Course: 1.13 基于规则的职能图

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基于规则的函数图

Graphs of Functions based on Rules
::基于规则的函数图

Of course, we can always make a graph from a function rule by substituting values in for the variable and getting the corresponding output value.
::当然,我们总是可以用函数规则绘制图表,用变量的值替换变量的值,并获得相应的输出值。

Graphing a Function based on Rules
::根据规则绘制函数图

1. Graph the following function : $\begin{aligned} f (x) = | x - 2 | \end{aligned}$
::1. 图如下函数:f(x)x-2

Make a table of values. Pick a variety of negative and positive values for $\begin{aligned} x \end{aligned}$ . Use the function rule to find the value of $\begin{aligned} y \end{aligned}$ for each value of $\begin{aligned} x \end{aligned}$ . Then, graph each of the coordinate points.
::绘制数值表。为 x 选择各种负值和正值。使用函数规则查找 x 的每个值的y值。然后,绘制每个坐标点的图表。

$\begin{aligned} x \end{aligned}$	$\begin{aligned} y = f (x) = \| x - 2 \| \end{aligned}$
-4	$\begin{aligned} \| - 4 - 2 \| = \| - 6 \| = 6 \end{aligned}$
-3	$\begin{aligned} \| - 3 - 2 \| = \| - 5 \| = 5 \end{aligned}$
-2	$\begin{aligned} \| - 2 - 2 \| = \| - 4 \| = 4 \end{aligned}$
-1	$\begin{aligned} \| - 1 - 2 \| = \| - 3 \| = 3 \end{aligned}$
0	$\begin{aligned} ∣ 0 - 2 ∣=∣ - 2 ∣= 2 \end{aligned}$
1	$\begin{aligned} ∣ 1 - 2 ∣=∣ - 1 ∣= 1 \end{aligned}$
2	$\begin{aligned} ∣ 2 - 2 ∣=∣ 0 ∣= 0 \end{aligned}$
3	$\begin{aligned} ∣ 3 - 2 ∣=∣ 1 ∣= 1 \end{aligned}$
4	$\begin{aligned} ∣ 4 - 2 ∣=∣ 2 ∣= 2 \end{aligned}$
5	$\begin{aligned} ∣ 5 - 2 ∣=∣ 3 ∣= 3 \end{aligned}$
6	$\begin{aligned} ∣ 6 - 2 ∣=∣ 4 ∣= 4 \end{aligned}$
7	$\begin{aligned} ∣ 7 - 2 ∣=∣ 5 ∣= 5 \end{aligned}$
8	$\begin{aligned} ∣ 8 - 2 ∣=∣ 6 ∣= 6 \end{aligned}$

It is wise to work with a lot of values when you begin graphing. As you learn about different types of functions, you will start to only need a few points in the table of values to create an accurate graph.
::当您开始图形化时,使用许多数值开展工作是明智的。随着您了解不同类型的函数,你只需要在数值表中的几点即可创建准确的图表。

2. Graph the following function: $\begin{aligned} f (x) = \sqrt{x} \end{aligned}$
::2. 图如下函数:f(x)=x

Make a table of values. We know $\begin{aligned} x \end{aligned}$ can’t be negative because we can't take the square root of a negative number. The domain is all positive real numbers, so we pick a variety of positive integer values for $\begin{aligned} x \end{aligned}$ . Use the function rule to find the value of $\begin{aligned} y \end{aligned}$ for each value of $\begin{aligned} x \end{aligned}$ .
::绘制一个数值表。我们知道 x 不能是负数的平方根, 因为我们不能选择负数的平方根。域名都是正数, 所以我们为 x 选择了各种正整数。使用函数规则来查找 x 的每个值的 y 值。

$\begin{aligned} x \end{aligned}$	$\begin{aligned} y = f (x) = \sqrt{x} \end{aligned}$
0	$\begin{aligned} \sqrt{0} = 0 \end{aligned}$
1	$\begin{aligned} \sqrt{1} = 1 \end{aligned}$
2	$\begin{aligned} \sqrt{2} \approx 1.41 \end{aligned}$
3	$\begin{aligned} \sqrt{3} \approx 1.73 \end{aligned}$
4	$\begin{aligned} \sqrt{4} = 2 \end{aligned}$
5	$\begin{aligned} \sqrt{5} \approx 2.24 \end{aligned}$
6	$\begin{aligned} \sqrt{6} \approx 2.45 \end{aligned}$
7	$\begin{aligned} \sqrt{7} \approx 2.65 \end{aligned}$
8	$\begin{aligned} \sqrt{8} \approx 2.83 \end{aligned}$
9	$\begin{aligned} \sqrt{9} = 3 \end{aligned}$

