Finding The Y-intercept Of F(x) = 3^(x+2) A Step-by-Step Guide

In the realm of mathematics, understanding the behavior of functions is crucial. Among the various types of functions, exponential functions hold a significant place due to their unique properties and wide range of applications. One key aspect of analyzing a function is determining its intercepts, particularly the $y$-intercept. In this comprehensive guide, we will delve into the process of finding the $y$-intercept of the exponential function $f(x)=3^{x+2}$, while also exploring the underlying concepts and principles.

Understanding the $y$-intercept

To begin, let's clarify the concept of the $y$-intercept. The y-intercept of a function is the point where the graph of the function intersects the $y$-axis. At this point, the $x$-coordinate is always zero. Therefore, to find the $y$-intercept, we need to determine the value of the function when $x=0$. In other words, we need to evaluate $f(0)$. This simple yet fundamental concept forms the basis for our analysis of the given exponential function.

Analyzing the Exponential Function $f(x)=3^{x+2}$

The exponential function we are considering is $f(x)=3^{x+2}$. This function belongs to the family of exponential functions, which have the general form $f(x)=a^{x}$, where $a$ is a constant called the base. In our case, the base is 3. However, the function $f(x)=3^{x+2}$ is a slight variation of the basic exponential function due to the presence of the term $x+2$ in the exponent. This term represents a horizontal shift of the graph of the basic exponential function $f(x)=3^{x}$. Understanding the impact of this shift is crucial for accurately determining the $y$-intercept.

Evaluating $f(0)$

As we established earlier, to find the $y$-intercept, we need to evaluate $f(0)$. Let's substitute $x=0$ into the function:

$f(0) = 3^{0+2} = 3^2 = 9$

This calculation reveals that when $x=0$, the value of the function is 9. This means that the graph of the function intersects the $y$-axis at the point $(0,9)$. Therefore, the $y$-intercept of the function $f(x)=3^{x+2}$ is $(0,9)$. This straightforward calculation showcases the power of understanding function notation and applying it to determine key features of a function's graph.

Identifying the Correct Answer

Now that we have determined the $y$-intercept to be $(0,9)$, we can compare our result with the given options:

A. $(9,0)$ B. $(0,9)$ C. $(0,-9)$ D. $(9,-9)$

Clearly, option B, $(0,9)$, matches our calculated $y$-intercept. Therefore, option B is the correct answer. This step highlights the importance of accurately interpreting the results of our calculations and matching them to the appropriate answer choices.

Deeper Insights into Exponential Functions

While we have successfully found the $y$-intercept of the given function, it is beneficial to delve deeper into the properties of exponential functions. Exponential functions are characterized by their rapid growth or decay. The base of the exponential function determines whether the function is increasing or decreasing. If the base is greater than 1, the function is increasing, meaning that as $x$ increases, the value of the function also increases. Conversely, if the base is between 0 and 1, the function is decreasing. In our case, the base is 3, which is greater than 1, so the function $f(x)=3^{x+2}$ is an increasing exponential function. This understanding provides a broader context for interpreting the behavior of the function and its graph.

The Impact of Horizontal Shifts

The term $x+2$ in the exponent of our function represents a horizontal shift of the graph. Specifically, it shifts the graph 2 units to the left. To understand why, consider the basic exponential function $f(x)=3^{x}$. Its $y$-intercept is $(0,1)$, since $3^0=1$. Now, for the function $f(x)=3^{x+2}$, the value of the function at $x=-2$ is $f(-2)=3^{-2+2}=3^0=1$. This means that the point $(-2,1)$ on the graph of $f(x)=3^{x+2}$ corresponds to the point $(0,1)$ on the graph of $f(x)=3^{x}$. This horizontal shift affects the $y$-intercept, moving it from $(0,1)$ for the basic function to $(0,9)$ for the shifted function. Visualizing these transformations is key to mastering function analysis.

Generalizing the Process

The process we used to find the $y$-intercept of $f(x)=3^{x+2}$ can be generalized to any function. To find the $y$-intercept of any function, simply substitute $x=0$ into the function and evaluate. This simple rule applies to all types of functions, including polynomial, trigonometric, and logarithmic functions. Mastering this process provides a versatile tool for analyzing functions and understanding their behavior. This versatility underscores the importance of grasping fundamental concepts in mathematics.

