节: 赤道等量的垂直方格 | 10.7 二次赤道的螺旋形式

章节大纲

Vertex Form of a Quadratic Equation
::赤道等量的垂直方格

Probably one of the best applications of the method of is using it to rewrite a quadratic function in vertex form. The vertex form of a quadratic function is
::方法的最佳应用之一可能是使用它重写顶点形式的二次函数。二次函数的顶点形式是二次函数。

$\begin{aligned} y - k = a (x - h)^{2} \end{aligned}$

::y-k=a(x-h)2

This form is very useful for graphing because it gives the vertex of the parabola explicitly. The vertex is at the point $\begin{aligned} (h, k) \end{aligned}$ .
::此窗体对于图形化非常有用, 因为它明确给出了抛物线的顶点。顶点在( h, k) 。

It is also simple to find the $\begin{aligned} x - \end{aligned}$ intercepts from the vertex form: just set $\begin{aligned} y = 0 \end{aligned}$ and take the square root of both sides of the resulting equation .
::从顶端形态中查找 x - inter 点也很简单: 只需设置 y=0 并取出结果方程式两侧的平方根。

To find the $\begin{aligned} y - \end{aligned}$ intercept , set $\begin{aligned} x = 0 \end{aligned}$ and simplify.
::查找 y - 界面, 设置 x=0 并简化。

Finding the Vertex and Intercepts of Parabolas
::找到Parabolas的动脉和拦截器

Find the vertex, the $\begin{aligned} x - \end{aligned}$ intercepts and the $\begin{aligned} y - \end{aligned}$ intercept of the following parabolas:
::查找下列参数的顶部、 x - 截面和 y - 截面:

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

Vertex: (1, 2)
::顶点:1,2)

To find the $\begin{aligned} x - \end{aligned}$ intercepts,
::为了找到X - 拦截,

$\begin{aligned} Set y = 0 : & - 2 & = (x - 1)^{2} \\ Take the square root of both sides : & \sqrt{- 2} & = x - 1 & and & - \sqrt{- 2} = x - 1 \end{aligned}$

::设置 y= 0:- 2= (x- 1) 2 以两边的平方根 :-2=x- 1 和 @ @ @% 2=x- 1

The solutions are not real so there are no $\begin{aligned} x - \end{aligned}$ intercepts.
::解决方案是不真实的,因此存在新问题。

To find the $\begin{aligned} y - \end{aligned}$ intercept,
::为了找到y - intermission,

$\begin{aligned} Set x = 0 : & y - 2 & = (- 1)^{2} \\ Simplify : & y - 2 & = 1 \Rightarrow \underline{y = 3} \end{aligned}$

::设置 x=0:y- 2=(- 1) 2 简化: y- 2=1 @ =3_

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

$\begin{aligned} Rewrite : & y - (- 8) = 2 (x - 3)^{2} \\ Vertex : & \underline{(3, - 8)} \end{aligned}$

::重写: y- (- 8) = 2 (x-3) 2Vertex: (3) - 8_

To find the $\begin{aligned} x - \end{aligned}$ intercepts,
::为了找到X - 拦截,

$\begin{aligned} Set y = 0 : & 8 & = 2 (x - 3)^{2} \\ Divide both sides by 2 : & 4 & = (x - 3)^{2} \\ Take the square root of both sides : & 2 & = x - 3 & and & - 2 = x - 3 \\ Simplify : & \underline{\underline{x = 5}} & and & \underline{\underline{x = 1}} \end{aligned}$

::以 2 : 4 = (x-3) 2 将两边的平方根 : 2 = x-3 和 - 2 = x-3 简化: x= 5__ 和x= 1__ 。

To find the $\begin{aligned} y - \end{aligned}$ intercept,
::为了找到y - intermission,

$\begin{aligned} Set x = 0 : & y + 8 & = 2 (- 3)^{2} \\ Simplify : & y + 8 & = 18 \Rightarrow \underline{\underline{y = 10}} \end{aligned}$

::设置 x=0:y+8=2(- 3)2 简化:y+8=18_ y=10__

To graph a parabola, we only need to know the following information:
::要绘制抛物线图,我们只需要知道以下信息:

the vertex
::顶部

the $\begin{aligned} x - \end{aligned}$ intercepts
::x - 界面

the $\begin{aligned} y - \end{aligned}$ intercept
::y - 截取

whether the parabola turns up or down (remember that it turns up if $\begin{aligned} a > 0 \end{aligned}$ and down if $\begin{aligned} a < 0 \end{aligned}$ )
::月球是上升的,还是下降的,(如果有0)和(如果有0),都是下降的。

Graphing Parabolas
::图示图

1. Graph the parabola given by the function $\begin{aligned} y + 1 = (x + 3)^{2} \end{aligned}$ .
::1. 绘制函数y+1=(x+3)给出的抛物线图。

