Course: 5.1 确定一条线的平方

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确定一条线的平方
Determining the Equation of a Line
::确定一条线的平方

You are probably aware that many real-world situations can be described with linear graphs and equations. In this lesson, we’ll see how to find equations in a variety of situations.
::您可能知道许多真实世界的情况可以用线性图表和方程式来描述。在这个教训中,我们会看到如何在各种情况下找到方程式。

Write an Equation Given Slope and $\begin{aligned} y - \end{aligned}$ Intercept
::撰写一个有斜坡和y- 截取的方程式

Recall that you may write an equation in slope– intercept form with a few simple steps: start with the general equation for the slope-intercept form of a line, $\begin{aligned} y = m x + b \end{aligned}$ , and then substitute the given values of $\begin{aligned} m \end{aligned}$ and $\begin{aligned} b \end{aligned}$ into the equation. For example, a line with a of 4 and a $\begin{aligned} y - \end{aligned}$ intercept of –3 would have the equation $\begin{aligned} y = 4 x - 3 \end{aligned}$ .
::提醒您可以用一些简单的步骤在斜坡- interphysict 格式中写入一个方程式 : 以 y=mx+b 线的斜坡- interview 格式的通用方程开始, 然后将 m 和 b 的给定值替换到方程。例如, 以 y= 4 和 y- interview of - 3 的直线将具有 y= 4x-3 的等式。

If you are given just the graph of a line, you can identify the slope and $\begin{aligned} y - \end{aligned}$ intercept from the graph and write the equation from there. For example, on the graph below you can see that the line rises by 1 unit as it moves 2 units to the right, so its slope is $\begin{aligned} \frac{1}{2} \end{aligned}$ . Also, you can see that the $\begin{aligned} y - \end{aligned}$ intercept is –2, so the equation of the line is $\begin{aligned} y = \frac{1}{2} x - 2 \end{aligned}$ .
::如果只给出一条线的图形,您可以从图形中识别斜度和 y - intercept , 并从中写出方程。例如,在下图中,您可以看到线在向右移动2 个单位时上升1 个单位, 因此其斜度是 12 。另外,您可以看到 y - intercut is - 2, 所以线的方程式是 y= 12x-2 。

Write an Equation Given the Slope and a Point
::写一个基于斜坡和点的等量

Often, we don’t know the value of the $\begin{aligned} y - \end{aligned}$ intercept, but we know the value of $\begin{aligned} y \end{aligned}$ for a non-zero value of $\begin{aligned} x \end{aligned}$ . In this case, it’s often easier to write an equation of the line in point-slope form. An equation in point-slope form is written as $\begin{aligned} y - y_{0} = m (x - x_{0}) \end{aligned}$ , where $\begin{aligned} m \end{aligned}$ is the slope and $\begin{aligned} (x_{0}, y_{0}) \end{aligned}$ is a point on the line.
::通常,我们不知道y- intercept的价值,但我们知道y对于x的非零值的值。在这种情况下,以点- slope 形式写出线的方程式往往比较容易。点- slope 形式的方程式是y-y0=m(x-xx0), m 是斜度, (x0,y0) 是线上的点。

Writing the Equation of a Line in Point-Slope Form
::以点-斜体窗体写入线条的公式

A line has a slope of $\begin{aligned} \frac{3}{5} \end{aligned}$ , and the point (2, 6) is on the line. Write the equation of the line in point-slope form.
::线的斜坡为35, 点( 2, 6) 在线上。以点倾斜形式写入线的方程。

Start with the formula $\begin{aligned} y - y_{0} = m (x - x_{0}) \end{aligned}$ .
::以公式 y-y0=m(x-x0) 开始。

Plug in $\begin{aligned} \frac{3}{5} \end{aligned}$ for $\begin{aligned} m \end{aligned}$ , 2 for $\begin{aligned} x_{0} \end{aligned}$ and 6 for $\begin{aligned} y_{0} \end{aligned}$ .
::30米加35,x0加2,y0加6。

The equation in point-slope form is $\begin{aligned} y - 6 = \frac{3}{5} (x - 2) \end{aligned}$ .
::以点-斜体形式 isy-6=35(x-2)的方程。

