What Is The $y$-intercept Of The Exponential Function? (Hint: When $x=0$)${ F(x) = -32(2)^{x-3} + 3 }$A. $y = -2$B. $y = -1$C. $y = 3$D. $y = -3$

Introduction

In mathematics, the $y$-intercept of a function is the point at which the function intersects the $y$-axis. This occurs when the value of $x$ is equal to zero. In this article, we will explore the concept of the $y$-intercept of an exponential function and determine the $y$-intercept of the given function.

Understanding Exponential Functions

Exponential functions are a type of mathematical function that describes a relationship between two variables, typically $x$ and $y$. The general form of an exponential function is:

${ f(x) = ab^x + c }$

where $a$, $b$, and $c$ are constants, and $x$ is the independent variable. The base $b$ is a positive number, and the exponent $x$ is a real number.

The Given Function

The given function is:

${ f(x) = -32(2)^{x-3} + 3 }$

This function is an exponential function with a base of $2$, a coefficient of $-32$, and a constant term of $3$.

Finding the $y$-Intercept

To find the $y$-intercept of the function, we need to substitute $x = 0$ into the function and solve for $y$.

${ f(0) = -32(2)^{0-3} + 3 }$

Simplifying the Expression

To simplify the expression, we need to evaluate the exponent $0-3$.

${ 0-3 = -3 }$

Evaluating the Exponent

The exponent $-3$ is equal to $-3$.

Substituting the Value of the Exponent

Substituting the value of the exponent into the expression, we get:

${ f(0) = -32(2)^{-3} + 3 }$

Evaluating the Exponent

The exponent $-3$ is equal to $\frac{1}{2^3}$.

Simplifying the Expression

To simplify the expression, we need to evaluate the fraction $\frac{1}{2^3}$.

${ \frac{1}{2^3} = \frac{1}{8} }$

Substituting the Value of the Fraction

Substituting the value of the fraction into the expression, we get:

${ f(0) = -32 \left( \frac{1}{8} \right) + 3 }$

Simplifying the Expression

To simplify the expression, we need to multiply $-32$ by $\frac{1}{8}$.

${ -32 \left( \frac{1}{8} \right) = -4 }$

Substituting the Value of the Product

Substituting the value of the product into the expression, we get:

${ f(0) = -4 + 3 }$

Simplifying the Expression

To simplify the expression, we need to add $-4$ and $3$.

${ -4 + 3 = -1 }$

Conclusion

The $y$-intercept of the function is $-1$.

Final Answer

The final answer is:

${ y = -1 }$

Discussion

The $y$-intercept of an exponential function is the point at which the function intersects the $y$-axis. To find the $y$-intercept of a function, we need to substitute $x = 0$ into the function and solve for $y$. In this article, we used this method to find the $y$-intercept of the given function.

Related Topics

• Exponential functions

• y$-intercept

• Algebra
• Mathematics

References

• [1] "Exponential Functions" by Math Open Reference
• [2] "$y$-Intercept" by Khan Academy
• [3] "Algebra" by Wikipedia

Tags

• Exponential functions

• y$-intercept

• Algebra
• Mathematics

• y$-axis

• Independent variable
• Dependent variable
• Constants
• Base
• Exponent
• Fraction
• Product
• Sum
• Difference
  

Introduction

In our previous article, we explored the concept of the $y$-intercept of an exponential function and determined the $y$-intercept of the given function. In this article, we will answer some frequently asked questions about exponential functions and the $y$-intercept.

Q1: What is an exponential function?

A1: An exponential function is a type of mathematical function that describes a relationship between two variables, typically $x$ and $y$. The general form of an exponential function is:

${ f(x) = ab^x + c }$

where $a$, $b$, and $c$ are constants, and $x$ is the independent variable.

Q2: What is the $y$-intercept of an exponential function?

A2: The $y$-intercept of an exponential function is the point at which the function intersects the $y$-axis. This occurs when the value of $x$ is equal to zero.

Q3: How do I find the $y$-intercept of an exponential function?

A3: To find the $y$-intercept of an exponential function, you need to substitute $x = 0$ into the function and solve for $y$.

Q4: What is the base of an exponential function?

A4: The base of an exponential function is a positive number that is raised to a power. In the general form of an exponential function, the base is $b$.

Q5: What is the exponent of an exponential function?

A5: The exponent of an exponential function is a real number that is used to raise the base to a power. In the general form of an exponential function, the exponent is $x$.

Q6: How do I evaluate an exponential expression?

A6: To evaluate an exponential expression, you need to raise the base to the power of the exponent. For example, if you have the expression $2^3$, you would evaluate it as follows:

${ 2^3 = 2 \times 2 \times 2 = 8 }$

Q7: What is the difference between an exponential function and a linear function?

A7: An exponential function is a type of function that describes a relationship between two variables, typically $x$ and $y$, where the variable $y$ is raised to a power. A linear function, on the other hand, is a type of function that describes a relationship between two variables, typically $x$ and $y$, where the variable $y$ is not raised to a power.

Q8: Can you give an example of an exponential function?

A8: Yes, here is an example of an exponential function:

${ f(x) = 2^x + 3 }$

Q9: Can you give an example of a linear function?

A9: Yes, here is an example of a linear function:

${ f(x) = 2x + 3 }$

Q10: What is the importance of exponential functions in real-life applications?

A10: Exponential functions are used in a wide range of real-life applications, including finance, population growth, and chemical reactions. They are also used in modeling and predicting the behavior of complex systems.

Conclusion

In this article, we answered some frequently asked questions about exponential functions and the $y$-intercept. We hope that this article has provided you with a better understanding of these concepts and has helped you to develop your skills in working with exponential functions.

Final Answer

The final answer is:

• Q1: An exponential function is a type of mathematical function that describes a relationship between two variables, typically $x$ and $y$.
• Q2: The $y$-intercept of an exponential function is the point at which the function intersects the $y$-axis.
• Q3: To find the $y$-intercept of an exponential function, you need to substitute $x = 0$ into the function and solve for $y$.
• Q4: The base of an exponential function is a positive number that is raised to a power.
• Q5: The exponent of an exponential function is a real number that is used to raise the base to a power.
• Q6: To evaluate an exponential expression, you need to raise the base to the power of the exponent.
• Q7: An exponential function is a type of function that describes a relationship between two variables, typically $x$ and $y$, where the variable $y$ is raised to a power.
• Q8: An example of an exponential function is $f(x) = 2^x + 3$.
• Q9: An example of a linear function is $f(x) = 2x + 3$.
• Q10: Exponential functions are used in a wide range of real-life applications, including finance, population growth, and chemical reactions.

Discussion

Exponential functions are an important concept in mathematics and have many real-life applications. They are used to model and predict the behavior of complex systems, and are an essential tool for scientists and engineers.

Related Topics

• Exponential functions

• y$-intercept

• Algebra
• Mathematics

• y$-axis

• Independent variable
• Dependent variable
• Constants
• Base
• Exponent
• Fraction
• Product
• Sum
• Difference

References

• [1] "Exponential Functions" by Math Open Reference
• [2] "$y$-Intercept" by Khan Academy
• [3] "Algebra" by Wikipedia

Tags

• Exponential functions

• y$-intercept

• Algebra
• Mathematics

• y$-axis

• Independent variable
• Dependent variable
• Constants
• Base
• Exponent
• Fraction
• Product
• Sum
• Difference