Problem 28 - Algebra 1: Concepts and Skills

Question

Identify the growth rate and the growth factor in the exponential function. $$y=50(1+1)^{t}$$

Step-by-Step Solution

Answer

The growth factor for the given exponential function is 2 and the growth rate is 100%.

Step 1: Identify the Initial Quantity

By comparing the given equation \(y=50(1+1)^{t}\) with the general format \(y=ab^{x}\), it becomes evident that the initial quantity 'a' is 50.

Step 2: Identify the Growth Factor

The base of the power in the equation \(y=50(1+1)^{t}\) represents the growth factor 'b'. Hence, the growth factor here is equal to \(1+1=2\).

Step 3: Calculate the Growth Rate

The growth rate for exponential functions can be calculated using the formula: growth rate = 100 * (b - 1). Substituting the value of the growth factor 'b' from Step 2, the growth rate = 100 * (2 - 1) = 100%.

Key Concepts

Growth Rate

The growth rate is a key concept when dealing with exponential functions. It essentially tells you how much a quantity increases or decreases over a specific period of time. In the context of exponential growth, the growth rate is always positive, indicating an increase. Here, the growth rate can be found by using the formula:\[\text{Growth Rate} = 100 \times (b - 1)\]where \(b\) is the growth factor. From the provided solution, the growth factor \(b\) is determined as 2. Therefore, the growth rate is calculated as:\[100 \times (2 - 1) = 100\%\]This means the quantity doubles (which is a 100% increase) at each time interval. Bullet points help underline important points:

• Growth rate is determined by subtracting 1 from the growth factor.
• The result is then multiplied by 100 to convert it to a percentage.
• A positive growth rate indicates an increase in quantity.

Remember, understanding the growth rate as a percentage helps visualize how fast or slow something grows over time.

Growth Factor

In an exponential function, the growth factor is a crucial element that informs us about the multiplier applied to the initial quantity for each time interval. The growth factor is the base of the exponent in the equation. For the given equation:\[y = 50(1+1)^{t}\]The growth factor can be directly extracted as 2, because the expression inside the parentheses is \(1+1\), which simplifies to 2.It’s important to note:

• If the growth factor is greater than 1, it indicates growth.
• If it is between 0 and 1, it indicates decay (a reduction over time).
• The growth factor directly affects how sharply a quantity increases or decreases.

To ensure full clarity, always check:- That it's correctly calculated, as small errors can lead to large misunderstandings.This growing multiplier consistently changes the initial quantity, illustrating how dynamic exponential growth is.

Initial Quantity

The initial quantity in an exponential function is the starting point from which growth or decay begins. It is represented by the coefficient of the exponential expression, often denoted as \(a\) in the general form:\[y = ab^{x}\]From the equation:\[y = 50(1+1)^{t}\]we identify that the initial quantity is 50. This means before any growth occurs, we start with these 50 units. It's like the initial deposit in a savings account that grows with interest over time.Key points to consider:

• Ensures you know how much you start with before any increases happen.
• Crucial for predicting future outcomes.”
• Even if the growth factor is significant, the initial quantity impacts the future projections heavily.

Understanding the initial quantity is essential for tracking how much transformation occurs through the growth stages.