Notice that the equation in point-slope form is not solved for $\begin{aligned} y \end{aligned}$ . If we did solve it for $\begin{aligned} y \end{aligned}$ , we’d have it in $\begin{aligned} y - \end{aligned}$ intercept form. To do that, we would just need to distribute the $\begin{aligned} \frac{3}{5} \end{aligned}$ and add 6 to both sides. That means that the equation of this line in slope-intercept form is $\begin{aligned} y = \frac{3}{5} x - \frac{6}{5} + 6 \end{aligned}$ , or simply $\begin{aligned} y = \frac{3}{5} x + \frac{24}{5} \end{aligned}$ .
::请注意, 点窗体的等式对于y来说没有解答。如果我们为y解决了它, 我们就会用y- interphy的形式解决它。要做到这一点, 我们只需要向两边分配 35 和 6 。这意味着, 斜坡- intercept 格式中的这条线的等式是 y= 35x- 65+6 或 y= 35x+245 。

Write an Equation Given Two Points
::写出基于两点的等式

Point-slope form also comes in useful when we need to find an equation given just two points on a line.
::当我们需要找到一个公式时,当我们只需要在一条线上两点时,点窗体也是有用的。

For example, suppose we are told that the line passes through the points (-2, 3) and (5, 2). To find the equation of the line, we can start by finding the slope.
::例如,假设我们被告知该线穿过点(2、3)和点(5.2)。为了找到线的方程式,我们可以从找到斜坡开始。

Starting with the slope formula, $\begin{aligned} m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} \end{aligned}$ , we plug in the $\begin{aligned} x - \end{aligned}$ and $\begin{aligned} y - \end{aligned}$ values of the two points to get $\begin{aligned} m = \frac{2 - 3}{5 - (- 2)} = \frac{- 1}{7} \end{aligned}$ . We can plug that value of $\begin{aligned} m \end{aligned}$ into the point-slope formula to get $\begin{aligned} y - y_{0} = - \frac{1}{7} (x - x_{0}) \end{aligned}$ .
::从斜坡公式 m=y2-y1x2-x1 开始,我们插入两个点的 x- 和 y- 值,以获得 m=2- 35- (- 2)\\\ 17。我们可以将 m 值插入点斜坡公式,以获得 y- y0_ 17(x- xx0) 。

Now we just need to pick one of the two points to plug into the formula. Let’s use (5, 2); that gives us $\begin{aligned} y - 2 = - \frac{1}{7} (x - 5) \end{aligned}$ .
::现在我们只需要选择其中的一个点来插入公式。让我们使用( 5, 2 ) ; 这让我们有了 y-217(x)-5 ) 。

What if we’d picked the other point instead? Then we’d have ended up with the equation $\begin{aligned} y - 3 = - \frac{1}{7} (x + 2) \end{aligned}$ , which doesn’t look the same. That’s because there’s more than one way to write an equation for a given line in point-slope form. But let’s see what happens if we solve each of those equations for $\begin{aligned} y \end{aligned}$ .
::如果我们选择了另一个点呢? 那么我们最终会选择一个看起来不一样的等式 y - 3 17 (x+2 ) 。这是因为有不止一种方法可以用点倾斜形式写出给定线的等式。但是让我们看看如果我们为y 解决了这些等式中的每一个等式,会发生什么。

Starting with $\begin{aligned} y - 2 = - \frac{1}{7} (x - 5) \end{aligned}$ , we distribute the $\begin{aligned} - \frac{1}{7} \end{aligned}$ and add 2 to both sides. That gives us $\begin{aligned} y = - \frac{1}{7} x + \frac{5}{7} + 2 \end{aligned}$ , or $\begin{aligned} y = - \frac{1}{7} x + \frac{19}{7} \end{aligned}$ .
::从 y - 2 17 (x- 5) 开始, 我们分发 ~ 17 和向双方添加 2。这让我们有 y 17x+ 57+2 或 y 17x+ 197 。

On the other hand, if we start with $\begin{aligned} y - 3 = - \frac{1}{7} (x + 2) \end{aligned}$ , we need to distribute the $\begin{aligned} - \frac{1}{7} \end{aligned}$ and add 3 to both sides. That gives us $\begin{aligned} y = - \frac{1}{7} x - \frac{2}{7} + 3 \end{aligned}$ , which also simplifies to $\begin{aligned} y = - \frac{1}{7} x + \frac{19}{7} \end{aligned}$ .
::另一方面,如果我们从y - 317(x+2)开始,我们需要向双方分发 ~ 17, 并增加 3。这使得我们得到 y 17x- 27+3, 也简化到 y 17x+197 。

