Course: 1.3 变化率-interactive

Section outline

Select section General

Collapse Expand
General

Collapse all Expand all
- Select activity 新闻通告
  
  新闻通告 Forum

变化率变化率

The Purpose of This Lesson
::本课程的目的

In this lesson you will explore the rate of change for a linear function , called the slope. You'll see that the rate of change is visible in tables, equations, and graphs of .
::在此课程中, 您将探索线性函数的变速率, 称为斜坡。您将会看到变化率在表格、方程式和图表中可见。

Introduction: Rate of Change
::导言:变化率

The rate of change is one of the essential characteristics of a function . A function describes what the relationship is between two quantities. The rate of change tells us how one quantity changes as the other changes. R ates of change are everywhere, every day: the miles per hour at which you are traveling, the number of liters of water you consume per week, your heart rate in beats per minute. For a linear function, the rate of change is represented by the parameter $\begin{aligned} m \end{aligned}$ in the slope-intercept form for a line: $\begin{aligned} y = m x + b \end{aligned}$ , and is visible in a table or on a graph.
::变化速率是函数的基本特征之一。函数描述两个数量之间的关系。变化速率告诉我们一个数量是如何变化的。变化速率告诉我们一个数量是如何变化的。变化速率无处不在,每天都有:旅行时每小时的里程、每周消耗的水升数、每分钟消耗的水的心率。对于线性函数来说,变化速率以Y=mx+b线的斜坡界面中的参数 m 表示,并在表格或图表中可见。

Activity 1: R epresenting Rate of Change
::活动1:代表变化率

Example 1 - 1
::例1-1

Mike is driving from Portland to San Diego. After 3 hours, he has travelled 150 miles. After 7 hours, he has travelled 350 miles. What does this mean? How far does Mike travel in 10 hours? Why? Make a table, equation , and graph to represent the distance Mike has traveled by different time periods.
::麦克从波特兰开车到圣地亚哥。 3小时后, 他已经走了150英里。 7小时后, 他已经走了350英里。这是什么意思 ? 迈克在10小时后要走多远? 为什么? 制作一张表格、方程和图表来代表迈克在不同时间段旅行的距离。

Solution: Here is a table representing the distance Mike has traveled by different times. Can you determine Mike's speed from these two points? How? How can you use the table to determine how far Mike has gone after $\begin{aligned} 8 \end{aligned}$ hours?
::解答: 这是一张显示Mike在不同时间行走距离的桌子。你能从这两个点上确定Mike的速度吗? 如何? 您如何使用这张桌子来确定Mike在8小时后走了多远?

\begin{aligned} \begin{array}{cc} time (hours) & distance (miles) \\ 3 & 150 \\ 7 & 350 \end{array} \end{aligned}

::距离(英里) 31507350

Speed is the rate of change of distance over time. There are several ways to think about and compute the rate of change:
::速度是随时间推移距离变化的速度。有几种方法可以思考和计算变化速度:

\begin{aligned} \begin{array}{l} \frac{rise}{run} = \frac{200}{4} \\ \frac{Δ y}{Δ x} = \frac{200}{4} \\ \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{350 - 150}{7 - 3} \end{array} \end{aligned}

::上上浮=2004 yx=2004y2-y1x2-x1=350-1507-3

All of these representations show that Mike is moving at a speed of 50 miles per hour. An increase of 1 hour of time implies a corresponding increase of 50 miles of distance. Use that to extend the table:
::所有这些表述都表明迈克的移动速度是每小时50英里。增加1小时意味着相应增加50英里的距离。用这个来延长表格的长度 :

\begin{aligned} \begin{array}{cc} time (hours) & distance (miles) \\ 3 & 150 \\ 7 & 350 \\ 8 & 400 \end{array} \end{aligned}

::距离(英里) 315073508400

Here is a graph representing the distance Mike has traveled by different times. Can you see how you could determine Mike's speed from the graph? According to the graph, about how far has Mike gone after 2 hours and 20 minutes?
::这是一张显示Mike在不同时间行走的距离的图表。您能看到如何从图表中确定Mike的速度吗? 根据图表, Mike在2小时20分钟后走了多远?

We can draw a slope triangle between any two lattice points ( points at the intersection of gridlines, and observe the rise over run :
::我们可以在任何两个网格点(网格线交叉点)之间绘制斜度三角形,

To find the rate of change:
::为了找到变化率:

$\begin{aligned} \frac{rise}{run} = \frac{100}{2} = 50 \end{aligned}$
::上升=1002=50

$\begin{aligned} m = 50 \end{aligned}$
::m=50

That's the same slope found by other methods previously.
::和以前其他方法一样的斜坡

You can see the y-intercept is 0.
::你可以看到 Y 界面是 0 。

$\begin{aligned} b = 0 \end{aligned}$
::b=0 (b=0)

Slope-intercept form of a line is $\begin{aligned} y = m x + b \end{aligned}$ . Mike's distance, $\begin{aligned} y, \end{aligned}$ as a function of time , $\begin{aligned} x, \end{aligned}$ is described by the equation:
::线Isy=mx+b的斜坡截住形式。 Mike的距离, y, 作为时间的函数, x, 用方程式描述 :

