3.5 赤道不平等-interactive

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  赤道不平等

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  Lesson Objectives
  ::经验教训目标

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  • Solve real-world problems involving a quadratic inequality graphically.
    ::解决现实世界的问题 涉及二次不平等的图形化。
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  • Solve real-world problems by solving a quadratic inequality algebraically.
    ::解决现实世界的问题 解决二次不平等的代数。
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  • Graph a quadratic inequality.
    ::绘制二次不平等的图。

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  Introduction: Fear of Heights
  ::导言:对高地的恐惧

  

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  Q uadratic equations have either 0, 1, or 2  valid solutions in the context of the problem. H owever, problems often have a range of acceptable solutions that can be expressed using inequalities.
  ::在问题的背景下,赤道方程式有0、1或2个有效的解决办法,但问题往往有一系列可以接受的解决办法,可以用不平等来表达。

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  Example
  ::示例示例示例示例

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  Chase is a stunt man who is base jumping from a bridge 500 feet above a river for a movie. He needs to pull the cord on his parachute between 400 and 275 feet. His height during the fall is defined by the function  $\begin{array}{r}h(t)=500-16t^2.\end{array}$  Write an inequality for the times during which he should pull his parachute.
  ::大通是一个特技演员,他正在从河上500英尺高的桥上跳跃,看电影。他需要拉起降落伞的绳索,在400至275英尺之间。他跌倒时的高度由函数 h(t)=500-16t2确定。写下他拉降落伞时的不平等。

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  Discuss how you might begin to approach this problem.  The next activity will explore this problem further.
  ::讨论您可能如何开始处理这一问题。 下一步活动将进一步探讨这个问题 。

   

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  Activity 1: Solving Quadratic Inequalities Graphically
  ::活动1:用图形方式解决赤道不平等问题

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  Graphing a function is a good way to help  to visualize every possible solution.  A  graph can be used to identify the solutions that will satisfy  certain conditions.
  ::绘制函数图是帮助想象每一种可能的解决方案的好方法。 图表可用于确定符合某些条件的解决方案。

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  To solve the problem in the introduction, look at a graph of the function  $\begin{array}{r}h(t)=500-16t^2.\end{array}$
  ::要解决导言中的问题,请查看函数 h(t) = 500- 16t2 的图示。

  

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  The condition that Chase  needs to pull the cord on his parachute between 400 and 275 feet  can be expressed using  inequalities: 
  ::大通需要将降落伞的绳索拉到400至275英尺之间,

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  $$\begin{array}{r}275\le 500-16t^2\le 400\end{array}$$

  
  ::275 - 500 - 16t-2400

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  T he inequality can be broken apart into three pieces to get a better idea of what the conditions look like on a graph:
  ::不平等可以分成三部分, 以便更清楚地了解图表上的状况:

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  • $\begin{array}{r}h(t)=500-16t^2\end{array}$  
    ::h(t)=500-16t2
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  • $\begin{array}{r}h(t)\ge 275\end{array}$  
    :t)_____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
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  • $\begin{array}{r}h(t)\le 400\end{array}$  
    :t)4000

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  Use the  graph above to answer the following questions.
  ::使用上图回答下列问题。

   

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  Activity 2: Solving Quadratic Inequalities Algebraically
  ::活动2:用代数法解决赤道不平等问题

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  While graphs are a useful method for solving inequalities, an algebraic approach can also be taken. The lesson  Roots of Quadratic Functions introduced this example : 
  ::虽然图表是解决不平等问题的有用方法,但也可以采用代数法。

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  A firework is launched from a platform 29 feet off the ground with an initial velocity of 112 ft/s. The height of the firework can be modeled as a function of time using the function  $\begin{array}{r}h(x)=-16x^2+112x+29.\end{array}$  The firework must go off at 200 feet, to ensure the safety of the onlookers.
  ::从离地29英尺的平台上发射烟花,最初速度为112英尺/秒。烟花的高度可以用h(x)16x2+112x+29的功能作为时间函数模型。烟花必须以200英尺的速度起落,以确保旁观者的安全。

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  Suppose the firework had to go off at least 200 feet from the ground. A t what times could the firework go off?
  ::如果烟花离地至少200英尺 烟花什么时候会熄灭呢?

