Identifying Functions With A Y-Intercept Of (0 5) A Comprehensive Guide

Finding functions with a specific y-intercept is a fundamental concept in mathematics. This article will explore how to identify exponential functions that intersect the y-axis at the point (0, 5). We'll dissect each function, providing step-by-step explanations and insights to solidify your understanding. Let's delve into the world of exponential functions and their y-intercepts.

Understanding Y-Intercepts

The y-intercept of a function is the point where its graph intersects the y-axis. This occurs when the input value, x, is equal to 0. Therefore, to find the y-intercept of a function, we substitute x = 0 into the function's equation and solve for y. This resulting y value represents the y-coordinate of the y-intercept, and the point is expressed as (0, y).

In the context of exponential functions, the y-intercept holds significant importance. It represents the initial value of the function, the starting point before any exponential growth or decay takes place. Understanding how to determine the y-intercept allows us to quickly analyze and compare different exponential functions. Moreover, the y-intercept plays a crucial role in real-world applications, such as modeling population growth, compound interest, and radioactive decay, where the initial amount or starting point is a key factor.

To master the concept of y-intercepts, it's essential to practice with various types of functions, including linear, quadratic, and exponential functions. This will help you develop a strong foundation for identifying and interpreting y-intercepts in different mathematical contexts. By understanding the significance of the y-intercept, you can gain valuable insights into the behavior and characteristics of functions.

Analyzing the Given Functions

Let's examine the given functions one by one to determine which have a y-intercept of (0, 5). Remember, we'll substitute x = 0 into each function and check if the resulting y value is equal to 5.

Function 1: $f(x) = 5(b)^x - 1$

To find the y-intercept, we substitute x = 0:

$f(0) = 5(b)^0 - 1$

Since any non-zero number raised to the power of 0 is 1, we have:

$f(0) = 5(1) - 1$

$f(0) = 5 - 1$

$f(0) = 4$

The y-intercept for this function is (0, 4), which is not (0, 5). Therefore, this function is not a solution.

Function 2: $f(x) = 2(b)^x + 5$

Substituting x = 0, we get:

$f(0) = 2(b)^0 + 5$

$f(0) = 2(1) + 5$

$f(0) = 2 + 5$

$f(0) = 7$

The y-intercept for this function is (0, 7), which is not (0, 5). This function is also not a solution.

Function 3: $f(x) = -5(b)^x + 10$

Let's find the y-intercept by substituting x = 0:

$f(0) = -5(b)^0 + 10$

$f(0) = -5(1) + 10$

$f(0) = -5 + 10$

$f(0) = 5$

The y-intercept for this function is (0, 5), which matches our target. This function is a solution.

Function 4: $f(x) = 7(b)^x - 2$

Substituting x = 0, we have:

$f(0) = 7(b)^0 - 2$

$f(0) = 7(1) - 2$

$f(0) = 7 - 2$

$f(0) = 5$

The y-intercept for this function is (0, 5), which matches our target. This function is a solution.

Function 5: $f(x) = -3(b)^x - 5$

Finally, let's substitute x = 0 for the last function:

$f(0) = -3(b)^0 - 5$

$f(0) = -3(1) - 5$

$f(0) = -3 - 5$

$f(0) = -8$

The y-intercept for this function is (0, -8), which is not (0, 5). This function is not a solution.

Identifying the Correct Answers

After analyzing each function, we have identified the functions with a y-intercept of (0, 5):

• $f(x) = -5(b)^x + 10$
• $f(x) = 7(b)^x - 2$

These two functions are the correct answers. They both intersect the y-axis at the point (0, 5).

Key Takeaways and Further Exploration

This exercise demonstrates the importance of understanding how to find the y-intercept of a function. By substituting x = 0, we can easily determine where the function's graph intersects the y-axis. This skill is crucial for analyzing and comparing functions, especially in the context of exponential functions, where the y-intercept represents the initial value.

To further enhance your understanding, consider exploring the following:

• Graphing the functions: Visualizing the graphs of these functions will provide a clear understanding of their y-intercepts and overall behavior.
• Exploring different values of b: How does the value of b affect the shape and steepness of the exponential function? Does it affect the y-intercept?
• Real-world applications: Investigate how exponential functions are used to model real-world phenomena, such as population growth, compound interest, and radioactive decay.
• Transformations of exponential functions: How do vertical shifts, horizontal shifts, and reflections affect the y-intercept of an exponential function?

By delving deeper into these topics, you'll gain a more comprehensive understanding of exponential functions and their applications.

Conclusion

Identifying functions with a specific y-intercept is a fundamental skill in mathematics. In this article, we've explored how to determine the y-intercept of exponential functions by substituting x = 0. We've analyzed several functions and identified those with a y-intercept of (0, 5). Remember, the y-intercept represents the initial value of the function and plays a crucial role in various mathematical and real-world applications. Keep practicing and exploring to strengthen your understanding of this important concept.

By mastering the concept of y-intercepts, you equip yourself with a valuable tool for analyzing and interpreting functions, paving the way for deeper understanding and success in your mathematical journey. This article has provided a solid foundation, but the exploration doesn't end here. Continue to challenge yourself with new problems and concepts, and you'll be amazed at how far your mathematical abilities can take you. Remember, mathematics is a journey of discovery, and every step you take brings you closer to a deeper understanding of the world around you.

The functions $f(x) = -5(b)^x + 10$ and $f(x) = 7(b)^x - 2$ both have a y-intercept of (0, 5). Understanding how to determine y-intercepts is crucial for analyzing and comparing functions, particularly exponential functions, where the y-intercept often represents the initial value. By mastering this concept, you'll be well-equipped to tackle a wide range of mathematical problems and real-world applications involving functions.

This exploration of y-intercepts and exponential functions highlights the interconnectedness of mathematical concepts. Each concept builds upon the previous one, creating a strong foundation for further learning. So, keep exploring, keep questioning, and keep building your mathematical knowledge. The world of mathematics is vast and fascinating, and there's always something new to discover.