写入线性函数

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Writing Linear Functions
::写入线性函数

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To write a linear function , you need two pieces of information: the and the y-intercept . Once you have determined these two variables, you can substitute them in for $\begin{array}{r}m\end{array}$  and $\begin{array}{r}b\end{array}$  in the slope-intercept form $\begin{array}{r}y=mx+b\end{array}$ . As you saw in the previous section, slope intercept form can be written as a function by replacing the $\begin{array}{r}y\end{array}$  with an $\begin{array}{r}f(x).\end{array}$  There are many situations in which you might need to write a linear function.  You may need to write a function from a graph, write a function using the slope and y-intercept, and write a function given several input/output pairs.  You will be exploring all three of these situations over the course of this section.
::要写入线性函数, 您需要两个信息块: 和 y 界面。 一旦您确定了这两个变量, 您可以在 y=mx+b 中将其替换为 m 和 b 。 正如您在上一节中看到的那样, 斜度截停形式可以用 f(x) 替换 y 来写成函数。 在很多情况下, 您可能需要用 y(x) 来写入线性函数。 您可能需要从图表中写一个函数, 使用 斜度和 y 界面写一个函数, 并写入一个函数, 给多个输入/ 输出对来写一个函数 。 您将会在本节中探索所有三种情况 。

 

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Writing Linear Functions From a Graph
::从图表写入线性函数

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In Linear Relationships , you practiced identifying the slope and y-intercept of a linear relationship displayed in a graph.  You will use these skills to write a linear equation . Keep in mind that this will only work for writing a linear function, which you know will appear as a straight and non- vertical line on the graph.
::在线性关系中,您练习了在图形中显示的线性关系的斜度和 Y 界面。您将使用这些技能来写出线性方程。 记住, 这只会用于写入线性函数, 您知道它会作为直线和非垂直线出现在图形中 。

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1. Find the slope by measuring the rise and run of the linear function.
   ::通过测量线性函数的上升和运行来查找斜坡 。
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2. Find the y-intercept by identifying where the function crosses the y-intercept.
   ::查找 Y 界面,通过识别函数交叉到 y 界面的位置来查找 y 界面 。
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3. Substitute the slope and y-intercept into the function  $\begin{array}{r}f(x)=mx+b\end{array}$
   ::在函数 f(x) =mx+b 中替换斜度和 Y 界面

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Example
::示例示例示例示例

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Write a linear function equation to model the function displayed in the graph below:
::写入线性函数方程式以模拟下图中显示的函数:

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For steps 1 and 2, you can determine the slope and y-intercept directly from the equation.
::对于步骤1和步骤2,您可以从方程中直接确定斜坡和 Y 接口。

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You  know that the y-intercept is 2 and the slope is ½. From here  you just have to follow step 2 and substitute these into the function equation  to obtain our answer.
::你知道y-interview是2, 斜坡是1⁄2, 从这里你只需要遵循第2步, 取而代之, 取而代之, 以获得我们的答案。

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Answer:  $\begin{array}{r}f(x)=\frac{1}{2}x+2\end{array}$
::答复:f(x)=12x+2

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Use the interactive below to explore this idea further.
::利用以下互动方式进一步探讨这一想法。

 

 

 

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Writing Linear Functions With the Slope and an Ordered Pair
::书写有斜坡和有顺序对等线性函数的线性函数

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I f we are given the slope and an ordered pair , then we are given half the information. We already have the slope, but we need to figure out the y-intercept. We can use the slope- intercept form  to figure this out by using the following steps:
::如果我们得到斜坡和一对定货,那么我们就会得到一半的信息。我们已经有了斜坡,但我们需要弄清楚Y界面。我们可以使用斜坡界面来用以下步骤来解决这个问题:

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1. Find the y-intercept by substituting the slope and ordered pair (input, output) into $\begin{array}{r}f(x)=mx+b\end{array}$  and solving for  $\begin{array}{r}b.\end{array}$  
   ::通过将斜坡和定购配对(投入、输出)替换成 f(x)=mx+b 并解决 b 来查找 Y 界面。
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2. Substitute the slope and y-intercept into the function  $\begin{array}{r}f(x)=mx+b.\end{array}$  
   ::在函数 f(x) =mx+b 中替换斜度和 Y 界面。

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Example
::示例示例示例示例

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Write an equation for the linear function, $\begin{array}{r}f(x),\end{array}$  with a slope of  $\begin{array}{r}\frac{3}{2}\end{array}$  and given that  $\begin{array}{r}f(-4)=5.\end{array}$
::f(x) 线性函数写公式, f(x), 斜度为32, 且考虑到 f(- 4) = 5 。

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1. Find the y-intercept by substituting the slope and ordered pair (input, output) into $\begin{array}{r}f(x)=mx+b\end{array}$  and solving for  $\begin{array}{r}b.\end{array}$  
   ::通过将斜坡和定购配对(投入、输出)替换成 f(x)=mx+b 并解决 b 来查找 Y 界面。