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  Example
  ::示例示例示例示例

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  Solve the inequality  $\begin{array}{r}200\le -16x^2+112x+29.\end{array}$
  ::解决不平等问题 20016x2+112x29。

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  The process for solving  quadratic inequalities is similar to  quadratic equations. First, subtract 200 from both sides of the inequality.
  ::解决二次不平等的过程与二次等式相似。 首先,从两边减去200个不平等。

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  $$\begin{array}{rl}200& \le -16x^2+112x+29\\ -200& -200\\ 0& \le -16x^2+112x-171\end{array}$$

  
  ::20016x2+112x+29-200-2000}16x2+112x-171

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  Next, factor the expression  $\begin{array}{r}-16x^2+112x-171.\end{array}$
  ::下一个乘数表示式 - 16x2+112x-171。

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  $$\begin{array}{r}0\le -16x^2+112x-1710\le -(4x-9)(4x-19)\end{array}$$

  
  ::016x2+112x-1710(4x-9)(4x-19)

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  T he expression  $\begin{array}{r}-(4x-9)(4x-19)\end{array}$  must be greater than or equal to zero. T he values that make the expression  $\begin{array}{r}-(4x-9)(4x-19)\end{array}$  equal to zero are  $\begin{array}{r}\frac{9}{4}=2.25\text{ and }\frac{19}{4}=\mathrm{4.75.}\end{array}$  Since this function forms a parabola when graphed, all positive values must either be inside or outside the $\begin{array}{r}x\end{array}$ - intercepts .
  ::表达式 -( 4x- 9) ( 4x-19) 必须大于或等于零。 使表达式 -( 4x- 9) ( 4x-19) 等于零的值为 94= 2. 25 和 194= 4. 75。 由于此函数在图形化时构成抛物线, 所有正值都必须在 x 界面内或外方。

  

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  T est a value  between 2.25 and 4.75  to see if it will make  $\begin{array}{r}-(4x-9)(4x-19)\end{array}$  positive. If the answer is positive,  all values between 2.25 and 4.75 will make the  inequality true. Otherwise,  all values outside of 2.25 and 4.75 will make the  inequality true.
  ::测试值在 2. 25 和 4. 75 之间, 以确定它是否会使 -( 4x- 9) ( 4x-19) 呈正数。 如果答案是正数, 则2. 25 和 4. 75 之间的所有数值都会使不平等成为事实。 否则, 2. 25 和 4. 75 之外的所有数值都会使不平等成为事实。

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  Testing   $\begin{array}{r}x=3,\end{array}$
  ::测试x=3,测试x=3

  $$\begin{array}{rl}(-4(3)+9)(4(3)-19)& =(-12+9)(12-19)\\ & =(-3)(-7)\\ & =21\end{array}$$

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  Since 3 makes the expression positive, all values between 2.25 and 4.75 will make the expression  $\begin{array}{r}(-4x+9)(4x-19)\end{array}$  positive. This means that  all values between 2.25 and 4.75 will be above 200 feet in the function  $\begin{array}{r}h(x)=-16x^2+112x+29.\end{array}$
  ::由于3 表示为正, 2. 25 至 4. 75 之间的所有值将使表达式( - 4x+9) (4x-19) 变为正,这意味着2. 25 至 4. 75 之间所有值在函数 h(x)\\ 16x2+112x+29 中将超过200英尺。

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  Answer:   $\begin{array}{r}2.25\le x\le 4.75\end{array}$
  ::答复:2.25x4.75

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  The firework can go off any time between 2.25 seconds and 4.75 seconds
  ::烟花可以随时在2.25秒到4.75秒之间熄灭

   

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  Discussion Question : Could  you have  determined that any $\begin{array}{r}x\end{array}$ -value between the $\begin{array}{r}x\end{array}$ -intercepts would result in a positive output based on the $\begin{array}{r}a\end{array}$ - coefficient ?
  ::讨论问题:你是否能够确定,在X接口之间有任何x值会导致基于系数效率的正产出?