So whichever point we choose to get an equation in point-slope form, the equation is still mathematically the same, and we can see this when we convert it to $\begin{aligned} y - \end{aligned}$ intercept form.
::所以无论我们选择什么点-斜方形的方程式, 方程式在数学上还是一样, 当我们把它转换成 y- 截取形式时, 我们可以看到这一点。

Writing an Equation in $\begin{aligned} y - \end{aligned}$ intercept form
::以 y- intercy 形式写入方程式

A line contains the points (3, 2) and (-2, 4). Write an equation for the line in point-slope form; then write an equation in $\begin{aligned} y - \end{aligned}$ intercept form.
::A 行包含点(3, 2) 和 (2, 4) 。用点- splope 格式写出线的方程式; 然后用 y- inter- intercept 格式写出方程式。

Find the slope of the line: $\begin{aligned} m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{4 - 2}{- 2 - 3} = - \frac{2}{5} \end{aligned}$
::查找线的斜坡: m=y2-y1x2-x1=4-2-2-325

Plug in the value of the slope: $\begin{aligned} y - y_{0} = - \frac{2}{5} (x - x_{0}) \end{aligned}$ .
::斜坡值中的插件:y-y025(x-x0)。

Plug point (3, 2) into the equation: $\begin{aligned} y - 2 = - \frac{2}{5} (x - 3) \end{aligned}$ .
::等式中的插点(3, 2) : y-225(x-3) 。

The equation in point-slope form is $\begin{aligned} y - 2 = - \frac{2}{5} (x - 3) \end{aligned}$ .
::以点-斜体形式 isy-225(x-3)的方程。

To convert to $\begin{aligned} y - \end{aligned}$ intercept form, simply solve for $\begin{aligned} y \end{aligned}$ :
::要转换为 y - 界面窗体,只需为 y 求解 :

$\begin{aligned} y - 2 = - \frac{2}{5} (x - 3) & \to y - 2 = - \frac{2}{5} x + \frac{6}{5} \\ \to y = - \frac{2}{5} x + \frac{6}{5} + 2 \\ \to y = - \frac{2}{5} x + 3 \frac{1}{5} . \end{aligned}$

::y2% 25(x- 3) y2% 25x+65y 25x+65+2y}25x+315。

The equation in $\begin{aligned} y - \end{aligned}$ intercept form is $\begin{aligned} y = - \frac{2}{5} x + 3 \frac{1}{5} \end{aligned}$ .
::y- 界面形式的方程式是 y25x+315。

Graph an Equation in Point-Slope Form
::a 点窗体的方形图形方形

Another useful thing about point-slope form is that you can use it to graph an equation without having to convert it to slope-intercept form. From the equation $\begin{aligned} y - y_{0} = m (x - x_{0}) \end{aligned}$ , you can just read off the slope $\begin{aligned} m \end{aligned}$ and the point $\begin{aligned} (x_{0}, y_{0}) \end{aligned}$ . To draw the graph, all you have to do is plot the point, and then use the slope to figure out how many units up and over you should move to find another point on the line.
::点窗体的另一个有用之处是,您可以使用它来绘制方程式,而不必将其转换为斜度截面。从公式 y-y0=m(x-x0) 中,您可以读取斜度 m 和点 (x0. y0) 。要绘制图形,您只需要绘制点,然后用斜度来确定您上下移动多少个单位来找到线上的另一个点。

Graphing an Equation of a Line
::绘制线条的方形

Make a graph of the line given by the equation $\begin{aligned} y + 2 = \frac{2}{3} (x - 2) \end{aligned}$ .
::绘制公式y+2=23(x-2)给出的线条的图形。

To read off the right values, we need to rewrite the equation slightly: $\begin{aligned} y - (- 2) = \frac{2}{3} (x - 2) \end{aligned}$ . Now we see that point (2, -2) is on the line and that the slope is $\begin{aligned} \frac{2}{3} \end{aligned}$ .
::要读取正确的值, 我们需要略微重写方程 : y - (-2) = 23 (x-2) 。现在我们看到点(2, 2) 在线上, 斜坡是 23 。

First plot point (2, -2) on the graph:
::图中的第一个绘图点(2, 2) :

