Problem 55 - Algebra Form and Function

Question

The number of butterflies in a collection \(x\) years after 1960 is given by \(B(x)=50+2(x-20)\) (a) What is the practical interpretation of the constants \(20,50,\) and \(2 ?\) (b) Express \(B\) in a form that clearly shows the size of the collection when it started in 1960 .

Step-by-Step Solution

Answer

Answer: In the given function, the constant 20 represents the starting point of the linear growth (1980), the constant 50 represents the number of butterflies in the collection at the starting point of the linear growth (1980), and the constant 2 represents the rate at which the number of butterflies increases every year after 1980. The initial size of the butterfly collection in 1960 was 10.

Step 1: Analyze the function B(x)

The function B(x) given is: B(x) = 50 + 2(x - 20) By observing the function, we can see that it's a linear function with x being the variable.

Step 2: Interpret the constants

We have three constants in the function: 20, 50, and 2. 1. The constant 20 represents the shift in the x-axis. Since x represents the years after 1960 (x=0 in 1960), when x=20, it means it's now 1980 (1960+20). This constant indicates the starting point of the linear growth of the collection. 2. The constant 50 is the value of B(x) when x=20. In other words, it represents the number of butterflies in the collection at the starting point of the linear growth, which is 1980. 3. The constant 2 is the slope or rate at which the number of butterflies increases every year after 1980.

Step 3: Express B in a form that shows the initial collection size in 1960

We will now rearrange the function B(x) to express it in a form showing the initial collection size in 1960. First, we need to find the value of B(x) when x = 0. B(0) = 50 + 2(0 - 20) B(0) = 50 - 40 B(0) = 10 Now we can rewrite the given function B(x) in a form that shows the initial collection size: B(x) = 10 + 2(x - 0) The function B(x) = 10 + 2(x - 0) clearly shows that in 1960 (when x=0), the size of the butterfly collection was 10.

Key Concepts

Constants Interpretation

When dealing with linear functions, constants play a crucial role in understanding the nature and transformation of the function. In the function given: \[B(x) = 50 + 2(x - 20)\] we have three constants: 20, 50, and 2. Let's break down their significance:

• **Constant 20:** This represents the shift along the x-axis. Here, it tells us that the calculation begins 20 years after 1960, which is 1980. Thus, this constant serves as a reference year, indicating the start of this linear progression of butterfly collection.
• **Constant 50:** This shows the number of butterflies in 1980, which is the starting point for this linear rate in the function. It is important for anchoring the function at that point in time.
• **Constant 2:** This is the slope, indicating how many butterflies are added each year. It paints the picture of the growth rate of the butterfly collection after 1980.

Understanding the role of these constants helps in interpreting the linear function's real-world application, providing clarity on events over time.

Slope

In linear functions, the slope is synonymous with the rate of change. In our function \(B(x) = 50 + 2(x - 20)\), the slope is represented by the constant 2. Here's what the slope does:

• The slope indicates that for every year increase in time (from 1980 onward), the number of butterflies increases by 2. This means the collection grows steadily with the passage of time.
• This constant gradient or slope provides insight into the steady accumulation of butterflies. It is a fixed number because linear functions depict a consistent rate of change.

Understanding the slope is vital for analyzing data over time, as it helps predict future outcomes and recognize patterns. It directly determines how steep or gradual the change is in the context of the problem, offering a glimpse into the constant growth dynamic in the butterfly collection.

Function Transformation

Function transformation refers to changes made to the graph of a function. These transformations can be shifts, reflections, stretches, or compressions. In our linear function, the transformations are seen mainly in the form of a horizontal shift and the set of growth. In transforming \(B(x) = 50 + 2(x - 20)\), our focus is to shift the function back to see its original state in 1960:

• **Vertical Transformation:** This involves the "50" constant, setting the count of butterflies at the 1980 starting point.
• **Horizontal Transformation:** The term \((x-20)\) affects how the function is interpreted over time. It shows the timeline beginning in 1980 instead of 1960.

To transform it to showcase the starting value in 1960, we rearrange to \(B(x) = 10 + 2(x)\). This presents us with the initial count of butterflies as 10, illustrating a shift back to where x equals zero (1960).Such transformations allow us to visualize and interpret a function's behavior throughout different periods and conditions, making the analysis of historical data straightforward and intuitive.