What Is The { Y$} − I N T E R C E P T O F T H E E X P O N E N T I A L F U N C T I O N ? -intercept Of The Exponential Function? − In T Erce Pt O F T H Ee X P O N E N T Ia L F U N C T I O N ? { F(x) = -0.25(6)^{x+2} - 1 \} Enter Your Answer In The Box. { Y$}$-intercept { = \square$}$

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Introduction

In mathematics, the {y$}$-intercept of a function is the point at which the graph of the function intersects the {y$}$-axis. This point is also known as the initial value or starting value of the function. In this article, we will explore the concept of the {y$}$-intercept of an exponential function and how to find it.

What is an Exponential Function?

An exponential function is a mathematical function of the form f(x)=abx{f(x) = ab^x}, where a{a} and b{b} are constants, and x{x} is the variable. The base b{b} is a positive number, and the exponent x{x} is a real number. Exponential functions are used to model a wide range of phenomena, including population growth, chemical reactions, and financial investments.

The Given Exponential Function

In this problem, we are given an exponential function of the form f(x)=−0.25(6)x+2−1{f(x) = -0.25(6)^{x+2} - 1}. To find the {y$}$-intercept of this function, we need to substitute x=0{x = 0} into the function and evaluate the result.

Finding the {y$}$-intercept

To find the {y$}$-intercept of the given exponential function, we need to substitute x=0{x = 0} into the function and evaluate the result. This is because the {y$}$-intercept is the point at which the graph of the function intersects the {y$}$-axis, and the {y$}$-axis is the vertical line where x=0{x = 0}.

import math

def find_y_intercept():
a = -0.25
b = 6
c = 2
x = 0
y_intercept = a * (b ** (x + c)) - 1
return y_intercept


y_intercept = find_y_intercept()
print("The y-intercept of the function is:", y_intercept)

Calculating the {y$}$-intercept

Now that we have the function to find the {y$}$-intercept, let's calculate the value. We substitute x=0{x = 0} into the function and evaluate the result.

f(0)=−0.25(6)0+2−1{f(0) = -0.25(6)^{0+2} - 1}

f(0)=−0.25(6)2−1{f(0) = -0.25(6)^{2} - 1}

f(0)=−0.25(36)−1{f(0) = -0.25(36) - 1}

f(0)=−9−1{f(0) = -9 - 1}

f(0)=−10{f(0) = -10}

Therefore, the {y$}$-intercept of the given exponential function is −10{-10}.

Conclusion

In this article, we explored the concept of the {y$}$-intercept of an exponential function and how to find it. We used the given exponential function f(x)=−0.25(6)x+2−1{f(x) = -0.25(6)^{x+2} - 1} to find the {y$}$-intercept by substituting x=0{x = 0} into the function and evaluating the result. We calculated the value of the {y$}$-intercept to be −10{-10}. This demonstrates the importance of understanding the concept of the {y$}$-intercept in mathematics and how it can be applied to real-world problems.

Frequently Asked Questions

Q: What is the {y$}$-intercept of an exponential function?

A: The {y$}$-intercept of an exponential function is the point at which the graph of the function intersects the {y$}$-axis.

Q: How do I find the {y$}$-intercept of an exponential function?

A: To find the {y$}$-intercept of an exponential function, substitute x=0{x = 0} into the function and evaluate the result.

Q: What is the formula for an exponential function?

A: The formula for an exponential function is f(x)=abx{f(x) = ab^x}, where a{a} and b{b} are constants, and x{x} is the variable.

Q: What is the base of an exponential function?

A: The base of an exponential function is a positive number.

Q: What is the exponent of an exponential function?

A: The exponent of an exponential function is a real number.

Q: What is the {y$}$-axis?

A: The {y$}$-axis is the vertical line where x=0{x = 0}.

Q: What is the {y$}$-intercept of the given exponential function?

Q: What is the {y$}$-intercept of an exponential function?

A: The {y$}$-intercept of an exponential function is the point at which the graph of the function intersects the {y$}$-axis. This point is also known as the initial value or starting value of the function.

Q: How do I find the {y$}$-intercept of an exponential function?

A: To find the {y$}$-intercept of an exponential function, substitute x=0{x = 0} into the function and evaluate the result. This is because the {y$}$-intercept is the point at which the graph of the function intersects the {y$}$-axis, and the {y$}$-axis is the vertical line where x=0{x = 0}.

Q: What is the formula for an exponential function?

A: The formula for an exponential function is f(x)=abx{f(x) = ab^x}, where a{a} and b{b} are constants, and x{x} is the variable.

Q: What is the base of an exponential function?

A: The base of an exponential function is a positive number. This means that the base b{b} is always greater than 0.

Q: What is the exponent of an exponential function?

A: The exponent of an exponential function is a real number. This means that the exponent x{x} can be any real number, including positive and negative numbers, as well as fractions and decimals.

Q: What is the {y$}$-axis?

A: The {y$}$-axis is the vertical line where x=0{x = 0}. This is the line that separates the positive and negative x-values on a graph.

Q: What is the {y$}$-intercept of the given exponential function?

A: The {y$}$-intercept of the given exponential function is −10{-10}. This is the point at which the graph of the function intersects the {y$}$-axis.

Q: How do I calculate the {y$}$-intercept of an exponential function?

A: To calculate the {y$}$-intercept of an exponential function, substitute x=0{x = 0} into the function and evaluate the result. This will give you the value of the {y$}$-intercept.

Q: What is the significance of the {y$}$-intercept of an exponential function?

A: The {y$}$-intercept of an exponential function is significant because it represents the starting value or initial value of the function. This value is important in many real-world applications, such as modeling population growth, chemical reactions, and financial investments.

Q: Can I use the {y$}$-intercept of an exponential function to make predictions?

A: Yes, you can use the {y$}$-intercept of an exponential function to make predictions. By understanding the behavior of the function and the value of the {y$}$-intercept, you can make informed predictions about the future behavior of the function.

Q: How do I use the {y$}$-intercept of an exponential function in real-world applications?

A: You can use the {y$}$-intercept of an exponential function in real-world applications such as modeling population growth, chemical reactions, and financial investments. By understanding the behavior of the function and the value of the {y$}$-intercept, you can make informed decisions and predictions about the future behavior of the function.

Conclusion

In this article, we have explored the concept of the {y$}$-intercept of an exponential function and how to find it. We have also answered some frequently asked questions about the {y$}$-intercept of an exponential function. By understanding the behavior of the function and the value of the {y$}$-intercept, you can make informed decisions and predictions about the future behavior of the function.