Problem 37 - Algebra 2

Question

For each function, find the percent increase or decrease that the function models. $$ y=0.65(1.3)^{x} $$

Step-by-Step Solution

Answer

The function models a 30% increase.

Step 1: Identify The Form of The Function

The given function \(y=0.65(1.3)^{x}\), represents an exponential growth model. Here, \(a=0.65\) represents the initial value and \(b=1.3\) is the growth factor.

Step 2: Calculate The Percentage Increase

Apply the formula for calculating the percentage increase or decrease, \((b-1) \times 100%\). Here, substitute \(b = 1.3\) in the formula.

Step 3: Calculate and Answer

Calculate \((1.3-1) \times 100% = 0.3 \times 100% = 30%\). So, the function models a 30% increase.

Key Concepts

Understanding Percent Increase

In the context of exponential functions, the percent increase tells us how much a quantity grows relative to its original amount over each period. When dealing with these functions, the percent increase is derived from the growth factor. Here's how to spot it:

• If the growth factor (often denoted by 'b') is more than 1, we have an exponential growth situation, indicating a percent increase.
• Calculate it using the formula: \((b-1) \times 100\%\). This formula effectively transforms the growth factor minus one into a percentage.

For our exercise, the growth factor is 1.3. Hence, the percent increase calculation looks like this: \((1.3 - 1) \times 100\% = 30\%\). Thus, each unit increase in 'x' results in a 30% increase in 'y'.

Deciphering the Growth Factor

The growth factor in an exponential function is key in determining how a quantity changes over time. In a function of the form \( y = a(b)^x \), 'b' represents this key value. It captures the base ratio by which the quantity changes for every single unit increment in 'x'.

• When \(b > 1\), it signifies growth, indicating the quantity is increasing.
• When \( 0 < b < 1 \), it indicates decay, or a decrease over time.

For our given function \( y = 0.65(1.3)^x \), the growth factor is 1.3. This tells us the quantity grows by a factor of 1.3 for each added 'x'. In simple terms, it grows 30% bigger with each step, as shown by the previous percent increase calculation.

Exploring Exponential Growth

Exponential growth is a process where a quantity increases by a constant percentage over equal time intervals. It's characterized by its quick escalation; small initial increases can compound into steep changes over time.

• This growth is modeled mathematically by functions like \( y = a(b)^x \).
• Exponential growth is prevalent in numerous real-world scenarios, including population growth, financial investments, and biological processes.

In our exercise's context, \( y = 0.65(1.3)^x \), we see exponential growth since \(b = 1.3\), which signifies the 30% increase per unit as discussed. Each increment in 'x' causes the output 'y' to grow significantly.

Recognizing Initial Value

The initial value in an exponential function like \( y = a(b)^x \) is represented by 'a'. It forms the starting point from which the growth or decay begins. Recognizing and understanding this value is crucial:

• 'a' provides the baseline quantity before any growth factor affects it.
• Think of it as the value when \( x = 0 \), since any number raised to the power of zero equals one. Thus, \( y = a \times 1 = a \).

In the function \( y = 0.65(1.3)^x \), the initial value 'a' is 0.65. This means at the start (when \( x = 0 \)), 'y' equals 0.65. From here, the function's exponential behavior unfolds according to the growth factor.