Note that the range is all positive real numbers.
::注意这个范围都是正数

Real-World Application
::真实世界应用

The post office charges 41 cents to send a letter that is one ounce or less and an extra 17 cents for each additional ounce or fraction of an ounce. This rate applies to letters up to 3.5 ounces.
::邮局收取41美分,以寄出一盎司或以下的信件,每多寄一盎司或盎司的分数,再寄17美分。此费率适用于最多3.5盎司的字母。

Make a table of values. We can’t use negative numbers for $\begin{aligned} x \end{aligned}$ because it doesn’t make sense to have negative weight. We pick a variety of positive values for $\begin{aligned} x \end{aligned}$ , making sure to include some decimal values because prices can be decimals too. Then we use the function rule to find the value of $\begin{aligned} y \end{aligned}$ for each value of $\begin{aligned} x \end{aligned}$ .
::绘制数值表。我们无法对 x 使用负数, 因为负数是没有道理的。我们为 x 选择了各种正数, 以确保包含一些十进制值, 因为价格也可以是十进制值。然后我们使用函数规则来为 x 的每个值找到 y 值。

\begin{aligned} x 0 0.2 0.5 0.8 1 1.2 1.5 1.8 2 2.2 2.5 2.8 3 3.2 3.5 \\ y 0 41 41 41 41 58 58 58 58 75 75 75 75 92 92 \end{aligned}

::X00.20.20.50.811.21.21.51.822.22.52.833.23.55.ny041 41 414115858585858585858587575 75759292

Example
::示例示例示例示例

Example 1
::例1

Graph the following function: $\begin{aligned} f (x) = \sqrt{x^{2}} \end{aligned}$
::如下函数图示: f( x) =x2

Make a table of values. Even though $\begin{aligned} x \end{aligned}$ can’t be negative inside the square root, because we are squaring $\begin{aligned} x \end{aligned}$ first, the domain is all real numbers. So we integer values for $\begin{aligned} x \end{aligned}$ which are on either side of zero. Use the function rule to find the value of $\begin{aligned} y \end{aligned}$ for each value of $\begin{aligned} x \end{aligned}$ .
::绘制一个数值表。即使 x 在平方根中不能为负值, 因为我们先对 x 进行对比, 域是全部真实数字。因此, 我们将位于零的两边的 x 的数值整数。使用函数规则来查找 x 的每个值的 y 值。

$\begin{aligned} x \end{aligned}$	$\begin{aligned} y = f (x) = \sqrt{x^{2}} \end{aligned}$
-2	$\begin{aligned} \sqrt{(- 2)^{2}} = 2 \end{aligned}$
-1	$\begin{aligned} \sqrt{(- 1)^{2}} = 1 \end{aligned}$
0	$\begin{aligned} \sqrt{0^{2}} = 0 \end{aligned}$
1	$\begin{aligned} \sqrt{1^{2}} = 1 \end{aligned}$
2	$\begin{aligned} \sqrt{2^{2}} = 2 \end{aligned}$

Note that the range is all positive real numbers, and that this looks like an absolute value function.
::注意这个范围都是正实际数字, 这看起来像一个绝对值函数。

Review
::回顾

Graph the following functions.
::如下图所示函数。

Vanson spends $20 a month on his cat.
::Vanson每月花20美元在他的猫身上

Brandon is a member of a movie club. He pays a $50 annual membership and $8 per movie.
::布兰登是电影俱乐部的会员他每年支付50美元和每部电影8美元

$\begin{aligned} f (x) = (x - 2)^{2} \end{aligned}$
:xx)=(x-2)2

$\begin{aligned} f (x) = {3.2}^{x} \end{aligned}$
:xx)=3.2x

$\begin{aligned} f (t) = 27 t - t^{2} \end{aligned}$
:t)=27t-t2

$\begin{aligned} f (w) = \frac{w}{4} + 5 \end{aligned}$
::f(w)=w4+5

$\begin{aligned} f (x) = t + 2 t^{2} + 3 t^{3} \end{aligned}$
:x) =t+2t2+3t3

$\begin{aligned} f (x) = (x - 1) (x + 3) \end{aligned}$
:xx)=(x-1)(x+3)

$\begin{aligned} f (x) = \frac{x}{3} + \frac{x^{2}}{5} \end{aligned}$
:xx)=x3+x25

$\begin{aligned} f (x) = \sqrt{2 x} \end{aligned}$
:fx)=2x

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。

Section outline

General

基于规则的函数图

Graphs of Functions based on Rules ::基于规则的函数图

Graphing a Function based on Rules ::根据规则绘制函数图

Real-World Application ::真实世界应用

Example ::示例示例示例示例

Example 1 ::例1

Review ::回顾

Review (Answers) ::回顾(答复)