Section: 线线函数图 | 4.10 线性函数图

Section outline

Graphs of Linear Functions
::线线函数图

The highly exclusive Fellowship of the Green Mantle allows in only a limited number of new members a year. In its third year of membership it has 28 members, in its fourth year it has 33, and in its fifth year it has 38. How many members are admitted a year, and how many founding members were there?
::绿色Mantle的高度排他性研究金每年只允许数量有限的新成员,在成员构成的第三个年头,它有28个成员,在第四个年头有33个成员,在第五年有38个成员,每年有多少成员被接纳,有多少创始成员?

Functions
::职能职能职能职能职能职能职能职能职能职能职能职能职能职能职能

So far we’ve used the term function to describe many of the equations we’ve been graphing, but in mathematics it’s important to remember that not all equations are functions. In order to be a function, a relationship between two variables, $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ , must map each $\begin{aligned} x - \end{aligned}$ value to exactly one $\begin{aligned} y - \end{aligned}$ value.
::到目前为止,我们已经用术语函数来描述我们一直在绘制的许多方程式,但在数学中,重要的是要记住并非所有方程式都是函数。要成为函数,两个变量( x 和 y ) 之间的关系必须绘制每个 x - 值的图示,使其与一值完全一致。

Visually this means the graph of $\begin{aligned} y \end{aligned}$ versus $\begin{aligned} x \end{aligned}$ must pass the vertical line test , meaning that a vertical line drawn through the graph of the function must never intersect the graph in more than one place:
::从视觉上看,这意味着 y 和 x 的图形必须通过垂直线测试, 意指通过函数的图形绘制的垂直线绝不能在一个以上的地方交叉图形 :

Use Function Notation
::使用函数符号

When we write functions we often use the notation “ $\begin{aligned} f (x) = \end{aligned}$ ” in place of “ $\begin{aligned} y = \end{aligned}$ ”. $\begin{aligned} f (x) \end{aligned}$ is pronounced “ $\begin{aligned} f \end{aligned}$ of $\begin{aligned} x \end{aligned}$ ”.
::当我们写入函数时,我们经常使用“f(x)=”的符号,代替“y=”。 f(x)是“f of x”。

Writing Equations as Functions of $\begin{aligned} x \end{aligned}$
::x 函数的书写方形

Rewrite the following equations so that $\begin{aligned} y \end{aligned}$ is a function of $\begin{aligned} x \end{aligned}$ and is written $\begin{aligned} f (x) \end{aligned}$ :
::重写以下方程式,使 Y 是 x 的函数, 并写入 f( x) :

a) $\begin{aligned} y = 2 x + 5 \end{aligned}$
::a) y=2x+5

Simply replace $\begin{aligned} y \end{aligned}$ with $\begin{aligned} f (x) : f (x) = 2 x + 5 \end{aligned}$
::简单替换 y 为 f( x) f( x) = 2x+5

b) $\begin{aligned} y = - 0.2 x + 7 \end{aligned}$
:b) y=0.2x+7

Again, replace $\begin{aligned} y \end{aligned}$ with $\begin{aligned} f (x) : f (x) = - 0.2 x + 7 \end{aligned}$
::再次将 y 替换为 f( x): f( x) =======0. 2x+7

c) $\begin{aligned} x = 4 y - 5 \end{aligned}$
:c) x=4y-5

First we need to solve for $\begin{aligned} y \end{aligned}$ . Starting with $\begin{aligned} x = 4 y - 5 \end{aligned}$ , we add 5 to both sides to get $\begin{aligned} x + 5 = 4 y \end{aligned}$ , divide by 4 to get $\begin{aligned} \frac{x + 5}{4} = y \end{aligned}$ , and then replace $\begin{aligned} y \end{aligned}$ with $\begin{aligned} f (x) : f (x) = \frac{x + 5}{4} \end{aligned}$ .
::首先我们需要解决y。从 x=4y-5 开始, 我们向两边增加 5, 以获得 x+5=4, 以 4 除以 4 以获得 x+54 =y, 然后用 f( x): f( x) =x+54 替换 y 。