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The slope is $\begin{array}{r}\frac{3}{2}\end{array}$  . The statement $\begin{array}{r}f(-4)=5\end{array}$  represents the ordered pair (-4, 5). The input value is -4 and the output value is 5. When you substitute this into  $\begin{array}{r}f(x)=mx+b\end{array}$ you get $\begin{array}{r}5=-4(\frac{3}{2})+b\end{array}$
::斜度为 32 。 语句 f(- 4)=5 表示定购对( 4) 。 输入值为 - 4, 输出值为 5 。 当您将其替换为 f( x) =mx+b 时, 您可以得到 5 @ 4( 32)+b 。

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$\begin{array}{rl}f(x)& =mx+b\\ 5& =-4(\frac{3}{2})+b\\ 5& =-6+b\\ 11& =b\end{array}$
::f(x) =mx+b5*4(32)+b5=6+b11=b

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The y-intercept is 11.
::y 界面是11号

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2. Substitute the slope and y-intercept into the function $\begin{array}{r}f(x)=mx+b\end{array}$  
   ::在函数 f(x) =mx+b 中替换斜度和 Y 界面

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Answer:  $\begin{array}{r}f(x)=\frac{3}{2}x+11\end{array}$
::答复:f(x)=32x+11

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Use the interactive below to write the function of a linear equation given the slope and a point.
::使用下面的交互式线性方程式写入线性方程式的函数, 以给定斜度和点数 。

 

 

 

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Writing Linear Functions With Two Ordered Pairs
::具有两个有顺序对等的书写线性函数

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If you are given two ordered pairs from the function, then you have neither the slope nor the y-intercept.  In order to write an equation, you will need to use the steps below:
::如果从函数中给您两对顺序,那么您既无斜坡,也无 Y 干扰。为了写一个方程,您需要使用以下步骤:

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1. Use the two ordered pairs to find the slope using the formula $\begin{array}{r}m=\frac{y_2-y_1}{x_2-x_1}\end{array}$ .
   ::使用公式 m=y2-y1x2-x1 查找斜坡。
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2. Find the y-intercept by substituting the slope and one of the ordered pairs into $\begin{array}{r}f(x)=mx+b\end{array}$  and solving for  $\begin{array}{r}b.\end{array}$  
   ::以 f( x) =mx+b 替换斜坡和定单对中的一对并解决 b 来查找 Y 界面 。
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3. Substitute the slope and y-intercept into the function $\begin{array}{r}f(x)=mx+b\end{array}$ .
   ::在函数 f(x) =mx+b 中替换斜度和 Y 界面。

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Example
::示例示例示例示例

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Write an equation for the linear function, $\begin{array}{r}f(x),\end{array}$  given the ordered pairs (-2,6) and (4,0)
::写入线性函数的公式 f( x), 以给定对对数( 2, 6) 和 ( 4,0) 为条件

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1. Use the two ordered pairs to find the slope using the formula $\begin{array}{r}m=\frac{y_2-y_1}{x_2-x_1}\end{array}$
   ::使用公式 m=y2-y1x2-x1 查找斜度

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$\begin{array}{r}m=\frac{y_2-y_1}{x_2-x_1}=\frac{0-6}{4-(-2)}=\frac{-6}{6}=-1\end{array}$
::my2-y1x2-x1=0-64-(-2)________________________________________________________________________________________________

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2. Find the y-intercept by substituting the slope and an ordered pair (input, output) into $\begin{array}{r}f(x)=mx+b\end{array}$  and solving for  $\begin{array}{r}b.\end{array}$  
   ::以 f( x) =mx+b 替换斜坡和定购对( 输入、 输出) 来查找 Y 界面,并解决 b 。

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The  pair  (4,0) is easier to substitute since the numbers are simpler.
::配对(4,0)比较容易替代,因为数字比较简单。

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$\begin{array}{rl}f(x)& =mx+b\\ 0& =4(-1)+b\\ 0& =-4+b\\ 4& =b\end{array}$
::f(x) =mx+b0=4(- 1)+b0 @%4+b4=b

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The y-intercept is 4.
::y 界面为 4 。

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3. Substitute the slope and y-intercept into the function $\begin{array}{r}f(x)=mx+b\end{array}$ .
   ::在函数 f(x) =mx+b 中替换斜度和 Y 界面。

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Answer:  $\begin{array}{r}f(x)=-4x+4\end{array}$
::答复:f(x)4x+4

 

Summary

播放段落 Linear functions are in the form f(x)=mx+b. ::线性函数在 f(x) = mx+b 表中。 播放段落 To find the equation of a linear function from its graph: 播放段落 Find the slope by finding the rise and run. ::寻找斜坡 找到上升和运行。 播放段落 Find the y-intercept (the point where the line crosses the y-axis) ::查找 Y 界面( 线横跨 Y 轴的点) 播放段落 Substitute the value into the function f(x)=mx+b ::将值替换为函数 f( x) =mx+b ::从图表中查找线性函数的方程式 : 通过查找上升并运行来查找斜度 。 查找 y 界面( 线条横跨 Y 轴的点) , 将值替换到函数 f( x) = mx+b 中 。