   

    

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  Activity 3: Quadratic Inequalities With Two Variables
  ::活动3:具有两个变量的赤道不平等

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  T he  graph of a linear inequality will display all possible coordinates that satisfy the conditions of the inequality.
  ::线性不平等图将显示满足不平等条件的所有可能的坐标。

  

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  To get a sense of how the graphs of quadratic inequalities with two variables will look, test a few points in the interactive below to see if they satisfy the conditions of the inequality.
  ::为了了解带有两个变量的二次不平等的图表将如何看, 测试下面互动的几点, 看看它们是否符合不平等的条件。

  

  INTERACTIVE

  Quadratic Inequalities With Two Variables

  

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  Use the interactive to graph five test points on the inequality  @$y \geq x^2 - 5x + 7@$ .
  ::使用互动来图示不平等的五个测试点@$y\geq x%2 - 5x+7@$。

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  • For each point, fill in the first blank with the y-coordinate of the point. Fill in the second blank by plugging the x-coordinate into the equality.
    ::对于每个点, 请用点的 Y 坐标填入第一个空格。 将 x 坐标插入等距 。
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  • Compare the two values and decide if each point satisfies the equality.
    ::比较两种价值观,决定每一点是否符合平等。
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  • Generate more random coordinates to finish the graph.
    ::生成更多随机坐标以完成图形 。

  
  

  
  

  
  

  
  

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  To graph an inequality with two variables, first graph the function. As  shown  in the interactive above, all solutions will either be inside or outside of the parabola.
  ::要用两个变量绘制不平等图,请先绘制函数图。如上互动图所示,所有解决方案要么在抛物线内,要么在抛物线外。

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  To determine whether to shade inside or outside the parabola, choose one point that is either inside or outside of the parabola and check it.
  ::为了确定抛物线内外的阴影,选择一个在抛物线内外的点并检查它。

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  • If the coordinate makes the inequality true, then shade the same side of the parabola that the coordinate is on.
    ::如果坐标使不平等成为真实,则将坐标所在的抛物线的同一面遮蔽。
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  • If the coordinate makes the inequality false, then shade the opposite side of the parabola that the coordinate it on.
    ::如果坐标显示不平等是假的, 则将协调的抛物线的反面遮蔽 。

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  Recall from linear inequalities that the < or > symbols  indicate  using  a dotted line to graph the linear function because it  excludes  points  that  would make both sides of the function equal. The same holds for quadratic inequalities: the < or > symbols will result in a dotted parabola and the ≤ or ≥ symbols will result in a solid parabola.
  ::从线性不平等中回顾, < 或 > 符号表示用虚线来显示线性函数,因为它排除了使函数两侧相等的点。对二次不平等来说, " 或 > 符号将产生虚线抛物线,而 或 或 符号将产生坚固的抛物线。

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  Discussion Question : A firework that is launched can be modeled using the function $\begin{array}{r}h(x)=-16x^2+112x+29.\end{array}$   Will every height produce a positive and negative answer? Which times and heights can you exclude from your solution set ? What would the graph look like with these taken into account?
  ::讨论问题: 使用函数 h( x)\\\ 16x2+112x+29可以模拟启动的烟火。 每个高度是否都产生正反回答? 您能将哪些时间和高度排除在解决方案集之外吗? 图表将如何看待这些考虑?

   

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  Wrap-Up: Review Questions
  ::总结:审查问题

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     Summary
  ::摘要

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  When solving a quadratic inequality rewrite the equation so it is in the form: 
  ::当解决二次不平等时重写方程式时, 它以形式出现 :

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  • $\begin{array}{r}0\le a{x}^2+bx+c\end{array}$  and find the values that make the inequality positive.
    ::0ax2+bx+c 并找到使不平等成为正数的价值观。
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  • $\begin{array}{r}0\ge a{x}^2+bx+c\end{array}$  and find the values that make the inequality negative.
    ::0ax2+bx+c 并找到不平等负值 。

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  When graphing a quadratic inequality of one variable, the solutions can be found either  between  the roots or outside of the roots.
  ::当绘制一个变量的二次不平等图时,可以在根或根之外找到解决办法。

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  When graphing a quadratic inequality of two variables, the solutions can be found either inside or outside the parabola.
  ::当绘制两个变量的二次不平等图时,可以在抛物线内外找到解决办法。