Section: 关节、截取和标准表格 | 5.16 口服、拦截和标准表格

Section outline

The profit on your school fundraiser is represented by the quadratic expression $\begin{aligned} - 5 p^{2} + 400 p - 8000 \end{aligned}$ , where $\begin{aligned} p \end{aligned}$ is your price point. What price point will result in a maximum profit and what is that profit?
::您的学校募捐活动的利润由四边表达式 5p2+400p-800表示, p是您的价格点。什么价格点能带来最大利润,什么利润?

Vertex Intercept and Standard Form
::口服拦截和标准表格

So far, we have only used the standard form of a quadratic equation , $\begin{aligned} y = a x^{2} + b x + c \end{aligned}$ to graph a parabola . From standard form, we can find the vertex and either factor or use the Quadratic Formula to find the $\begin{aligned} x - \end{aligned}$ intercepts . The intercept form of a quadratic equation is $\begin{aligned} y = a (x - p) (x - q) \end{aligned}$ , where $\begin{aligned} a \end{aligned}$ is the same value as in standard form, and $\begin{aligned} p \end{aligned}$ and $\begin{aligned} q \end{aligned}$ are the $\begin{aligned} x - \end{aligned}$ intercepts. This form looks very similar to a factored quadratic equation.
::到目前为止,我们只使用四方形的标准形式 y= ax2+bx+c 来绘制抛物线。从标准形式中,我们可以找到顶点和因子,或者使用二次方程来寻找 x- intercuts。四方形的截取形式是 y=a(x- p)(x- q),其值与标准形式相同,而 p和 q 是 x- intercuts。这个形式看起来与参数的二次方程非常相似。

Let's change $\begin{aligned} y = 2 x^{2} + 9 x + 10 \end{aligned}$ to intercept form and find the vertex. Then we will graph the parabola.
::让我们将 y= 2x2+9x+10 更改为截取表单并找到顶点。然后我们将绘制抛物线图。

First, let’s change this equation into intercept form by factoring. $\begin{aligned} a c = 20 \end{aligned}$ and the factors of 20 that add up to 9 are 4 and 5. Expand the $\begin{aligned} x - \end{aligned}$ term .
::首先,让我们将这个方程式改变为截取形式,通过保理。 ac=20, 20的系数加到9是4和5。扩大x- term。

$\begin{aligned} y & = 2 x^{2} + 9 x + 10 \\ y & = 2 x^{2} + 4 x + 5 x + 10 \\ y & = 2 x (x + 2) + 5 (x + 2) \\ y & = (2 x + 5) (x + 2) \end{aligned}$

::y=2x2+9x+10y=2x2+4x+5x+10y=2x(x+2)+5(x+2)=2x+2y=2x+5(x+5)(x+2)

Notice, this does not exactly look like the definition. The factors cannot have a number in front of $\begin{aligned} x \end{aligned}$ . Pull out the 2 from the first factor to get $\begin{aligned} y = 2 (x + \frac{5}{2}) (x + 2) \end{aligned}$ . Now, let's find the vertex. Recall that all parabolas are symmetrical . This means that the axis of is halfway between the $\begin{aligned} x - \end{aligned}$ intercepts or their average.
::注意, 这与定义不完全相似。系数不能在 x 前面有一个数字。从第一个系数中拔出 2 来获取 y=2( x+52)( x+2) 。现在, 让我们找到顶点。回顾所有的 parabolas 都是对称的。这意味着 X- inter 之间的轴是 X- inter 或平均值的一半。

$\begin{aligned} axis of symmetry = \frac{p + q}{2} = \frac{- \frac{5}{2} - 2}{2} = - \frac{9}{2} \div 2 = - \frac{9}{2} \cdot \frac{1}{2} = - \frac{9}{4} \end{aligned}$

::对称轴=p+q252-2292292921294

This is also the $\begin{aligned} x - \end{aligned}$ coordinate of the vertex. To find the $\begin{aligned} y - \end{aligned}$ coordinate, plug the $\begin{aligned} x - \end{aligned}$ value into either form of the quadratic equation. We will use Intercept form.
::这也是顶端的 x 坐标。要找到 y - 坐标, 请将 x - 值插入二次方程的两种形式中。我们将使用截取形式。

$\begin{aligned} y & = 2 (- \frac{9}{4} + \frac{5}{2}) (- \frac{9}{4} + 2) \\ y & = 2 \cdot \frac{1}{4} \cdot - \frac{1}{4} \\ y & = - \frac{1}{8} \end{aligned}$

