Given The Equation: $f(x) = 4^{x+2} + 2$, Answer The Following:A. Growth Or Decay? $\square$ B. $h =$ $\square$ C. $k =$ $\square$ D. Horizontal Asymptote: $\square$ E. Find

Feb 27, 2025 by ADMIN 186 views

Understanding Exponential Functions: A Comprehensive Analysis of the Given Equation

Exponential functions are a fundamental concept in mathematics, describing growth and decay phenomena in various fields, including finance, biology, and physics. In this article, we will delve into the given equation: $f(x) = 4^{x+2} + 2$ , and answer the following questions: A. Growth or Decay, B. $h =$ , C. $k =$ , D. Horizontal Asymptote, and E. Find.

To determine whether the function represents growth or decay, we need to analyze the coefficient of the exponential term. In the given equation, the coefficient is 4, which is greater than 1. This indicates that the function will exhibit exponential growth.

Why Exponential Growth?

When the coefficient of the exponential term is greater than 1, the function will grow exponentially. This is because the value of the function will increase rapidly as the input variable (x) increases. In contrast, if the coefficient were between 0 and 1, the function would exhibit exponential decay.

To find the value of $h$ , we need to rewrite the equation in the form $f(x) = a \cdot b^x + c$ . Comparing this form with the given equation, we can see that $a = 4^2 = 16$ , $b = 4$ , and $c = 2$ . Therefore, the value of $h$ is the x-coordinate of the vertex of the parabola represented by the function.

Finding the Vertex

The x-coordinate of the vertex of a parabola in the form $f(x) = a \cdot b^x + c$ is given by the formula $h = -\frac{1}{\ln(b)} \cdot \ln(a)$ . Substituting the values of $a$ and $b$ into this formula, we get:

$h = -\frac{1}{\ln(4)} \cdot \ln(16)$

Using a calculator to evaluate this expression, we get:

$h \approx -\frac{1}{\ln(4)} \cdot 4.0$

$h \approx -1.0$

Therefore, the value of $h$ is approximately -1.0.

To find the value of $k$ , we need to rewrite the equation in the form $f(x) = a \cdot b^x + c$ . Comparing this form with the given equation, we can see that $a = 4^2 = 16$ , $b = 4$ , and $c = 2$ . Therefore, the value of $k$ is the y-coordinate of the vertex of the parabola represented by the function.

Finding the Vertex

The y-coordinate of the vertex of a parabola in the form $f(x) = a \cdot b^x + c$ is given by the formula $k = c$ . In this case, $c = 2$ , so the value of $k$ is 2.

To find the horizontal asymptote of the function, we need to analyze the behavior of the function as x approaches positive or negative infinity. In this case, the function is an exponential function with a base greater than 1, so the horizontal asymptote is given by the formula $y = \lim_{x \to \infty} f(x)$ .

Evaluating the Limit

To evaluate the limit, we can use the fact that the exponential function grows rapidly as x increases. Therefore, the limit as x approaches positive infinity is given by:

$\lim_{x \to \infty} f(x) = \lim_{x \to \infty} (4^{x+2} + 2)$

Using the properties of limits, we can rewrite this expression as:

$\lim_{x \to \infty} f(x) = \lim_{x \to \infty} 4^{x+2} + \lim_{x \to \infty} 2$

The first term on the right-hand side approaches infinity as x approaches positive infinity, while the second term remains constant at 2. Therefore, the limit as x approaches positive infinity is given by:

$\lim_{x \to \infty} f(x) = \infty$

In conclusion, the given equation $f(x) = 4^{x+2} + 2$ represents an exponential function with a base greater than 1, indicating exponential growth. The value of $h$ is approximately -1.0, and the value of $k$ is 2. The horizontal asymptote of the function is given by the formula $y = \lim_{x \to \infty} f(x)$ , which approaches infinity as x approaches positive infinity.

Exponential functions with a base greater than 1 exhibit exponential growth.
The value of $h$ is the x-coordinate of the vertex of the parabola represented by the function.
The value of $k$ is the y-coordinate of the vertex of the parabola represented by the function.
The horizontal asymptote of an exponential function is given by the formula $y = \lim_{x \to \infty} f(x)$ .

Exponential functions are a fundamental concept in mathematics, describing growth and decay phenomena in various fields. Understanding the properties of exponential functions is essential for analyzing and modeling real-world phenomena. In this article, we have delved into the given equation $f(x) = 4^{x+2} + 2$ and answered the following questions: A. Growth or Decay, B. $h =$ , C. $k =$ , D. Horizontal Asymptote, and E. Find. We hope that this article has provided a comprehensive understanding of exponential functions and their properties.
Exponential Functions: A Q&A Guide

Exponential functions are a fundamental concept in mathematics, describing growth and decay phenomena in various fields, including finance, biology, and physics. In our previous article, we delved into the given equation $f(x) = 4^{x+2} + 2$ and answered the following questions: A. Growth or Decay, B. $h =$ , C. $k =$ , D. Horizontal Asymptote, and E. Find. In this article, we will provide a Q&A guide to help you better understand exponential functions and their properties.

A: An exponential function is a mathematical function of the form $f(x) = a \cdot b^x + c$ , where $a$ , $b$ , and $c$ are constants, and $b$ is the base of the exponential function.

A: Exponential growth occurs when the base of the exponential function is greater than 1, indicating that the function will grow rapidly as the input variable (x) increases. Exponential decay occurs when the base of the exponential function is between 0 and 1, indicating that the function will decrease rapidly as the input variable (x) increases.

A: To determine whether a function is exponential growth or decay, you need to analyze the coefficient of the exponential term. If the coefficient is greater than 1, the function will exhibit exponential growth. If the coefficient is between 0 and 1, the function will exhibit exponential decay.

A: The value of $h$ is the x-coordinate of the vertex of the parabola represented by the function. It is given by the formula $h = -\frac{1}{\ln(b)} \cdot \ln(a)$ .

A: The value of $k$ is the y-coordinate of the vertex of the parabola represented by the function. It is given by the formula $k = c$ .

A: The horizontal asymptote of an exponential function is given by the formula $y = \lim_{x \to \infty} f(x)$ . This represents the value that the function approaches as x approaches positive infinity.

A: To evaluate the limit of an exponential function as x approaches positive infinity, you can use the properties of limits. If the base of the exponential function is greater than 1, the limit will approach infinity. If the base is between 0 and 1, the limit will approach 0.

A: Exponential functions have many applications in various fields, including finance, biology, and physics. Some common applications include:

Modeling population growth and decay
Describing the behavior of chemical reactions
Analyzing the growth of investments and savings
Modeling the spread of diseases

In conclusion, exponential functions are a fundamental concept in mathematics, describing growth and decay phenomena in various fields. Understanding the properties of exponential functions is essential for analyzing and modeling real-world phenomena. We hope that this Q&A guide has provided a comprehensive understanding of exponential functions and their properties.

Exponential functions are a fundamental concept in mathematics.
Exponential growth occurs when the base of the exponential function is greater than 1.
Exponential decay occurs when the base of the exponential function is between 0 and 1.
The value of $h$ is the x-coordinate of the vertex of the parabola represented by the function.
The value of $k$ is the y-coordinate of the vertex of the parabola represented by the function.
The horizontal asymptote of an exponential function is given by the formula $y = \lim_{x \to \infty} f(x)$ .

Exponential functions are a powerful tool for modeling and analyzing real-world phenomena. Understanding the properties of exponential functions is essential for making informed decisions in various fields. We hope that this Q&A guide has provided a comprehensive understanding of exponential functions and their properties.