A slope of $\begin{aligned} \frac{2}{3} \end{aligned}$ tells you that from that point you should move 2 units up and 3 units to the right and draw another point:
::23的斜坡表示,从那个角度,你应该向上移动2个单位,向右移动3个单位,再画另一个点:

Now draw a line through the two points and extend it in both directions:
::现在通过这两个点划一条线,并将它延伸至两个方向:

Examples
::实例

A line contains the points (1, -2) and (0, 0). Write an equation for the line in point-slope form; then write an equation in $\begin{aligned} y - \end{aligned}$ intercept form.
::直线包含点(1, 2) 和点( 0, 0) 。用点- spope 格式为直线写一个方程式; 然后用 y- inter- intercept 格式写一个方程式。

Find the slope of the line: $\begin{aligned} m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{- 2 - 0}{1 - 0} = \frac{- 2}{1} = - 2 \end{aligned}$
::查找线的斜坡: m=y2-y1x2-x1______01-0_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Plug in the value of the slope: $\begin{aligned} y - y_{0} = - 2 (x - x_{0}) \end{aligned}$ .
::斜坡值中的插件:y-y02(x-x0)。

Plug point (1, -2) into the equation: $\begin{aligned} y - (- 2) = - 2 (x - 1) \end{aligned}$ .
::等式中的插点(1, 2) : y - (-2) 2 (x-1) 。

The equation in point-slope form is $\begin{aligned} y + 2 = - 2 (x - 1) \end{aligned}$ .
::点斜形等式是 y+2 2(x-1) 等式。

To convert to $\begin{aligned} y - \end{aligned}$ intercept form, simply solve for $\begin{aligned} y \end{aligned}$ :
::要转换为 y - 界面窗体,只需为 y 求解 :

$\begin{aligned} y + 2 = - 2 (x - 1) & \to y + 2 = - 2 x + 2 \\ \to y = - 2 x + 2 - 2 \\ \to y = - 2 x . \end{aligned}$

::y+2+22x+22x+22x+2+2+2+2+2+2+2+2+2}~2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2

The equation in $\begin{aligned} y - \end{aligned}$ intercept form is $\begin{aligned} y = - 2 x \end{aligned}$ .
::y - 界面的方程式是 y - 2x 。

Review
::回顾

Find the equation of each line in slope–intercept form.
::查找每条线的方程,以斜坡-截取形式显示。

The line has a slope of 7 and a $\begin{aligned} y - \end{aligned}$ intercept of -2.
::这条线的斜坡为7, y - interviews of - 2。

The line has a slope of -5 and a $\begin{aligned} y - \end{aligned}$ intercept of 6.
::这条线的斜坡为 -5, y - interviews为 6 。

The line has a slope of $\begin{aligned} - \frac{1}{4} \end{aligned}$ and contains the point (4, -1).
::该线的斜坡为-14, 包含点( 4, - 1 ) 。

The line contains points (3, 5) and (-3, 0).
::该行包含点数(3,5)和点数(3,0)。

The line contains points (10, 15) and (12, 20).
::该行包含点(10、15)和点(12、20)。

Write the equation of each line in slope-intercept form.
::将每行的方程写成斜坡界面。

Find the equation of each linear function in slope–intercept form.
::在斜度 - interphysict 窗体中查找每个线性函数的方程式。

$\begin{aligned} m = 5, f (0) = - 3 \end{aligned}$
::m=5,f(0) 3

$\begin{aligned} m = - 7, f (2) = - 1 \end{aligned}$
::m7,f(2)1

$\begin{aligned} m = \frac{1}{3}, f (- 1) = \frac{2}{3} \end{aligned}$
::m=13,f(-1)=23

$\begin{aligned} m = 4.2, f (- 3) = 7.1 \end{aligned}$
::m=4.2,f(-3)=7.1

$\begin{aligned} f (\frac{1}{4}) = \frac{3}{4}, f (0) = \frac{5}{4} \end{aligned}$
::f(14)=34,f(0)=54

$\begin{aligned} f (1.5) = - 3, f (- 1) = 2 \end{aligned}$
::f(1.5)3,f(-1)=2

Write the equation of each line in point-slope form.
::将每行的方程写成点倾斜形。

The line has slope $\begin{aligned} - \frac{1}{10} \end{aligned}$ and goes through the point (10, 2).
::该线有斜坡-110,经过点(10,2)。

The line has slope -75 and goes through the point (0, 125).
::该线有斜坡- 75, 穿过点( 0, 125)。

