Problem 32 - Algebra Form and Function

Question

The following functions describe four different collections of baseball cards. The collections begin with different numbers of cards and cards are bought and sold at different rates. The number, \(B\), of cards in each collection is a function of the number of years, \(t,\) that the collection has been held. Describe each of these collec- tions in words. (a) \(B=200+100 t\) (b) \(\quad B=100+200 t\) (c) \(\quad B=2000-100 t\) (d) \(\quad B=100-200 t\)

Step-by-Step Solution

Answer

Question: Describe the rate of change in the number of cards for each given function representing collections of baseball cards as a function of time. Answer: For Function (a), the rate of change is 100 cards added per year. For Function (b), it is 200 cards added per year. For Function (c), the rate of change is 100 cards removed per year. For Function (d), it is 200 cards removed per year.

Step 1: Function (a) Analysis: B = 200 + 100t

For this function, when the time (t) is 0, the number of cards, B, is 200. This means there were initially 200 cards in the collection. The rate of change in the number of cards is 100 cards per year, which indicates that the collection grows by 100 cards each year.

Step 2: Function (a) Description

This collection starts with 200 baseball cards, and 100 cards are being added each year.

Step 3: Function (b) Analysis: B = 100 + 200t

For this function, when the time (t) is 0, the number of cards (B) is 100. This implies that there were initially 100 cards in the collection. The rate of change in the number of cards is 200 cards per year, meaning the collection grows by 200 cards each year.

Step 4: Function (b) Description

This collection starts with 100 baseball cards, and 200 cards are being added each year.

Step 5: Function (c) Analysis: B = 2000 - 100t

For this function, the initial number of cards (B) is 2000 when the time (t) is 0. In this case, the rate of change in the number of cards is -100 cards per year. This indicates that the collection is decreasing by 100 cards each year.

Step 6: Function (c) Description

This collection starts with 2000 baseball cards, and 100 cards are being removed each year.

Step 7: Function (d) Analysis: B = 100 - 200t

For this function, the initial number of cards (B) is 100 when the time (t) is 0. Here, the rate of change in the number of cards is -200 cards per year, meaning the collection decreases by 200 cards each year.

Step 8: Function (d) Description

This collection starts with 100 baseball cards, and 200 cards are being removed each year.

Key Concepts

Rate of Change

When discussing linear functions, the rate of change is a critical concept. It tells us the amount by which a variable changes over a certain period. Think of it as the "pace" at which something occurs.
In the context of the baseball card collections, it refers to how many cards are either added to or removed from the collections every year.

• For function (a) and function (b), we see a positive rate: 100 cards per year and 200 cards per year, respectively. This means these collections are growing over time.
• In contrast, function (c) and function (d) have negative rates: -100 cards per year and -200 cards per year, respectively. This indicates a reduction in cards each year.

Understanding the rate of change in a function enables you to visualize and predict the behavior of that function over time.

Initial Value

Another fundamental concept in understanding linear functions is the initial value. This represents the point where the function begins before any changes (like growth or reduction) happen.
In the baseball card problem, the initial value is the number of cards at the start, when time, \( t \), is zero.

• For function (a), the initial value is 200 cards.
• Function (b) starts with 100 cards.
• Function (c) has a high initial value of 2000 cards.
• Lastly, function (d) begins with 100 cards.

Understanding the initial value is important because it gives insight into the starting point of a scenario or function, setting the stage for how changes will proceed from there.

Function Analysis

Analyzing a function involves understanding both its rate of change and initial value, allowing us to describe how a function behaves over time.
Let's take a look at each function from our problem:

• Function (a): Starts with 200 cards, adding 100 cards each year. This indicates a steady increase in the collection size over time.
• Function (b): Begins with 100 cards. It has a faster growth rate, adding 200 cards per year, which results in more rapid collection growth.
• Function (c): Starts at 2000 cards but decreases by 100 cards each year. Here, there's a steady decline, gradually diminishing the collection.
• Function (d): Initial collection size is 100 cards, and it shrinks rapidly, losing 200 cards each year.

Function analysis is a powerful tool to predict future outcomes, compare scenarios, and make informed decisions based on how things are expected to change over time.

Word Problem

When tackling word problems with linear functions, it's important to extract the relevant information and translate it into a mathematical model.
In our baseball card problem, each statement provides clues:

• The starting number of cards is the initial value.
• The mention of adding or removing cards annually helps determine the rate of change.

Approaching word problems involves recognizing patterns, identifying constants and variables, and setting up equations like \(B = 200 + 100t\). Once a function is established, you can analyze and use it to describe the situation. This practice helps build a strong foundation for problem-solving and analytical skills that are crucial across various academic subjects.