::y=2(- 94+52)(- 94+2)y=2141418

The vertex is $\begin{aligned} (- 2 \frac{1}{4}, - \frac{1}{8}) \end{aligned}$ . Now, we will plot the $\begin{aligned} x - \end{aligned}$ intercepts and the vertex to graph.
::顶点是 (- 214 ) - 18。现在, 我们将绘制 x - intercuts 和顶点到图形。

The last form is vertex form. Vertex form is written $\begin{aligned} y = a (x - h)^{2} + k \end{aligned}$ , where $\begin{aligned} (h, k) \end{aligned}$ is the vertex and $\begin{aligned} a \end{aligned}$ is the same is in the other two forms. Notice that $\begin{aligned} h \end{aligned}$ is negative in the equation, but positive when written in coordinates of the vertex.
::最后一个窗体是顶点形式。 Vertex 窗体是 y=a( x-h) 2+k, 其中(h,k) 是顶点, a 是相同的, 在另外两个窗体中。请注意, h 在方程中为负值, 但以顶点坐标写时为正值。

Now, let's find the vertex of $\begin{aligned} y = \frac{1}{2} (x - 1)^{2} + 3 \end{aligned}$ and graph the parabola.
::现在,让我们找到y=12(x- 1)2+3的顶点,然后绘制抛物线图。

The vertex is going to be (1, 3). To graph this parabola, use the symmetric properties of the function . Pick a value on the left side of the vertex. If $\begin{aligned} x = - 3 \end{aligned}$ , then $\begin{aligned} y = \frac{1}{2} (- 3 - 1)^{2} + 3 = 11. \end{aligned}$ -3 is 4 units away from 1 (the $\begin{aligned} x - \end{aligned}$ coordinate of the vertex). 4 units on the other side of 1 is 5. Therefore , the $\begin{aligned} y - \end{aligned}$ coordinate will be 11. Plot (1, 3), (-3, 11), and (5, 11) to graph the parabola.
::顶点将是(1, 3) 。要绘制此抛物线, 请使用函数的对称属性。在顶点左侧选择一个值。如果 x3, 那么y= 12 (- 3- 1) 2+3=11。 - 3 是 1 的 4 个单位( 顶点的x- 坐标) 。 1 的对面的 4 个单位是 5 。因此, y- 坐标将是 11. Plot ( 1, 3, (3, 3, 11) 和 (5, 11) 来绘制 parbola 。

Finally, let's change $\begin{aligned} y = x^{2} - 10 x + 16 \end{aligned}$ into vertex form.
::最后,让我们将 y=x2 - 10x+16 变成顶点形式。

To change an equation from standard form into vertex form, you must complete the square. The major difference is that you will not need to solve this equation.
::要将方程式从标准窗体修改为顶点窗体, 您必须完成正方形。主要区别在于您不需要解析此方程式。

$\begin{aligned} y & = x^{2} - 10 x + 16 \\ y - 16 + 25 & = x^{2} - 10 x + 25 Move 16 to the other side and add {(\frac{b}{2})}^{2} to both sides . \\ y + 9 & = (x - 5)^{2} Simplify left side and factor the right side \\ y & = (x - 5)^{2} - 9 Subtract 9 from both sides to get y by itself . \end{aligned}$

::y=x2- 10x+16y- 16+25=x2- 10x+25+25 将 16 移动到另一侧,并将 (b2) 2 添加到两侧。y+9 = (x-5) 2 简化左侧并乘以右侧=(x-5) 2- 9 从两侧抽取9 以获得 y。

To solve an equation in vertex form, set $\begin{aligned} y = 0 \end{aligned}$ and solve for $\begin{aligned} x \end{aligned}$ .
::要解析顶点形式的方程式,请设置 y=0 并解析 x。

$\begin{aligned} (x - 5)^{2} - 9 & = 0 \\ (x - 5)^{2} & = 9 \\ x - 5 & = \pm 3 \\ x & = 5 \pm 3 o r 8 a n d 2 \end{aligned}$

:x-5)2-9=0(x-5)2=9x-53x=53或8和2

Examples
::实例

Example 1
::例1

Earlier, you were asked to find the price point that will result in a maximum profit and to find that profit.
::早些时候,你被要求找到能带来最大利润的价格点, 并找到这一利润。