$\begin{aligned} y = 50 x \end{aligned}$
::y=50x y=50x

To determine Mike's distance after 2 hours and 40 minutes, convert 40 minutes to a fraction of an hour: $\begin{aligned} x = 2 \frac{2}{3} . \end{aligned}$ Now substitute :
::为确定麦克在2小时40分后距离,将40分钟转换成一个小时的分数:x=223。现在替换 :

\begin{aligned} \begin{array}{l} y = 50 (2 \frac{2}{3}) \\ y = 50 (\frac{8}{3}) \\ y = \frac{400}{3} \\ y = 133 \frac{1}{3} miles \end{array} \end{aligned}

::yy=50(83y=4003y=13313英里)

PLIX Interactive
::PLIX 交互式互动

Activity 2: Model Constant Rate of Change
::活动2:示范不变变化率

Linear functions feature a constant rate of change. This is called the slope . The slope can be seen in the table, the equation, and the graph for a line. Here is a table showing the different ways that slope can be represented and calculated. An example of each representation is also given. Discuss the advantages and disadvantages of each.
::线性函数具有固定的变速率。这称为斜坡。斜坡可以在表格、方程和直线的图表中看到。下面是一张表格, 显示斜坡的表达和计算方式, 并给出每个表达方式的示例。讨论每个表达方式的利弊。

Work it Out
::工作出来

Suppose Kyrie and Jessica live in New York. They are both driving cross-country on a road trip but drive separate cars. Kyrie leaves early, and by the time Jessica gets on the road, Kyrie has already driven 200 miles. Kyrie drives at 50 miles per hour, while Jessica drives at 65 miles per hour. Create a table, equation, and graph to model each driver. Graph both on the same coordinate axes. Determine when Jessica passes Kyrie.
::假设Kyrie和Jessica都住在纽约。他们两人都是在公路旅行中驾驶跨国汽车,但驾驶不同的汽车。Kyrie提前离开,到Jessica上路时,Kyrie已经驾驶了200英里。Kyrie以每小时50英里的速度驾驶,Jessica以每小时65英里的速度驾驶。创建一张表格、方程和图表来模拟每个司机。在同一条坐标轴上绘制两个图。确定Jessica何时通过Kyrie。

Let $\begin{aligned} x \end{aligned}$ be the number of hours that have passed since Jessica starts driving. Let $\begin{aligned} y \end{aligned}$ be the distance traveled. When Jessica starts driving, Kyrie is already at the 200-mile mark. So Kyrie's table begins like this:
::让 x 成为自Jessica 开始驾驶以来经过的小时数。让 y 成为所经过的距离。当Jessica 开始驾驶时, Kyrie 已经到达200英里的标记。所以 Kyrie 的桌子开始像这样:

\begin{aligned} Kyrie \\ \begin{array}{cc} x & y \\ 0 & 200 \end{array} \end{aligned}

::Kyriexy0200

Use his speed to add more values.
::使用他的速度添加更多的值。

Jessica's table begins like this:
::杰西卡的桌子是这样开始的:

\begin{aligned} Jessica \\ \begin{array}{cc} x & y \\ 0 & 0 \end{array} \end{aligned}

::杰西卡 xy00

Use her speed to add more values, then find the equation and graph for each.
::使用她的速度添加更多的值, 然后找到每一个方程式和图形。

Write and solve an equation to determine when Jessica passes Kyrie. Determine the distance traveled by each at that time.
::写入并解析一个方程以确定 Jessica 何时通过 Kyrie 。确定当时每个人的距离。

These two functions show us the distance traveled by each person as a function of time:
::这两个功能显示了每个人作为时间函数所走的距离:

\begin{aligned} \begin{array}{ll} Kyrie & y = 50 x + 200 \\ Jessica & y = 65 x \end{array} \end{aligned}

::Kyriey = 50x + 200 Jessicay = 65x

The desired equation will answer the question: What $\begin{aligned} x \end{aligned}$ value returns the same $\begin{aligned} y \end{aligned}$ value for both? The objective is to find the point where Kyrie's distance ( his $\begin{aligned} y \end{aligned}$ value) is the same as Jessica's distance ( her $\begin{aligned} y \end{aligned}$ value). Since you want the $\begin{aligned} y \end{aligned}$ 's from each equation to be equal, you want the part of each equation that is equal to $\begin{aligned} y \end{aligned}$ to be the same also. Y ou can use the following equation to find the $\begin{aligned} x \end{aligned}$ -value that makes the sides match:
::想要的方程式将解答一个问题: 什么是 x 值返回两个数值相同的y 值? 目标是找到 Kyrie 的距离( y 值) 与 Jessica 的距离( her y 值) 相同的点。由于您希望每个方程式的 y 值相等, 您想要每个方程式中与 y 相等的部分也相同。您可以使用以下方程式找到使两边匹配的 x 值 :

\begin{aligned} 50 x + 200 = 65 x \end{aligned}

::50x50+200=65x

Use the value you find to determine the distance traveled by each.
::使用您找到的值来确定每个人的距离。