Common Pitfalls to Avoid

When finding the $y$-intercept, there are a few common pitfalls to avoid. One common mistake is confusing the $x$-intercept with the $y$-intercept. The $x$-intercept is the point where the graph intersects the $x$-axis, and it is found by setting $f(x)=0$ and solving for $x$. This is a different process than finding the $y$-intercept, which involves setting $x=0$. Another common mistake is incorrectly evaluating the function at $x=0$. Be sure to carefully substitute $x=0$ into the function and perform the calculations accurately. Attention to detail is crucial in mathematical problem-solving.

The Significance of the $y$-intercept

The $y$-intercept is a significant feature of a function's graph. It provides valuable information about the function's behavior and its relationship to the coordinate axes. In the context of real-world applications, the $y$-intercept often represents an initial value or a starting point. For example, in a model of population growth, the $y$-intercept might represent the initial population size. In a financial model, the $y$-intercept might represent the initial investment. Understanding the significance of the $y$-intercept allows us to interpret mathematical models in meaningful ways.

Visualizing the Function and its Intercept

To further enhance our understanding, let's visualize the function $f(x)=3^{x+2}$ and its $y$-intercept. The graph of the function is an exponential curve that increases rapidly as $x$ increases. The $y$-intercept is the point $(0,9)$, where the curve intersects the $y$-axis. This visual representation provides a clear picture of the function's behavior and its key features. Visualizing mathematical concepts can often lead to deeper understanding and retention. Graphical representations are powerful tools in mathematics.

Using Graphing Tools

Modern technology provides powerful tools for graphing functions and visualizing their properties. Graphing calculators and online graphing tools can quickly generate accurate graphs of functions, allowing us to explore their behavior and identify key features like intercepts. These tools can be invaluable for students and professionals alike. Using technology effectively enhances mathematical exploration and discovery.

Practice Problems

To solidify your understanding of finding $y$-intercepts, let's consider a few practice problems:

1. Find the $y$-intercept of $g(x)=2^{x-1}$.
2. Find the $y$-intercept of $h(x)=5^{x}+3$.
3. Find the $y$-intercept of $k(x)=4^{x+1}-2$.

Working through these problems will reinforce the concepts we have discussed and build your problem-solving skills. Practice is essential for mastering any mathematical skill.

Solutions to Practice Problems

Here are the solutions to the practice problems:

1. For $g(x)=2^{x-1}$, the $y$-intercept is found by evaluating $g(0)=2^{0-1}=2^{-1}=\frac{1}{2}$. So the $y$-intercept is $(0,\frac{1}{2})$.

2. For $h(x)=5^{x}+3$, the $y$-intercept is found by evaluating $h(0)=5^{0}+3=1+3=4$. So the $y$-intercept is $(0,4)$.

3. For $k(x)=4^{x+1}-2$, the $y$-intercept is found by evaluating $k(0)=4^{0+1}-2=4^1-2=4-2=2$. So the $y$-intercept is $(0,2)$.

These solutions demonstrate the consistent application of the process for finding $y$-intercepts. Reviewing these solutions will solidify your understanding and confidence.

Conclusion

In conclusion, finding the $y$-intercept of a function is a fundamental skill in mathematics. For the exponential function $f(x)=3^{x+2}$, the $y$-intercept is $(0,9)$. This was determined by substituting $x=0$ into the function and evaluating. We have also explored the properties of exponential functions, the impact of horizontal shifts, and the general process for finding $y$-intercepts. By understanding these concepts and practicing problem-solving, you can confidently analyze functions and their graphs. Mastering these skills provides a solid foundation for further mathematical exploration.

This comprehensive guide has provided a thorough exploration of finding the $y$-intercept of exponential functions. By understanding the concepts, practicing the process, and visualizing the functions, you can confidently tackle similar problems and deepen your understanding of mathematics. Remember, mathematics is a journey of discovery, and each step builds upon the previous one.