$\begin{aligned} Rewrite : & y - (- 1) = (x - (- 3))^{2} \\ Vertex : & \underline{(- 3, - 1)} & vertex : (- 3, - 1) \end{aligned}$

::重写: y- (- 1) = (x- (- 3)) 2Vertex: (- 3) 3, - 1)_ vertex: (- 3) - 1

To find the $\begin{aligned} x - \end{aligned}$ intercepts,
::为了找到X - 拦截,

$\begin{aligned} Set y = 0 : & 1 = (x + 3)^{2} \\ Take the square root of both sides : & 1 = x + 3 and - 1 = x + 3 \\ Simplify : & \underline{\underline{x = - 2}} and \underline{\underline{x = - 4}} \\ x - intercepts : (- 2, 0) and (- 4, 0) \end{aligned}$

::设置 y= 0: 1 = (x+3) 2 选择两边的平方根 :1= x+3and- 1= x+3 简化: x @ @ @ @ 2_ 和 x%4_ x- 截取-2,0) 和 (- 4,0)

To find the $\begin{aligned} y - \end{aligned}$ intercept,
::为了找到y - intermission,

$\begin{aligned} Set x = 0 : & y + 1 = (3)^{2} \\ Simplify: & \underline{\underline{y = 8}} & y - intercept : (0, 8) \end{aligned}$

::设置 x=0:y+1=(3)2 简化: y=8_ y- 截取( 0. 8)

And since $\begin{aligned} a > 0 \end{aligned}$ , the parabola turns up.
::从零开始抛物线出现了

Graph all the points and connect them with a smooth curve:
::绘制所有点的图, 并用平滑曲线将其连接 :

2. Graph the parabola given by the function $\begin{aligned} y = - \frac{1}{2} (x - 2)^{2} \end{aligned}$ .
::2. 绘制函数 y12(x-2) 给出的抛物线图。

$\begin{aligned} Rewrite & y - (0) = - \frac{1}{2} (x - 2)^{2} \\ Vertex: & \underline{(2, 0)} & vertex: (2, 0) \end{aligned}$

::重写 - (0)\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\Vertex:\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\去\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\去\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

To find the $\begin{aligned} x - \end{aligned}$ intercepts,
::为了找到X - 拦截,

$\begin{aligned} Set y = 0 : & 0 & = - \frac{1}{2} (x - 2)^{2} \\ Multiply both sides by - 2 : & 0 & = (x - 2)^{2} \\ Take the square root of both sides : & 0 & = x - 2 \\ Simplify : & \underline{\underline{x = 2}} & x - intercept: (2, 0) \end{aligned}$

::将 y= 0: 0\% 12 (x- 2) 2 设置为 y= 0: 0\% 12 (x- 2) 。将两边的平方根 : 0= (x- 2) 2 选择两边的平方根 : 0=x- 2 简化: x= 2__ x- 截取 : (2, 0)

Note: there is only one $\begin{aligned} x - \end{aligned}$ intercept, indicating that the vertex is located at this point, (2, 0).
::注:只有一个 x- 界面,表示顶点位于此点(2,0) 。

To find the $\begin{aligned} y - \end{aligned}$ intercept,
::为了找到y - intermission,

$\begin{aligned} Set x = 0 : & y & = - \frac{1}{2} (- 2)^{2} \\ Simplify: & y & = - \frac{1}{2} (4) \Rightarrow \underline{\underline{y = - 2}} & y - intercept: (0, - 2) \end{aligned}$

::设置 x=0: y12(- 2) 2 简化: y\\ 12(4)\ y2- y- 截取 : (0) -0- 2

Since $\begin{aligned} a < 0 \end{aligned}$ , the parabola turns down.
::从零开始,抛物线就掉下来了

Graph all the points and connect them with a smooth curve:
::绘制所有点的图, 并用平滑曲线将其连接 :

Example
::示例示例示例示例

Example 1
::例1

Graph the parabola given by the function $\begin{aligned} y = 4 (x + 2)^{2} - 1 \end{aligned}$ .
::y=4( x+2)2-2-1 函数给出的参数图。

$\begin{aligned} Rewrite & y - (- 1) = 4 (x + 2)^{2} \\ Simplify & y + 1 = 4 (x + 2)^{2} \\ Vertex: & \underline{(- 2, - 1)} & vertex: (- 2, - 1) \end{aligned}$

::重写 - (- 1) = 4( x+2) 2Simplify+1= 4( x+2) 2Vertex: (- 2) 2- 1)_ verftex: (- 2) - 1