d) $\begin{aligned} 9 x + 3 y = 6 \end{aligned}$
:d) 9x+3y=6

Solve for $\begin{aligned} y : \end{aligned}$ take $\begin{aligned} 9 x + 3 y = 6 \end{aligned}$ , subtract $\begin{aligned} 9 x \end{aligned}$ from both sides to get $\begin{aligned} 3 y = 6 - 9 x \end{aligned}$ , divide by 3 to get $\begin{aligned} y = \frac{6 - 9 x}{3} = 2 - 3 x \end{aligned}$ , and express as a function: $\begin{aligned} f (x) = 2 - 3 x \end{aligned}$ .
::y: 使用 9x+3y=6, 减去9x, 从两边取 3y=6- 9x, 除以 3 以 y=6- 9x3=2- 3x, 以 y=6- 9x3=2- 3x 表达函数 : f(x)=2- 3x 。

Using the functional notation in an equation gives us more information. For instance, the expression $\begin{aligned} f (x) = m x + b \end{aligned}$ shows clearly that $\begin{aligned} x \end{aligned}$ is the independent variable because you plug in values of $\begin{aligned} x \end{aligned}$ into the function and perform a series of operations on the value of $\begin{aligned} x \end{aligned}$ in order to calculate the values of the dependent variable, $\begin{aligned} y \end{aligned}$ .
::在方程式中使用函数符号给我们提供更多信息。例如,表达式 f( x) =mx+b 清楚地显示, x 是独立的变量, 因为您在函数中插入 x 的值, 并在 x 值上进行一系列操作, 以便计算依赖变量的值, y 。

We can also plug in expressions rather than just numbers. For example, if our function is $\begin{aligned} f (x) = x + 2 \end{aligned}$ , we can plug in the expression $\begin{aligned} (x + 5) \end{aligned}$ . We would express this as $\begin{aligned} f (x + 5) = (x + 5) + 2 = x + 7 \end{aligned}$ .
::我们也可以插入表达式而非数字。例如, 如果我们的函数是 f( x) =x+2, 我们可以插入表达式 (x+5) 。我们将表示为 f( x+5) = (x+5) +2=x+7 。

Evaluating Functions
::评价职能

A function is defined as $\begin{aligned} f (x) = 6 x - 36 \end{aligned}$ . Evaluate the following:
::A 函数被定义为 f( x)=6x-36. 评价如下:

a) $\begin{aligned} f (2) \end{aligned}$
::a) f(2)

Substitute $\begin{aligned} x = 2 \end{aligned}$ into the function $\begin{aligned} f (x) : f (2) = 6 \cdot 2 - 36 = 12 - 36 = - 24 \end{aligned}$
::函数 f( x) 的替代 x=2 : f(2)= 62-36= 12- 36 = 24

b) $\begin{aligned} f (0) \end{aligned}$
::b) f(0)

Substitute $\begin{aligned} x = 0 \end{aligned}$ into the function $\begin{aligned} f (x) : f (0) = 6 \cdot 0 - 36 = 0 - 36 = - 36 \end{aligned}$
::函数 f( x) 的替代 x=0 : f( 0) = 60 - 36= 0- 36 =36

c) $\begin{aligned} f (z) \end{aligned}$
::c) f(z)

Substitute $\begin{aligned} x = z \end{aligned}$ into the function $\begin{aligned} f (x) : f (z) = 6 z + 36 \end{aligned}$
::函数 f(x): f(z)=6z+36中的替代 x=z

d) $\begin{aligned} f (x + 3) \end{aligned}$
:d) f(x+3)

Substitute $\begin{aligned} x = (x + 3) \end{aligned}$ into the function $\begin{aligned} f (x) : f (x + 3) = 6 (x + 3) + 36 = 6 x + 18 + 36 = 6 x + 54 \end{aligned}$
::函数 f( x) 的替代 x= (x+3) : f( x+3) = 6( x+3) +36= 6x+18+36=6x+54

e) $\begin{aligned} f (2 r - 1) \end{aligned}$
:e) f(2r-1)

