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7.1 Modeling Linear Relationships
7.1 线性关系建模

Essential Question: How can you model linear relationships given limited information?
基本问题：在给定有限信息的情况下，如何对线性关系进行建模？

Explore Modeling Linear Relationships with Slope-Intercept Form
探索使用 Slope-Intercept 形式对线性关系进行建模

A department store offers a frequent-buyers card to earn rewards for purchases customers make at the store. Each transaction is worth 12 points, and customers automatically earn 25 points when they sign up.
百货公司提供常客卡，客户在商店购物时可以赚取奖励。每笔交易价值 12 积分，客户在注册时自动获得 25 积分。
Write an equation for the function that gives the card value based on the number of transactions that have occurred.
为函数编写一个方程式，该方程式根据已发生的交易数量给出卡值。
(A) What units would be associated with the variables in this function? ◻ ◻ ◻\square
（A） 哪些单位将与此函数中的变量相关联？ ◻ ◻ ◻\square
(B) Complete the verbal model for the frequent-buyers card function. Include units.
（B） 完成常客卡功能的口头模型。包括单位。
Card Value (points) = = == Initial Value ( ( (( points ) + ) + )+)+ ◻ ◻ ◻\square
卡值 （积分） = = == 初始值 ( ( (( 积分 ) + ) + )+)+ ◻ ◻ ◻\square
© Write the function rule for the card-value function C C CC.
© 为 card-value 函数 C C CC 编写函数规则 。
C ( t ) = ◻ + t C ( t ) = ◻ + t C(t)=◻quad+quad tC(t)=\square \quad+\quad t, where t t tt is the number of transactions.
C ( t ) = ◻ + t C ( t ) = ◻ + t C(t)=◻quad+quad tC(t)=\square \quad+\quad t ，其中 t t tt 是事务数。
◻ ◻ ◻\square
◻ ◻ ◻\square
(D) For each 100 points, the customer receives a gift certificate. How many transactions will it take for the customer to earn the first gift certificate? ◻ ◻ ◻\square
（D） 每获得 100 积分，客户将收到一张礼券。客户需要进行多少笔交易才能获得第一张礼品券？ ◻ ◻ ◻\square
(E) What is the y y yy-intercept for this linear function, and what does it represent? ◻ ◻ ◻\square
（E） 这个线性函数的 y y yy -intercept 是什么，它代表什么？ ◻ ◻ ◻\square
(F) What is the slope for this linear function, and what does it represent? ◻ ◻ ◻\square
（F） 这个线性函数的斜率是多少，它代表什么？ ◻ ◻ ◻\square

Reflect  反映

  1. Discussion Use the function rule to show that the units for C ( t ) C ( t ) C(t)C(t) are points.
    讨论 使用函数规则来显示 的 C ( t ) C ( t ) C(t)C(t) 单位是点。
  2. Critical Thinking What types of number are appropriate for the domain of C ( t ) C ( t ) C(t)C(t) ?
    批判性思维 哪些类型的数字适合 C ( t ) C ( t ) C(t)C(t) 的领域 ？
  3. Using inequalities, express the restrictions on the range of C ( t ) C ( t ) C(t)C(t).
    使用不等式，表示对 的范围的限制 C ( t ) C ( t ) C(t)C(t) 。

Explain Creating and Interpreting Linear Models
解释创建和解释线性模型

You can create linear equations and inequalities to model some real-world situations.
您可以创建线性方程和不等式来模拟某些实际情况。

Example Given the real-world situation, solve the problem.
示例 给定实际情况，解决问题。

Fundraising The Band Booster Club is selling T-shirts and blanket wraps to raise money for a trip. The band director has asked the club to raise at least $ 1000 $ 1000 $1000\$ 1000.
筹款 Band Booster Club 正在出售 T 恤和毛毯包裹物，以筹集旅行资金。乐队总监已要求俱乐部至少 $ 1000 $ 1000 $1000\$ 1000 筹集 .
The booster club president wants to know how many T-shirts and how many blanket wraps the club needs to sell to meet their goal of $ 1000 $ 1000 $1000\$ 1000. The T-shirts cost $ 10 $ 10 $10\$ 10 each, and the blanket wraps cost $ 25 $ 25 $25\$ 25 each. Write a linear equation that describes the problem, and then graph the linear equation. How can the booster club president use the sales price of each item to meet the goal?
助推器俱乐部主席想知道俱乐部需要销售多少 T 恤和多少毛毯包装才能达到他们的目标 $ 1000 $ 1000 $1000\$ 1000 。T 恤每个要花 $ 10 $ 10 $10\$ 10 钱，毛毯包每个要花 $ 25 $ 25 $25\$ 25 钱。编写一个描述问题的线性方程，然后绘制线性方程。助推器俱乐部总裁如何使用每件商品的销售价格来达到目标？