The line has slope 10 and goes through the point (8, -2).
::该线有斜度 10 并经过点( 8, - 2) 。

The line goes through the points (-2, 3) and (-1, -2).
::这条线穿过点数(-2,3)和点数(-1,2)。

The line contains the points (10, 12) and (5, 25).
::该行包含点数(10、12)和点数(5、25)。

The line goes through the points (2, 3) and (0, 3).
::这条线穿过点数(2、3)和点数(0、3)。

The line has a slope of $\begin{aligned} \frac{3}{5} \end{aligned}$ and a $\begin{aligned} y - \end{aligned}$ intercept of -3.
::这条线的斜坡为35, y - interviews of - 3。

The line has a slope of -6 and a $\begin{aligned} y - \end{aligned}$ intercept of 0.5.
::这条线的斜坡为-6,Y-拦截为0.5。

Write the equation of each linear function in point-slope form.
::以点倾斜窗体写出线性函数的方程式。

$\begin{aligned} m = - \frac{1}{5} \end{aligned}$ and $\begin{aligned} f (0) = 7 \end{aligned}$
::15和f(0)=7

$\begin{aligned} m = - 12 \end{aligned}$ and $\begin{aligned} f (- 2) = 5 \end{aligned}$
::m*12和f(-2)=5

$\begin{aligned} f (- 7) = 5 \end{aligned}$ and $\begin{aligned} f (3) = - 4 \end{aligned}$
:f-7)=5和f(3)4

$\begin{aligned} f (6) = 0 \end{aligned}$ and $\begin{aligned} f (0) = 6 \end{aligned}$
::f(6)=0和f(6)=6

$\begin{aligned} m = 3 \end{aligned}$ and $\begin{aligned} f (2) = - 9 \end{aligned}$
::m=3和f(2)=9

$\begin{aligned} m = - \frac{9}{5} \end{aligned}$ and $\begin{aligned} f (0) = 32 \end{aligned}$
::m*95和f(0)=32

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。

5.1 确定一条线的平方

Section outline

General

确定一条线的平方

Determining the Equation of a Line
::确定一条线的平方

Write an Equation Given Slope and $\begin{aligned} y - \end{aligned}$ Intercept
::撰写一个有斜坡和y- 截取的方程式

Write an Equation Given the Slope and a Point
::写一个基于斜坡和点的等量

Writing the Equation of a Line in Point-Slope Form
::以点-斜体窗体写入线条的公式

Write an Equation Given Two Points
::写出基于两点的等式

Writing an Equation in $\begin{aligned} y - \end{aligned}$ intercept form
::以 y- intercy 形式写入方程式

Graph an Equation in Point-Slope Form
::a 点窗体的方形图形方形

Graphing an Equation of a Line
::绘制线条的方形

Examples
::实例

Review
::回顾

Review (Answers)
::回顾(答复)

Section outline

General

确定一条线的平方

Determining the Equation of a Line ::确定一条线的平方

Write an Equation Given Slope and y − Intercept ::撰写一个有斜坡和y- 截取的方程式

Write an Equation Given the Slope and a Point ::写一个基于斜坡和点的等量

Writing the Equation of a Line in Point-Slope Form ::以点-斜体窗体写入线条的公式

Write an Equation Given Two Points ::写出基于两点的等式

Writing an Equation in y − intercept form ::以 y- intercy 形式写入方程式

Graph an Equation in Point-Slope Form ::a 点窗体的方形图形方形

Graphing an Equation of a Line ::绘制线条的方形

Examples ::实例

Review ::回顾

Review (Answers) ::回顾(答复)

Determining the Equation of a Line
::确定一条线的平方

Write an Equation Given Slope and $\begin{aligned} y - \end{aligned}$ Intercept
::撰写一个有斜坡和y- 截取的方程式

Write an Equation Given the Slope and a Point
::写一个基于斜坡和点的等量

Writing the Equation of a Line in Point-Slope Form
::以点-斜体窗体写入线条的公式

Write an Equation Given Two Points
::写出基于两点的等式

Writing an Equation in $\begin{aligned} y - \end{aligned}$ intercept form
::以 y- intercy 形式写入方程式

Graph an Equation in Point-Slope Form
::a 点窗体的方形图形方形

Graphing an Equation of a Line
::绘制线条的方形

Examples
::实例

Review
::回顾

Review (Answers)
::回顾(答复)