Substitute $\begin{aligned} x = (2 r + 1) \end{aligned}$ into the function $\begin{aligned} f (x) : f (2 r + 1) = 6 (2 r + 1) + 36 = 12 r + 6 + 36 = 12 r + 42 \end{aligned}$
::函数 f( x) 的替代 x= (2r+1) = f( x): f( 2r+1) = 6( 2r+1) +36= 12r+6+36= 12r+42

Graphing a Linear Function
::绘制线性函数图

Since the notations “ $\begin{aligned} f (x) = \end{aligned}$ ” and “ $\begin{aligned} y = \end{aligned}$ ” are interchangeable, we can use all the concepts we have learned so far to graph functions.
::由于“f(x)=”和“y=”两词可以互换,我们可以使用我们迄今学到的所有概念来绘制函数图。

Graph the function $\begin{aligned} f (x) = \frac{3 x + 5}{4} \end{aligned}$ .
::函数 f( x) = 3x+54 的图形。

We can write this function in slope- intercept form:
::我们可以以斜坡界面的形式写入此函数 :

\begin{aligned} f (x) = \frac{3}{4} x + \frac{5}{4} = 0.75 x + 1.25 \end{aligned}

::f(x)=34x+54=0.75x+1.25

So our graph will have a $\begin{aligned} y - \end{aligned}$ intercept of (0, 1.25) and a of 0.75.
::因此,我们的图表将有一个y- 截取器( 0, 1. 25) 和一个 0. 75 。

Arithmetic Progressions
::相对进量

You may have noticed that with , when you increase the $\begin{aligned} x - \end{aligned}$ value by 1 unit, the $\begin{aligned} y - \end{aligned}$ value increases by a fixed amount, equal to the slope. For example, if we were to make a table of values for the function $\begin{aligned} f (x) = 2 x + 3 \end{aligned}$ , we might start at $\begin{aligned} x = 0 \end{aligned}$ and then add 1 to $\begin{aligned} x \end{aligned}$ for each row:
::您可能注意到, 当您将 x - 值增加 1 个单位时, y - 值增加一个固定的值, 等于斜度。例如, 如果我们为函数 f( x) = 2x+ 3 绘制一个数值表, 我们可以从 x=0 开始, 然后为每行添加 1 到 x :

$\begin{aligned} x \end{aligned}$	$\begin{aligned} f (x) \end{aligned}$
0	3
1	5
2	7
3	9
4	11

Notice that the values for $\begin{aligned} f (x) \end{aligned}$ go up by 2 (the slope) each time. When we repeatedly add a fixed value to a starting number, we get a sequence like {3, 5, 7, 9, 11....}. We call this an arithmetic progression , and it is characterized by the fact that each number is bigger (or smaller) than the preceding number by a fixed amount. This amount is called the common difference . We can find the common difference for a given sequence by taking 2 consecutive terms in the sequence and subtracting the first from the second.
::注意 f( x) 的值每次都会上升 2 ( 斜度) 。当我们反复将固定值添加到起始数时, 我们得到的顺序类似 { 3, 5, 7, 9, 9, 11....}。我们称之为算术过程, 其特点是每个数字比前一个数字大( 或小) , 加上一个固定数额。这个数额被称为共同差。我们可以在序列中连续两个条件, 从第二个顺序中减去第一个条件, 找到给定序列的共同差。

Finding Common Differences
::寻找共同差异

Find the common difference for the following arithmetic progressions:
::查找下列算术进度的常见差 :

a) {7, 11, 15, 19, ...}
::a) {7、11、15、19、......}

$\begin{aligned} 11 - 7 = 4; 15 - 11 = 4; 19 - 15 = 4 \end{aligned}$ . The common difference is 4 .
::11-7=4;15-11=4;19-15=4,共同差为4。

b) {12. 1, -10, -21}
:b) {12.1,1-10,-21}

$\begin{aligned} 1 - 12 = - 11 \end{aligned}$ . The common difference is -11 .
::1-1211. 共同的区别是-11。

c) {7, __, 12, __. 17, ...}
:c) {7,___,12,___。 17,___}

There are not 2 consecutive terms here, but we know that to get the term after 7 we would add the common difference, and then to get to 12 we would add the common difference again. So twice the common difference is $\begin{aligned} 12 - 7 = 5 \end{aligned}$ , and so the common difference is 2.5.
::这里没有连续两个任期,但我们知道,为了在7年后获得任期,我们将增加共同差数,然后再增加12个差数。因此,共同差数的两倍是12-7=5,共同差数是2.5。

