Which Of The Following Correctly Rewrites The Function $f(x)=7 \\cdot 2^{2x+3} \\cdot 2^{-(x+5)}+3$ In The Form $f(x)=a(b)^{x-h}+k$ Using The Properties Of Exponents?

In the realm of mathematical functions, exponential functions hold a significant place, particularly in modeling phenomena that exhibit rapid growth or decay. These functions, characterized by a constant base raised to a variable exponent, possess unique properties that make them indispensable in diverse fields such as finance, biology, and physics. To effectively manipulate and analyze exponential functions, a solid grasp of exponent properties is crucial. In this article, we delve into the application of these properties to rewrite exponential functions into a more standard and insightful form. Our focus will be on transforming a given function, $f(x) = 7 \\cdot 2^{2x+3} \\cdot 2^{-(x+5)} + 3$, into the form $f(x) = a(b)^{x-h} + k$, where a, b, h, and k are constants. This standard form provides valuable insights into the function's behavior, including its vertical stretch, base, horizontal shift, and vertical shift. By meticulously applying the properties of exponents, we will unravel the steps involved in this transformation, shedding light on the underlying principles and techniques.

The given function, $f(x) = 7 \\cdot 2^{2x+3} \\cdot 2^{-(x+5)} + 3$, presents an interesting challenge. It is an exponential function, but it's not immediately apparent in the standard form $f(x) = a(b)^{x-h} + k$. Our mission is to rewrite the function using the properties of exponents to match this standard form. This transformation will allow us to easily identify the key parameters of the function, such as the base, vertical stretch, horizontal shift, and vertical shift. These parameters provide valuable information about the function's graph and behavior. Before we embark on the transformation, let's briefly revisit the properties of exponents that will be instrumental in this process. These properties include the product of powers rule, the quotient of powers rule, the power of a power rule, and the negative exponent rule. A clear understanding of these rules is essential for manipulating the exponential terms and achieving the desired form. By carefully applying these properties, we can systematically simplify the given function and reveal its underlying structure.

Before we dive into rewriting the function, let's take a moment to refresh our understanding of the properties of exponents. These properties are the fundamental tools we'll use to manipulate and simplify the exponential terms. Here's a brief overview:

1. Product of Powers: When multiplying exponents with the same base, we add the powers: $a^m \\cdot a^n = a^{m+n}$. This property will be crucial for combining the exponential terms in our function.
2. Quotient of Powers: When dividing exponents with the same base, we subtract the powers: $a^m / a^n = a^{m-n}$. Although not directly applicable in this specific problem, it's a valuable property to keep in mind for other exponential manipulations.
3. Power of a Power: When raising a power to another power, we multiply the exponents: $(a^m)^n = a^{m \\cdot n}$. This property can be used to simplify expressions where an exponential term is raised to a power.
4. Negative Exponent: A negative exponent indicates the reciprocal of the base raised to the positive exponent: $a^{-n} = 1/a^n$. This property will be useful for dealing with the negative exponent in our function.

With these properties in mind, we're now ready to tackle the task of rewriting the given exponential function.

Now, let's embark on the step-by-step transformation of the function $f(x) = 7 \\cdot 2^{2x+3} \\cdot 2^{-(x+5)} + 3$ into the standard form $f(x) = a(b)^{x-h} + k$. Our journey will involve carefully applying the properties of exponents we've just reviewed.

1. Combine Exponential Terms: The first step is to combine the exponential terms using the product of powers property. We have $2^{2x+3}$ and $2^{-(x+5)}$. Adding the exponents, we get: $2^{2x+3} \\cdot 2^{-(x+5)} = 2^{(2x+3) + (-(x+5))} = 2^{2x+3-x-5} = 2^{x-2}$ So, our function now looks like this: $f(x) = 7 \\cdot 2^{x-2} + 3$
2. Identify the Standard Form: Comparing this to the standard form $f(x) = a(b)^{x-h} + k$, we can see that:
   • $a = 7$ (the vertical stretch factor)
   • $b = 2$ (the base of the exponential function)
   • $h = 2$ (the horizontal shift)
   • $k = 3$ (the vertical shift)

Therefore, the function is already in the standard form $f(x) = 7(2)^{x-2} + 3$. We have successfully rewritten the function using the properties of exponents. This form clearly reveals the key parameters of the exponential function.

Now that we have successfully rewritten the function in the standard form $f(x) = 7(2)^{x-2} + 3$, we can confidently identify the correct option from the given choices. Let's compare our result with the options provided:

A. $f(x) = 7 \\cdot 2^{x+8} + 3$ B. $f(x) = 7(2)^{x-2} + 3$

Clearly, option B matches our transformed function, $f(x) = 7(2)^{x-2} + 3$. Therefore, option B is the correct answer. Options A is incorrect as it has the wrong exponent value. By meticulously applying the properties of exponents, we have not only rewritten the function but also verified the correct option.

In this exploration, we successfully rewrote the exponential function $f(x) = 7 \\cdot 2^{2x+3} \\cdot 2^{-(x+5)} + 3$ into the standard form $f(x) = a(b)^{x-h} + k$ using the properties of exponents. This transformation allowed us to identify the key parameters of the function, such as the vertical stretch factor, base, horizontal shift, and vertical shift. The ability to rewrite exponential functions in this way is crucial for understanding their behavior and applications in various fields. By mastering the properties of exponents and practicing these transformations, you can gain a deeper understanding of exponential functions and their role in mathematical modeling. Remember, the key is to carefully apply the properties of exponents in a systematic manner, breaking down complex expressions into simpler forms. This approach will empower you to confidently tackle a wide range of problems involving exponential functions.

By understanding the properties of exponents, we can manipulate and simplify exponential expressions, revealing their underlying structure and facilitating their analysis. This skill is invaluable in various mathematical contexts and real-world applications.