesis, x, right parenthesis趋近于?

### 研究: 单项式的极端情况

2) 看看单项式 f, left parenthesis, x, right parenthesis, equals, x, squared.
x 取一个非常大的正值, 哪一个最好的描述了 f, left parenthesis, x, right parenthesis?

x 取一个非常大的负值, 哪一个最好的描述了 f, left parenthesis, x, right parenthesis?

3) 看看单项式 g, left parenthesis, x, right parenthesis, equals, minus, 3, x, squared.
x 为非常大的正值时, 哪一个最好的描述了 g, left parenthesis, x, right parenthesis?

x 为非常大的负值时, 哪一个最好的描述了 g, left parenthesis, x, right parenthesis?

4) 看看单项式 h, left parenthesis, x, right parenthesis, equals, x, cubed.
x 为非常大的正值时, 哪一个最好的描述了 h, left parenthesis, x, right parenthesis?

x 为非常大的负值时, 哪一个最好的描述了 h, left parenthesis, x, right parenthesis?

5) 看看单项式 j, left parenthesis, x, right parenthesis, equals, minus, 2, x, cubed.
x 为非常大的正值时, 哪一个最好的描述了 j, left parenthesis, x, right parenthesis?

x 为非常大的负值时, 哪一个最好的描述了 j, left parenthesis, x, right parenthesis?

### 总结这些研究

n 为偶数, 函数在两 "端" 的趋向是相同的. 首项系数的正负号决定了它们都趋向 plus, infinity 或都趋向 minus, infinity.
n 为奇数, 函数在两 "端" 的趋向是相反的. 首项系数的正负号决定了哪一端是 plus, infinity 和哪一端是 minus, infinity.

start color #11accd, n, end color #11accd 为偶数并且 start color #1fab54, a, end color #1fab54, is greater than, 0start color #11accd, n, end color #11accd 为偶数并且 start color #1fab54, a, end color #1fab54, is less than, 0
x, right arrow, minus, infinity, f, left parenthesis, x, right parenthesis, right arrow, plus, infinity, 当 x, right arrow, plus, infinity, f, left parenthesis, x, right parenthesis, right arrow, plus, infinity.
x, right arrow, minus, infinity, f, left parenthesis, x, right parenthesis, right arrow, minus, infinity, 当 x, right arrow, plus, infinity, f, left parenthesis, x, right parenthesis, right arrow, minus, infinity, point
start color #11accd, n, end color #11accd为奇数并且 start color #1fab54, a, end color #1fab54, is greater than, 0start color #11accd, n, end color #11accd 为奇数并且 start color #1fab54, a, end color #1fab54, is less than, 0
x, right arrow, minus, infinity, f, left parenthesis, x, right parenthesis, right arrow, minus, infinity, 当 x, right arrow, plus, infinity, f, left parenthesis, x, right parenthesis, right arrow, plus, infinity.
x, right arrow, minus, infinity, f, left parenthesis, x, right parenthesis, right arrow, plus, infinity, 当 x, right arrow, plus, infinity, f, left parenthesis, x, right parenthesis, right arrow, minus, infinity, point

#### 检查你是否理解

6) g, left parenthesis, x, right parenthesis, equals, 8, x, cubed 的末端走势是什么?

### 多项式的末端走势

#### 检查你是否理解

7) f, left parenthesis, x, right parenthesis, equals, 8, x, start superscript, 5, end superscript, minus, 7, x, squared, plus, 10, x, minus, 1 的末端走势是什么?

8) g, left parenthesis, x, right parenthesis, equals, minus, 6, x, start superscript, 4, end superscript, plus, 8, x, cubed, plus, 4, x, squared 的末端走势是什么?

## 为什么首项决定了末端走势?

x 为大的正值时，让我们通过分析函数 g, left parenthesis, x, right parenthesis, equals, minus, 3, x, squared, plus, 7, x 来做更进一步的探索 .
x 趋近于 plus, infinity, 我们知道 minus, 3, x, squared 趋近于 minus, infinity 同时 7, x 趋近于 plus, infinity.

xminus, 3, x, squared7, xminus, 3, x, squared, plus, 7, x
1minus, 374
10minus, 30070minus, 230
100minus, 30, comma, 000700minus, 29, comma, 300
1000start color #ca337c, minus, 3, comma, 000, comma, 000, end color #ca337c7000start color #ca337c, minus, 2, comma, 993, comma, 000, end color #ca337c

xminus, 3, x, squared999, xminus, 3, x, squared, plus, 999, x
10minus, 3009, comma, 9909, comma, 690
100minus, 30, comma, 00099, comma, 90069, comma, 900
1000minus, 3, comma, 000, comma, 000999, comma, 000minus, 2, comma, 001, comma, 000
10, comma, 000start color #ca337c, minus, 300, comma, 000, comma, 000, end color #ca337c9, comma, 990, comma, 000start color #ca337c, minus, 290, comma, 010, comma, 000, end color #ca337c

## 挑战题

9*) 下面哪一项是 h, left parenthesis, x, right parenthesis, equals, minus, 8, x, cubed, plus, 7, x, minus, 1? 的图像

10*) g, left parenthesis, x, right parenthesis, equals, left parenthesis, 2, minus, 3, x, right parenthesis, left parenthesis, x, plus, 2, right parenthesis, squared 的末端走势是什么？