To find the $\begin{aligned} x - \end{aligned}$ intercepts,
::为了找到X - 拦截,

$\begin{aligned} Set. y = 0 : & 0 & = 4 (x + 2)^{2} - 1 \\ Subtract 1 from each side : & 1 & = 4 (x + 2)^{2} \\ Divide both sides by 4 : & \frac{1}{4} & = (x + 2)^{2} \\ Take the square root of both sides : & \frac{1}{2} & = \pm (x + 2) \\ Separate : & \frac{1}{2} = - (x + 2) & \frac{1}{2} = x + 2) \\ Simplify : & \underline{\underline{x = - 2.5}} & \underline{\underline{x = - 1.5}} \end{aligned}$

::设置 y= 0: 0= 4( x+2) 2 - 1 从每侧排列的 1: 1= 4( x+2) 2, 2 将两边的平方根 4: 14= ( x+2) 2 切换两侧的平方根 : 12\ ( x+2) 分隔 : 12\ ( x+2) 12= x+2 简化 : x\\\ 2.5_ x\\\\ 1.5__

The $\begin{aligned} x - \end{aligned}$ intercepts are $\begin{aligned} (- 2.5, 0) \end{aligned}$ and $\begin{aligned} (- 1.5, 0) \end{aligned}$ .
::x- 拦截是(-2.5,0)和(-1.5,0)。

To find the $\begin{aligned} y - \end{aligned}$ intercept,
::为了找到y - intermission,

$\begin{aligned} Set x = 0 : & y & = 4 (0 + 2)^{2} - 1 \\ Simplify: & y & = 15 \Rightarrow \underline{\underline{y = 15}} & y - intercept: (0, 15) \end{aligned}$

::设置 x=0:y=4( 0+2) 2- 1 简化: y= 15_ y=15_ y- interview: (0, 15)

Since $\begin{aligned} a < 0 \end{aligned}$ , the parabola turns up.
::从零开始抛物线出现

Graph all the points and connect them with a smooth curve:
::绘制所有点的图, 并用平滑曲线将其连接 :

Review
::回顾

Rewrite each quadratic function in vertex form.
::以顶点形式重写每个二次函数。

$\begin{aligned} y = x^{2} - 6 x \end{aligned}$
::y=x2 - 6x

$\begin{aligned} y + 1 = - 2 x^{2} - x \end{aligned}$
::y+1%2x2 - x

$\begin{aligned} y = 9 x^{2} + 3 x - 10 \end{aligned}$
::y= 9x2+3x- 10

$\begin{aligned} y = - 32 x^{2} + 60 x + 10 \end{aligned}$
::y32x2+60x+10

For each parabola, find the vertex; the $\begin{aligned} x - \end{aligned}$ and $\begin{aligned} y - \end{aligned}$ intercepts; and if it turns up or down. Then graph the parabola.
::每个抛物线都能找到顶端; x - 和 y - intercuts; 以及如果它上升或下降。然后绘制 parbola 。

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

$\begin{aligned} y = - 4 x^{2} + 20 x - 24 \end{aligned}$
::y* y4x2+20x- 24

$\begin{aligned} y = 3 x^{2} + 15 x \end{aligned}$
::y= 3x2+15x

$\begin{aligned} y + 6 = - x^{2} + x \end{aligned}$
::y+6x2+x

$\begin{aligned} x^{2} - 10 x + 25 = y + 9 \end{aligned}$
::x2 - 10x+25=y+9

$\begin{aligned} x^{2} + 18 x + 81 = y + 1 \end{aligned}$
::x2+18x+81=y+1

$\begin{aligned} 4 x^{2} - 12 x + 9 = y + 16 \end{aligned}$
::4x2 - 12x+9=y+16

$\begin{aligned} x^{2} + 14 x + 49 = y + 3 \end{aligned}$
::x2+14x+49=y+3 x2+14x+49=y+3

$\begin{aligned} 4 x^{2} - 20 x + 25 = y + 9 \end{aligned}$
::4x2-20x+25=y+9

$\begin{aligned} x^{2} + 8 x + 16 = y + 25 \end{aligned}$
::x2+8x+16=y+25 x2+8x+16=y+25

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。

章节大纲

Vertex Form of a Quadratic Equation ::赤道等量的垂直方格

Finding the Vertex and Intercepts of Parabolas ::找到Parabolas的动脉和拦截器

Graphing Parabolas ::图示图

Example ::示例示例示例示例

Example 1 ::例1

Review ::回顾

Review (Answers) ::回顾(答复)

Vertex Form of a Quadratic Equation
::赤道等量的垂直方格

Finding the Vertex and Intercepts of Parabolas
::找到Parabolas的动脉和拦截器

Graphing Parabolas
::图示图

Example
::示例示例示例示例

Example 1
::例1

Review
::回顾

Review (Answers)
::回顾(答复)