Expressing G(t) = (4t)^2 In Exponential Form: Find A And B

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Let's dive into the problem of expressing the function G(t) = (4t)^2 in the form G(t) = a * b^t. This is a classic problem in mathematics that tests our understanding of exponential functions and algebraic manipulation. Guys, we'll break this down step by step to make sure everyone's on board. We'll first simplify the given function and then see if it fits the mold of an exponential function. If it does, we'll figure out the values of 'a' and 'b'. If not, we'll explain why and conclude that it can't be represented in the desired exponential form. So, let's put on our thinking caps and get started!

Understanding the Given Function: G(t) = (4t)^2

To start, we need to understand what the function G(t) = (4t)^2 represents. This function tells us how the output G(t) changes as the input t changes. The key operation here is squaring the term (4t). This means we're multiplying (4t) by itself. Let's simplify this algebraically to get a clearer picture. When we square (4t), we get (4t) * (4t), which equals 16t^2. So, the function can be rewritten as G(t) = 16t^2. Now, let's analyze this simplified form. We see that G(t) is a quadratic function, not an exponential function. The presence of the t^2 term is a dead giveaway. In an exponential function, the variable t appears in the exponent, not as the base being raised to a power. This distinction is crucial for our problem.

Identifying the Exponential Form: G(t) = a * b^t

Now, let's define what an exponential function looks like. The general form of an exponential function is G(t) = a * b^t, where:

  • a is the initial value or the coefficient that scales the exponential term.
  • b is the base, which is a constant raised to the power of t.
  • t is the variable, which represents time or any other independent variable.

In this form, the variable t is in the exponent, which means the function's value changes exponentially with t. The base b determines whether the function grows exponentially (b > 1) or decays exponentially (0 < b < 1). The value of a simply scales the function vertically. For example, if we have G(t) = 2 * 3^t, then a = 2 and b = 3. As t increases, G(t) will grow exponentially because the base 3 is greater than 1. The factor of 2 just stretches the graph vertically. Comparing this form with our simplified function G(t) = 16t^2, we can clearly see that they are fundamentally different. Our function has t squared, while an exponential function has a constant raised to the power of t.

Comparing G(t) = 16t^2 with the Exponential Form

Let's directly compare G(t) = 16t^2 with the exponential form G(t) = a * b^t. The function G(t) = 16t^2 is a polynomial function, specifically a quadratic function. It's characterized by the term t^2, which means the function's growth is polynomial, not exponential. The graph of this function is a parabola, a U-shaped curve. On the other hand, the function G(t) = a * b^t represents exponential growth or decay. Its graph is a curve that either increases rapidly (if b > 1) or decreases rapidly towards zero (if 0 < b < 1). The key difference lies in how the variable t is used. In G(t) = 16t^2, t is the base that is squared, whereas in G(t) = a * b^t, t is the exponent. This difference in structure leads to vastly different behaviors. For instance, if we plot both types of functions, we'll see that the exponential function will eventually outpace the quadratic function as t becomes large. This is because exponential growth is inherently faster than polynomial growth. Given this fundamental difference, we can conclude that G(t) = 16t^2 cannot be expressed in the exponential form G(t) = a * b^t.

Determining the Values of a and b

Since we've established that G(t) = 16t^2 cannot be written in the form G(t) = a * b^t, the question of finding values for a and b becomes irrelevant. There are no values of a and b that would make the exponential form equivalent to the quadratic form. The structure of a quadratic function and an exponential function are simply too different. A quadratic function's growth is determined by the square of the variable, while an exponential function's growth is determined by raising a constant to the power of the variable. This distinction is crucial in understanding why certain functions behave the way they do. For example, quadratic functions are often used to model projectile motion or the shape of a parabola, while exponential functions are used to model population growth or radioactive decay. So, in our case, because the given function is quadratic and not exponential, we cannot find suitable values for a and b that would fit the exponential form.

Conclusion: G(t) = (4t)^2 is Not Exponential

In conclusion, after simplifying G(t) = (4t)^2 to G(t) = 16t^2, we've determined that this function is not exponential. It's a quadratic function due to the presence of the t^2 term. Exponential functions have the form G(t) = a * b^t, where t is in the exponent. Since our function doesn't fit this form, we cannot express it in exponential terms. Therefore, the values for a and b in the exponential form are NONE. This exercise highlights the importance of recognizing different types of functions and understanding their unique properties. Guys, remember that exponential functions grow or decay much faster than polynomial functions in the long run, and this difference is due to the placement of the variable in the exponent. So, next time you encounter a function, take a moment to analyze its form and determine its type – it will help you understand its behavior and how it can be used to model real-world phenomena.

Final Answer:

  • a = NONE
  • b = NONE