Arithmetic sequences and linear functions are very closely related. To get to the next term in a arithmetic sequence, you add the common difference to the last term; similarly, when the $\begin{aligned} x - \end{aligned}$ value of a linear function increases by one, the $\begin{aligned} y - \end{aligned}$ value increases by the amount of the slope . So arithmetic sequences are very much like linear functions, with the common difference playing the same role as the slope.
::亚学序列和线性函数非常密切相关。要在算术序列中进入下一个术语, 您可以在最后一个术语中添加共同差值; 同样, 当线性函数的 x - 值增加一个, y - 值增加一个斜度。所以算术序列与线性函数非常相似, 共同差的作用与斜度相同。

The graph below shows the arithmetic progression {-2, 0, 2, 4, 6...} along with the function $\begin{aligned} y = 2 x - 4 \end{aligned}$ . The only major difference between the two graphs is that an arithmetic sequence is discrete while a linear function is continuous .
::下图显示算术进度 {-2, 0, 2, 4, 6...} 以及函数 y=2x-4。两个图表之间唯一的主要区别是算术序列是离散的, 而线性函数是连续的。

We can write a formula for an arithmetic progression: if we define the first term as $\begin{aligned} a_{1} \end{aligned}$ and $\begin{aligned} d \end{aligned}$ as the common difference, then the other terms are as follows:
::我们可以为算术进展写一个公式:如果我们将第一个术语定义为a1和d是共同差,那么其他术语如下:

\begin{aligned} a_{1} a_{2} a_{3} a_{4} a_{5} a_{n} \\ a_{1} a_{1} + d a_{1} + 2 d a_{1} + 3 d a_{1} + 4 d \dots a_{1} + (n - 1) \cdot d \end{aligned}

::a1a2a3a4 a5ana1a1+da1+2da1+3da1+4d...a1+(n-1)__d

Example
::示例示例示例示例

Example 1
::例1

Graph the function $\begin{aligned} f (x) = \frac{7 (5 - x)}{5} \end{aligned}$ .
::函数 f( x) = 7( 5- x) 5 的图形。

This time we’ll solve for the $\begin{aligned} x - \end{aligned}$ and $\begin{aligned} y - \end{aligned}$ intercepts .
::这次我们会解决X-和y-interviews的问题。

To solve for the $\begin{aligned} y - \end{aligned}$ intercept, plug in $\begin{aligned} x = 0 : f (0) = \frac{7 (5 - 0)}{5} = \frac{35}{5} = 7 \end{aligned}$ , so the $\begin{aligned} x - \end{aligned}$ intercept is (0, 7) .
::要解决 y- intercept, 插入 x= 0: f( 0) = 7( 5-0) 5= 355= 7, 所以 x- intercept is (0, 7) 。

To solve for the $\begin{aligned} x - \end{aligned}$ intercept, set $\begin{aligned} f (x) = 0 : 0 = \frac{7 (5 - x)}{5} \end{aligned}$ , so $\begin{aligned} 0 = 35 - 7 x \end{aligned}$ , therefore $\begin{aligned} 7 x = 35 \end{aligned}$ and $\begin{aligned} x = 5 \end{aligned}$ . The $\begin{aligned} y - \end{aligned}$ intercept is (5, 0) .
::要解析 x- intercept, 设置 f( x) = 0: 0= 7( 5- x) 5, 所以 0= 35- 7x, 所以 7x= 35 和 x= 5。 y- intercept is (5, 0) 。

We can graph the function from those two points:
::我们可以用这两点来图解函数 :

Review
::回顾

When an object falls under gravity, it gains speed at a constant rate of 9.8 m/s every second. An item dropped from the top of the Eiffel Tower, which is 300 meters tall, takes 7.8 seconds to hit the ground. How fast is it moving on impact?
::当物体处于重力之下时, 速度会以每秒9. 8米/秒的恒定速率增速。从埃菲尔铁塔顶部( 高度300米) 掉下来的物品, 需要7. 8秒才能撞击地面。撞击速度有多快 ?