Analyze Information  分析信息

Identify the important information.
确定重要信息。
  • T-shirts cost $ 10 $ 10 $10\$ 10 each.
    T 恤每个都要花 $ 10 $ 10 $10\$ 10 钱。
  • Blanket wraps cost $ 25 $ 25 $25\$ 25 each.
    毯子包装每个都要花钱 $ 25 $ 25 $25\$ 25 。
  • The booster club needs to raise a total of $ 1000 $ 1000 $1000\$ 1000.
    助推器俱乐部需要筹集总共 $ 1000 $ 1000 $1000\$ 1000 。

Formulate a Plan  制定计划

The total amount of revenue earned by selling T-shirts is $ 10 t $ 10 t $10 t\$ 10 t. The total amount of revenue earned from selling blanket wraps is $ 25 b $ 25 b $25 b\$ 25 b. These two results can be added and set equal to the sales goal to find the number of T-shirts and blanket wraps that need to be sold to reach $ 1000 $ 1000 $1000\$ 1000. Graph this function to find all of the possible combinations of T-rts and blanket wraps sold to reach $ 1000 $ 1000 $1000\$ 1000.
销售 T 恤所赚取的总收入为 $ 10 t $ 10 t $10 t\$ 10 t 。销售毛毯包装所赚取的总收入为 $ 25 b $ 25 b $25 b\$ 25 b 。可以将这两个结果相加并设置为等于销售目标，以查找需要销售才能达到 $ 1000 $ 1000 $1000\$ 1000 的 T 恤和毛毯包装的数量。绘制此函数的图表可查找 T-rts 和毛毯包装的所有可能组合以达到 $ 1000 $ 1000 $1000\$ 1000 。

Solve  解决

Write a linear equation for the sales goal.
为销售目标编写一个线性方程。
25 b + 10 t = 1000 25 b + 10 t = 1000 25 b+10 t=100025 b+10 t=1000
Calculate three pairs of values for t t tt and b b bb, and graph a line through those points to find possible solutions. Be sure to label the graph.
计算 t t tt 和 b b bb 的三对值，并在这些点上绘制一条线以找到可能的解决方案。请务必标记图表。
t t t\boldsymbol{t} b b b\boldsymbol{b}
0 40
50 20
100 0
t b 0 40 50 20 100 0| $\boldsymbol{t}$ | $\boldsymbol{b}$ | | :---: | :---: | | 0 | 40 | | 50 | 20 | | 100 | 0 |

Justify and Evaluate  Justify 和 Evaluate

The x x xx-intercept represents the number of T-shirts that need to be sold if no blanket wraps are sold. The y y yy-intercept represents the number of blanket wraps to be sold if no T-shirts are sold. The booster club president can use the line to find the possible combinations of T-shirts and blankets to reach $ 1000 $ 1000 $1000\$ 1000.
- x x xx intercept 表示在没有销售毛毯包装的情况下需要销售的 T 恤数量。- y y yy intercept 表示在没有售出 T 恤的情况下要出售的毯子包装的数量。助推器俱乐部主席可以使用这条线找到 T 恤和毯子的可能组合来到达 $ 1000 $ 1000 $1000\$ 1000 。

Reflect  反映

  1. Critical Thinking Technically, the graph of possible combinations of T-shirts and blanket wraps that reach the goal of $ 1000 $ 1000 $1000\$ 1000 should be discrete, but for convenience the graph is shown as a connected line. Explain why the solutions to this problem would be only the points on the line that have whole-number coordinates.
    批判性思维 从技术上讲，达到目标的 T 恤和毛毯包裹的可能组合图 $ 1000 $ 1000 $1000\$ 1000 应该是离散的，但为方便起见，图表显示为一条连接线。解释为什么这个问题的解决方案只是线上具有整数坐标的点。

Your Turn  该你了

  1. Business A sandwich shop sell sandwiches for $ 5 $ 5 $5\$ 5 each and bottles of water for $ 1 $ 1 $1\$ 1 each. The owner of this shop needs to earn a total of $ 100 $ 100 $100\$ 100 by the end of the day. Write a linear equation that describes the problem; then graph the linear equation. Make sure to label both axes with appropriate titles. Then use the graph to determine how many sandwiches the shop must sell if no waters are sold.
    商业 一家三明治店出售每个三 $ 5 $ 5 $5\$ 5 明治和 $ 1 $ 1 $1\$ 1 瓶装水。这家店的老板需要在一天结束时总共 $ 100 $ 100 $100\$ 100 赚到。编写一个描述问题的线性方程;然后绘制线性方程。确保使用适当的标题标记两个轴。然后使用图表来确定如果没有售出水，商店必须销售多少三明治。

Elaborate  精巧

  1. How can the graph of a linear function be used to find answers to a real-world problem?
    如何使用线性函数的图来查找实际问题的答案？
  2. Essential Question Check-In What is the first step when modeling linear relationships given limited information?
    基本问题签入 在给定有限信息的情况下对线性关系进行建模时，第一步是什么？