Section: 赤道不平等 | 3.5 赤道不平等-interactive

Section outline

Lesson Objectives
::经验教训目标

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。

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。

Graph a quadratic inequality.
::绘制二次不平等的图。

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

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。

Example
::示例示例示例示例

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 $\begin{aligned} h (t) = 500 - 16 t^{2} . \end{aligned}$ 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。

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。

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

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。

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

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,

$\begin{aligned} 275 \leq 500 - 16 t^{2} \leq 400 \end{aligned}$

::275 - 500 - 16t-2400

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:

$\begin{aligned} h (t) = 500 - 16 t^{2} \end{aligned}$
::h(t)=500-16t2

$\begin{aligned} h (t) \geq 275 \end{aligned}$
:t)_____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

$\begin{aligned} h (t) \leq 400 \end{aligned}$
:t)4000

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

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

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。

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 $\begin{aligned} h (x) = - 16 x^{2} + 112 x + 29. \end{aligned}$ 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。

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?

Example
::示例示例示例示例

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

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。

$\begin{aligned} 200 & \leq - 16 x^{2} + 112 x + 29 \\ - 200 & - 200 \\ 0 & \leq - 16 x^{2} + 112 x - 171 \end{aligned}$

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

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

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

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

T he expression $\begin{aligned} - (4 x - 9) (4 x - 19) \end{aligned}$ must be greater than or equal to zero. T he values that make the expression $\begin{aligned} - (4 x - 9) (4 x - 19) \end{aligned}$ equal to zero are $\begin{aligned} \frac{9}{4} = 2.25 and \frac{19}{4} = 4.75. \end{aligned}$ Since this function forms a parabola when graphed, all positive values must either be inside or outside the $\begin{aligned} x \end{aligned}$ - 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。

T est a value between 2.25 and 4.75 to see if it will make $\begin{aligned} - (4 x - 9) (4 x - 19) \end{aligned}$ 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。

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

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

Since 3 makes the expression positive, all values between 2.25 and 4.75 will make the expression $\begin{aligned} (- 4 x + 9) (4 x - 19) \end{aligned}$ positive. This means that all values between 2.25 and 4.75 will be above 200 feet in the function $\begin{aligned} h (x) = - 16 x^{2} + 112 x + 29. \end{aligned}$
::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。

Answer: $\begin{aligned} 2.25 \leq x \leq 4.75 \end{aligned}$
::答复:2.25x4.75

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

Discussion Question : Could you have determined that any $\begin{aligned} x \end{aligned}$ -value between the $\begin{aligned} x \end{aligned}$ -intercepts would result in a positive output based on the $\begin{aligned} a \end{aligned}$ - 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?

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

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。

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

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@$y\geq x%2 - 5x+7@$。

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。

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。

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。

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。

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。

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。

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。

Discussion Question : A firework that is launched can be modeled using the function $\begin{aligned} h (x) = - 16 x^{2} + 112 x + 29. \end{aligned}$ 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?

Wrap-Up: Review Questions
::总结:审查问题

Summary
::摘要

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 :

$\begin{aligned} 0 \leq a x^{2} + b x + c \end{aligned}$ 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。

$\begin{aligned} 0 \geq a x^{2} + b x + c \end{aligned}$ and find the values that make the inequality negative.
::0ax2+bx+c 并找到不平等负值。

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。

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。

Section outline

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

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

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

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

Wrap-Up: Review Questions ::总结:审查问题

Summary ::摘要

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

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

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

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

Wrap-Up: Review Questions
::总结:审查问题

Summary
::摘要