A prepaid phone card comes with $20 worth of calls on it. Calls cost a flat rate of $0.16 per minute.
1. Write the value left on the card as a function of minutes used so far.
  ::将卡片上留下的值作为到目前为止使用的分钟函数写入。
2. Use the function to determine how many minutes of calls you can make with the card.
  ::使用此函数来确定您可以用卡打多少分钟电话。
::预付电话卡的通话价值为20美元。通话费用为每分钟0. 16美元的统一费率。将卡片上留下的值作为目前所用分钟的函数写入。使用此功能来决定您可以用卡片打多少分钟电话。

For questions 3-5, evaluate the function for $\begin{aligned} f (- 3) \end{aligned}$ , $\begin{aligned} f (0) \end{aligned}$ , $\begin{aligned} f (z) \end{aligned}$ , $\begin{aligned} f (x + 3) \end{aligned}$ , $\begin{aligned} f (2 n) \end{aligned}$ , $\begin{aligned} f (3 y + 8) \end{aligned}$ , and $\begin{aligned} f (\frac{q}{2}) \end{aligned}$ .
::对于问题3-5,评价f(-3)、f(0)、f(z)、f(x+3)、f(2n)、f(3y+8)和f(q2)的功能。

$\begin{aligned} f (x) = - 2 x + 3 \end{aligned}$
:xx)%%2x+3)

$\begin{aligned} f (x) = 0.7 x + 3.2 \end{aligned}$
::f(x) = 0. 7x+3.2

$\begin{aligned} f (x) = \frac{5 (2 - x)}{11} \end{aligned}$
:xx)=5(2-x)11

For questions 6-9, determine whether the graph could be a function.
::对于问题6-9,请确定图表是否是一个函数。

The roasting guide for a turkey suggests cooking for 100 minutes plus an additional 8 minutes per pound.
1. Write a function for the roasting time the given the turkey weight in pounds $\begin{aligned} (x) \end{aligned}$ .
  ::给定火鸡重量( 磅) (x) 的烧烤时间写一个函数。
2. Determine the time needed to roast a 10 lb turkey.
  ::确定烤10磅火鸡所需的时间。
3. Determine the time needed to roast a 27 lb turkey.
  ::确定烤27磅火鸡所需的时间。
4. Determine the maximum size turkey you could roast in 4.5 hours.
  ::确定在4.5小时内可以烤的火鸡最大尺寸。
::火鸡的烤指南建议烹饪100分钟,外加每磅8分钟。写一个给定火鸡重量(磅)的烤时间函数。确定烤10磅火鸡所需的时间。确定烤27磅火鸡所需的时间。确定可在4.5小时内烤的最大火鸡尺寸。

For questions 11-13, determine the missing terms in the following arithmetic progressions.
::对于问题11-13,在下列算术进展中确定缺失的术语。

{-11, 17, __, 73}
{2, __, -4}
{13, __, __, __, 0}

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

Graphs of Linear Functions ::线线函数图

Functions ::职能职能职能职能职能职能职能职能职能职能职能职能职能职能职能

Use Function Notation ::使用函数符号

Writing Equations as Functions of x ::x 函数的书写方形

Evaluating Functions ::评价职能

Graphing a Linear Function ::绘制线性函数图

Arithmetic Progressions ::相对进量

Finding Common Differences ::寻找共同差异

Example ::示例示例示例示例

Example 1 ::例1

Review ::回顾

Review (Answers) ::回顾(答复)