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In the context of exponential functions, growth rate is a key concept that tells us how quickly the function's values are increasing. It is usually expressed as a percentage. For an exponential function of the form \( y = ab^x \), the growth rate can be determined from the base \( b \) of the exponent. Here’s how you can find it:

• First, identify the base \( b \). In our problem, \( b = 1.15 \).
• Subtract 1 from this base. So, \( 1.15 - 1 = 0.15 \).
• Convert the result into a percentage by multiplying by 100. Hence, \( 0.15 \times 100 = 15\text{\text{\text{\textperdigit}}} \).

A 15% growth rate means that for each unit increase in \( x \), our \( y \) value increases by 15% of its previous value. This concept is essential to understand how exponentially growing functions behave over time.

Exponential growth is when the increase of a quantity is proportional to its current value, leading to growth rapidly accelerating over time. This type of growth is commonly modeled with the formula \( y = ab^x \). In our example:

• \( a = 500.00 \), which is the initial value or starting point.
• \( b = 1.15 \), the growth factor.

Each step increases the quantity by a factor of \( 1.15 \), or 15%. This kind of growth is not linear; it multiplies the current value by a constant factor. Therefore, the rate of growth itself increases, making the growth accelerate. This is distinct from linear growth, where the amount added remains constant.
Exponential growth is seen in various real-world scenarios, like populations, finance, and certain types of technology advances. For example, if a population grows exponentially, it means it doubles its size at regular intervals, growing faster and faster with each passing period.

Solving exponential equations involves finding the unknown variable in an equation where the variable appears in the exponent. Given an exponential equation of the form \( y = ab^x \), solving this requires a few steps. In our given problem, we'll break it down:

• Start with the formula \( y = ab^x \). Identify points: \( (0, 500.00) \) and \( (1, 575.00) \)
• First, solve for \( a \): Using the point \( (0, 500.00) \), substitute into the equation: \( 500.00 = a \times b^0 \). Since any number raised to zero is 1, \( a = 500.00 \).
• Next, solve for \( b \): Using the point \( (1, 575.00) \), substitute \( a = 500.00 \) and solve: \( 575.00 = 500.00 \times b \). Dividing through by 500.00, \( b = \frac{575.00}{500.00} = 1.15 \).

Now we have all components needed: the exponential function is \( y = 500.00 \times 1.15^x \). With this function, any unknown value in our original data set can be determined by substituting \( x \) into the equation. This process helps us understand the dynamics of exponential models, which are frequently used in scientific and financial calculations.