linear models


Linear models are a part of everyday life, but many times they are not easily visible. They can be easily related to our life experiences. Read each of the three scenarios below to see how linear models show up in our lives and how we can use them to make decisions. Answer the related questions.

  1. Marco and his two younger sisters would like to purchase a silver charm bracelet for their mother’s birthday. They went to the mall and found what they were looking for at Store A. In Store A, the bracelet without charms costs $85 and each charm costs $15.
    1. Use function notation that models the total price of the bracelet and how that price is based on the number of charms. Explain the reasoning behind your equation.
    2. What would be a reasonable domain for this function based on this scenario? Explain why this is a proper domain.
    3. If Marco and his sisters have saved $250, make a graph to show all of the charms they could purchase with a bracelet if they had saved between $85 and $250.
    4. Marco found five charms that he likes.  Using your function, determine the cost of the bracelet he would make.
    5. As they walked down the hall of the mall, Marco saw this sales flyer for Store B on a bench.
Price for Bracelet is $100, each charm is $12.

Marco is wondering from which store he should buy the items so that he spends the least amount of money. Which store should Marco use if he wanted the five charms and silver bracelet all together? What advice can you tell Marco to help him shop wisely depending on the number of charms he wants to buy? Justify how you reasoned your answers.

  1. While in the United States visiting her grandmother, Kendra copied the famous family apple pie recipe. When she returns to England, she plans on making the same apple pie for her friends. To do this, Kendra needs to convert the flour from cups to grams. She knows that 0.5 cup of flour has a mass of 64 grams and 2 cups of flour has a mass of 256 grams.
    1. Write a function that shows grams as a function of cups. Use g(x).
    2. Given this scenario, determine a reasonable domain for this function.
    3. Graph the relationship between the number of cups and grams of flour.
    4. Evaluate g(x) for any value of x within the domain. Interpret the meaning of your solution in the context of this problem.
    5. Look at the method you used to evaluate your function in part D. Explain another way you can evaluate the function for the same input.
    6. Interpret what it means to have g(-2), and determine if this is reasonable given the situation.
  1. A coffee shop pays Coffee Delivery Company A, a certain price for each disposable cup it orders plus a weekly delivery charge to remain on the driver’s delivery route. The cups are purchased in increments of 500. To quickly determine how much the coffee shop will be spending on cups before their arrival, the owner created the following table:

    Cups Ordered Price
    0 $7
    500 $25.50
    1,000 $44
    1,500 $62.50
    2,000 $81
    2,500 $99.50
    3,000 $118

    1. The price of the cups, p(x), is a function of the number of x cups ordered. Using the table, determine the average rate of change for the first 1,000 cups ordered and then for all 3,000 cups ordered. What does this tell you about the function?
    2. Use the table to evaluate and interpret p(0). What is a possible explanation for this?
    3. Sketch a graph, labeling its key features, to show the price the coffee shop would pay Coffee Delivery A to have between 0 and 3,000 cups delivered each week.
    4. Create a model using function notation that represents how the two quantities, cups and cost, are related.
    5. The coffee shop found another delivery company that sells orders at increments of 500 cups, Coffee Delivery B. They charge $3.50 each week to be on their delivery route and charge 3.9 cents per disposable cup.  
      • Make a function using the information about the second delivery company.
      • Graph the price the coffee shop would pay for Coffee Delivery B to deliver the same amount of cups on the same graph from Part C, using a different color for the new line.
      • If the coffee shop can change delivery companies every three months, when should they consider Coffee Delivery A, and when should they consider Coffee Delivery B?


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