The vertex will give us the price point that will result in the maximum profit and that profit, so let’s change this equation into intercept form by factoring. First factor out -5.
::顶点将给我们带来最大利润和利润的价格点,因此让我们通过保理来改变这个方程式为截取形式。第一个因素是 - 5。

$\begin{aligned} - 5 p^{2} + 400 p - 8000 = - 5 (p^{2} - 80 p + 1600) \\ - 5 (p - 40) (p - 40) \end{aligned}$

::-5p2+400p-800005(p2-80p+1600)-5(p-40)(p-40)

From this we can see that the $\begin{aligned} x \end{aligned}$ -intercepts are 40 and 40. The average of 40 and 40 is 40 (of course). P lug 40 into the original equation:
::从这里可以看出, X 拦截是40和40,40和40的平均数是40(当然)。

$\begin{aligned} - 5 (40)^{2} + 400 (40) - 8000 = - 8000 + 16000 - 8000 = 0 \end{aligned}$

Therefore, the price point that results in a maximum profit is $40 and that price point results in a profit of $0. You're not making any money, so you better rethink your fundraising approach!
::因此,导致最大利润的价点是40美元,而价格点则产生0美元的利润。你没有赚任何钱,所以你最好重新思考你的筹资方法!

Example 2
::例2

Find the intercepts of $\begin{aligned} y = 2 (x - 7) (x + 2) \end{aligned}$ and change it to standard form.
::查找 y=2( x-7)( x+2) 的拦截, 并将其修改为标准格式。

The intercepts are the opposite sign from the factors; (7, 0) and (-2, 0). To change the equation into standard form, FOIL the factors and distribute $\begin{aligned} a \end{aligned}$ .
::截取的信号与因素相反;(7,0)和(2,0)。要将方程式改变为标准形式, 截取的信号与因素相反; (7,0) 和(2,0) 。

$\begin{aligned} y & = 2 (x - 7) (x + 2) \\ y & = 2 (x^{2} - 5 x - 14) \\ y & = 2 x^{2} - 10 x - 28 \end{aligned}$

::y=2x-7(x+2)y=2x2-5x-14y=2x2-10x-28

Example 3
::例3

Find the vertex of $\begin{aligned} y = - \frac{1}{2} (x + 4)^{2} - 5 \end{aligned}$ and change it to standard form.
::查找 y12 (x+4) 2 - 5 的顶点, 并将其修改为标准格式。

The vertex is (-4, -5). To change the equation into standard form, FOIL $\begin{aligned} (x + 4)^{2} \end{aligned}$ , distribute $\begin{aligned} a \end{aligned}$ , and then subtract 5.
::顶点是 (4 - 5) 。要将方程式转换为标准格式, FOIL (x+4) 2, 分配 a, 然后减去 5。

$\begin{aligned} y & = - \frac{1}{2} (x + 4) (x + 4) - 5 \\ y & = - \frac{1}{2} (x^{2} + 8 x + 16) - 5 \\ y & = - \frac{1}{2} x^{2} - 4 x - 13 \end{aligned}$

::y12(x+4)(x+4)- 5y12(x2+8x+16)- 5y12x2- 4x- 13

Example 4
::例4

Change $\begin{aligned} y = x^{2} + 18 x + 45 \end{aligned}$ to intercept form and graph.
::将 y=x2+18x+45 更改为截取窗体和图形。

To change $\begin{aligned} y = x^{2} + 18 x + 45 \end{aligned}$ into intercept form, factor the equation. The factors of 45 that add up to 18 are 15 and 3. Intercept form would be $\begin{aligned} y = (x + 15) (x + 3) \end{aligned}$ . The intercepts are (-15, 0) and (-3, 0). The $\begin{aligned} x - \end{aligned}$ coordinate of the vertex is halfway between -15 and -3, or -9. The $\begin{aligned} y - \end{aligned}$ coordinate of the vertex is $\begin{aligned} y = (- 9)^{2} + 18 (- 9) + 45 = - 36 \end{aligned}$ . Here is the graph:
::将 y=x2+18x+45 改为拦截形式, 乘以方程。 45 乘以 18 的因数是 15 和 3. 截取形式是 Y= (x+15) (x+3) 。截取形式是 (15) 0 和 (3) 0 。顶点的x- 坐标介于 - 15 和 - 3 之间, 或 - 9 。顶点的y- 坐标是 Y= (- 9) 2+18 (- 9)+ 45\ 。下图如下 :