Interactive
::交互式互动

Base jumpers jump from cliffs with parachutes. These days, base jumpers wear wingsuits which enable them to glide along desired paths. The interactive below shows the path of a wingsuit flier named Holly. The $\begin{aligned} y \end{aligned}$ -axis represents her height above the ground in meters, and the $\begin{aligned} x \end{aligned}$ -axis represents her distance in meters from the base of the cliff. The vertical segment represents a tower she wishes to fly right over, without crashing into it. Adjust the slope so that she just clears the tower.
::基底跳跃者用降落伞从悬崖跳下。这些天, 基底跳跃者穿戴翅膀衣, 可以滑翔在理想的道路上。下面的互动展示了一只叫霍莉的翅膀飞翔机的路径。 Y 轴代表她在地面上的高度, 以米计, x 轴代表她距离悬崖底部的距离。垂直段代表着一个她希望飞过而不会坠入的塔。调整斜坡, 以便她能够清空高塔。

INTERACTIVE

Wingsuit Tower

Move the blue point to adjust the slope of the base-jumpers path.
::移动蓝色点以调整基点偏移路径的斜度。

Interactive
::交互式互动

Holly wants to base jump from a new cliff, as shown in the interactive below. She wants to just barely clear two towers.
::霍莉想从一个新的悬崖上跳下跳下,下面的交互显示。她想刚刚清空两座塔。

Adjust the slope and $\begin{aligned} y \end{aligned}$ - intercept for this line using the sliders. Find the equation for this line.
::使用幻灯片调整此行的斜度和 Y 界面。查找此行的方程式。

If Holly deploys her parachute and keeps floating down along the same line, where will she land? Find this answer using two different methods. Explain which you prefer and why.
::如果霍莉部署降落伞并一直沿着同一条线漂浮,她会在哪里着陆?用两种不同的方法找到这个答案。解释你喜欢什么和为什么。

Do you think her slope is reasonable? Why or why not?
::你觉得她的斜坡合理吗?

After completing the problem, click on the reset button to experiment with different tower heights.
::完成问题后, 点击重置按钮来实验不同的塔高。

INTERACTIVE

Two Towers

Move the blue points to change the slope and elevation of the base-jumpers path.
::移动蓝色点以改变基点-弹道路径的斜度和高度。

Example 2-1
::例2-1

Three different pizza shops are competing to see who can sell the most pizzas by the end of the day. Here are tables representing the number of pizzas they've sold at different times.
::三个不同的比萨饼店在竞争看谁能在今天之前卖出最多的比萨饼。这里的表格代表了他们在不同时间卖出的比萨饼的数量。

Luigi's	Mario's	Francesca's
$\begin{aligned} \begin{array}{cc} x & y \\ 2 & 12 \\ 3 & 20 \\ 7 & 44 \end{array} \end{aligned}$	$\begin{aligned} \begin{array}{cc} x & y \\ 3 & 16 \\ 5 & 22 \\ 8 & 31 \end{array} \end{aligned}$	$\begin{aligned} \begin{array}{cc} x & y \\ 4 & 12 \\ 5 & 24 \\ 6 & 48 \end{array} \end{aligned}$

Summary
::摘要

A function describes what the relationship is between two quantities, while the rate of change describes how one quantity changes with respect to the other.
::函数描述两个数量之间的关系,而变化率则描述一个数量相对于另一个数量的变化。

The rate of change for a line is the slope , the rise over run , or the change in $\begin{aligned} y \end{aligned}$ over the change in $\begin{aligned} x . \end{aligned}$
::一条线的变动率是斜坡、递增率或相对于 x 的变动y 的变动率。

The slope can be calculated from two points in a table or from the slope triangle in a graph.
::斜度可以从表格中的两个点或从图表中的斜度三角点计算出来。

The slope is the parameter $\begin{aligned} m \end{aligned}$ in the slope=intercept form of a line: $\begin{aligned} y = m x + b . \end{aligned}$
::斜度是 y=mx+b 线的斜度= 界面形式的参数 m。

Section outline

General

变化率变化率

The Purpose of This Lesson ::本课程的目的

Introduction: Rate of Change ::导言:变化率

Activity 1: R epresenting Rate of Change ::活动1:代表变化率

Example 1 - 1 ::例1-1

PLIX Interactive ::PLIX 交互式互动

Activity 2: Model Constant Rate of Change ::活动2:示范不变变化率

Work it Out ::工作出来

Interactive ::交互式互动

Interactive ::交互式互动

Example 2-1 ::例2-1

Summary ::摘要

PLIX Interactive ::PLIX 交互式互动

The Purpose of This Lesson
::本课程的目的

Introduction: Rate of Change
::导言:变化率

Activity 1: R epresenting Rate of Change
::活动1:代表变化率

Example 1 - 1
::例1-1

PLIX Interactive
::PLIX 交互式互动

Activity 2: Model Constant Rate of Change
::活动2:示范不变变化率

Work it Out
::工作出来

Interactive
::交互式互动

Interactive
::交互式互动

Example 2-1
::例2-1

Summary
::摘要

PLIX Interactive
::PLIX 交互式互动