Example 5
::例5

Change $\begin{aligned} y = x^{2} - 6 x - 7 \end{aligned}$ to vertex form and graph.
::将 y=x2 - 6x-7 更改为顶点形式和图形。

To change $\begin{aligned} y = x^{2} - 6 x - 7 \end{aligned}$ into vertex form, complete the square.
::要将 y=x2 - 6x-7 改变为顶部形式,请填写正方形。

$\begin{aligned} y + 7 + 9 & = x^{2} - 6 x + 9 \\ y + 16 & = (x - 3)^{2} \\ y & = (x - 3)^{2} - 16 \end{aligned}$

::y+7+9=x2-6x+9y+16=(x-33)2y=(x-33)2-16

The vertex is (3, -16).
::顶部是 (3, -16) 。

For vertex form, we could solve the equation by using square roots or we could factor the standard form. Either way, we will get that the $\begin{aligned} x - \end{aligned}$ intercepts are (7, 0) and (-1, 0).
::对于顶点形式,我们可以使用平方根来解析方程,或者我们可以乘以标准方块。无论哪种方式,我们都会得到x- interviews是( 7, 0) 和( 1, 0) 。

Review
::回顾

Fill in the table below. Either describe how to find each entry or use a formula.
::填写下表。要么说明如何查找每个条目,要么使用公式。

Find the vertex and $\begin{aligned} x \end{aligned}$ -intercepts of each function below. Then, graph the function. If a function does not have any $\begin{aligned} x \end{aligned}$ -intercepts, use the symmetry property of parabolas to find points on the graph.
::查找以下每个函数的顶点和 X 界面。然后,绘制函数图。如果函数没有 X 界面,请使用parabolas的对称属性在图形中找到点。

$\begin{aligned} y = (x - 4)^{2} - 9 \end{aligned}$
::y= (x- 4) 2- 9

$\begin{aligned} y = (x + 6) (x - 8) \end{aligned}$
::y= (x+6)(x-8)

$\begin{aligned} y = x^{2} + 2 x - 8 \end{aligned}$
::y=x2+2x-8

$\begin{aligned} y = - (x - 5) (x + 7) \end{aligned}$
::y(x-5)(x+7)

$\begin{aligned} y = 2 (x + 1)^{2} - 3 \end{aligned}$
::y=2( x+1) 2- 3

$\begin{aligned} y = 3 (x - 2)^{2} + 4 \end{aligned}$
::y=3(x-2)2+4

$\begin{aligned} y = \frac{1}{3} (x - 9) (x + 3) \end{aligned}$
::y=13(x- 9) (x+3)

$\begin{aligned} y = - (x + 2)^{2} + 7 \end{aligned}$
::y(x+2)2+7

$\begin{aligned} y = 4 x^{2} - 13 x - 12 \end{aligned}$
::y= 4x2- 13x- 12

Change the following equations to intercept form.
::将以下方程式更改为截取形式。

$\begin{aligned} y = x^{2} - 3 x + 2 \end{aligned}$
::y=x2 - 3x+2

$\begin{aligned} y = - x^{2} - 10 x + 24 \end{aligned}$
::yx2 - 10x+24

$\begin{aligned} y = 4 x^{2} + 18 x + 8 \end{aligned}$
::y=4x2+18x+8

Change the following equations to vertex form.
::将以下方程式更改为顶点形式。

$\begin{aligned} y = x^{2} + 12 x - 28 \end{aligned}$
::y=x2+12x-28 y=x2+12x-28

$\begin{aligned} y = - x^{2} - 10 x + 24 \end{aligned}$
::yx2 - 10x+24

$\begin{aligned} y = 2 x^{2} - 8 x + 15 \end{aligned}$
::y=2x2 - 8x+15

Change the following equations to standard form.
::将以下方程式更改为标准格式。

$\begin{aligned} y = (x - 3)^{2} + 8 \end{aligned}$
::y=(x-3)2+8

$\begin{aligned} y = 2 (x - \frac{3}{2}) (x - 4) \end{aligned}$
::y=2(x-32)(x-4)

$\begin{aligned} y = - \frac{1}{2} (x + 6)^{2} - 11 \end{aligned}$
::y12( x+6) 2- 11

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

Vertex Intercept and Standard Form ::口服拦截和标准表格

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)

Vertex Intercept and Standard Form
::口服拦截和标准表格

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Review
::回顾

Review (Answers)
::回顾(答复)