What Is The { Y $} − I N T E R C E P T O F T H I S E X P O N E N T I A L F U N C T I O N ? -intercept Of This Exponential Function? − In T Erce Pt O F T Hi Se X P O N E N T Ia L F U N C T I O N ? { F(x) = 24(2)^{x-2} + 3 \} A. { Y = -2 $}$B. { Y = 3 $}$C. { Y = 9 $}$D. { Y = 2 $}$

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In mathematics, an exponential function is a function that has the form f(x) = ab^x, where a and b are constants, and x is the variable. The y-intercept of an exponential function is the point at which the function intersects the y-axis, i.e., when x = 0. In this article, we will explore the concept of the y-intercept of an exponential function and apply it to the given function f(x) = 24(2)^{x-2} + 3.

What is the y-Intercept of an Exponential Function?

The y-intercept of an exponential function is the value of the function when x = 0. To find the y-intercept, we need to substitute x = 0 into the function and evaluate it. The y-intercept is an important concept in mathematics, as it provides valuable information about the behavior of the function.

The Given Exponential Function

The given exponential function is f(x) = 24(2)^{x-2} + 3. To find the y-intercept of this function, we need to substitute x = 0 into the function and evaluate it.

Substituting x = 0 into the Function

To find the y-intercept, we need to substitute x = 0 into the function f(x) = 24(2)^{x-2} + 3. This gives us:

f(0) = 24(2)^{0-2} + 3

Evaluating the Expression

To evaluate the expression, we need to simplify the exponent. Since 0 - 2 = -2, we can rewrite the expression as:

f(0) = 24(2)^{-2} + 3

Simplifying the Exponent

To simplify the exponent, we need to apply the rule that a^(-n) = 1/a^n. In this case, we have:

f(0) = 24(1/2^2) + 3

Evaluating the Expression

To evaluate the expression, we need to simplify the fraction. Since 1/2^2 = 1/4, we can rewrite the expression as:

f(0) = 24(1/4) + 3

Simplifying the Fraction

To simplify the fraction, we need to multiply the numerator and denominator by 4. This gives us:

f(0) = 6 + 3

Evaluating the Expression

To evaluate the expression, we need to add the two numbers. This gives us:

f(0) = 9

Conclusion

In conclusion, the y-intercept of the exponential function f(x) = 24(2)^{x-2} + 3 is 9. This means that when x = 0, the value of the function is 9.

Answer

The correct answer is C. y = 9.

Discussion

The y-intercept of an exponential function is an important concept in mathematics, as it provides valuable information about the behavior of the function. In this article, we explored the concept of the y-intercept and applied it to the given function f(x) = 24(2)^{x-2} + 3. We found that the y-intercept of this function is 9.

Related Topics

  • Exponential functions
  • y-intercept
  • Mathematics

References

FAQs

  • Q: What is the y-intercept of an exponential function? A: The y-intercept of an exponential function is the value of the function when x = 0.
  • Q: How do I find the y-intercept of an exponential function? A: To find the y-intercept, substitute x = 0 into the function and evaluate it.
  • Q: What is the y-intercept of the function f(x) = 24(2)^x-2} + 3? A The y-intercept of the function f(x) = 24(2)^{x-2 + 3 is 9.
    Q&A: Exponential Functions and the y-Intercept =====================================================

In our previous article, we explored the concept of the y-intercept of an exponential function and applied it to the given function f(x) = 24(2)^{x-2} + 3. In this article, we will answer some frequently asked questions about exponential functions and the y-intercept.

Q: What is an exponential function?

A: An exponential function is a function that has the form f(x) = ab^x, where a and b are constants, and x is the variable.

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

A: The y-intercept of an exponential function is the value of the function when x = 0.

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

A: To find the y-intercept, substitute x = 0 into the function and evaluate it.

Q: What is the y-intercept of the function f(x) = 24(2)^{x-2} + 3?

A: The y-intercept of the function f(x) = 24(2)^{x-2} + 3 is 9.

Q: Can you explain the concept of the y-intercept in more detail?

A: The y-intercept is an important concept in mathematics, as it provides valuable information about the behavior of the function. When x = 0, the function intersects the y-axis, and the value of the function at this point is the y-intercept.

Q: How do I simplify the exponent in an exponential function?

A: To simplify the exponent, apply the rule that a^(-n) = 1/a^n.

Q: Can you provide an example of simplifying an exponent?

A: For example, consider the expression f(x) = 24(2)^{-2} + 3. To simplify the exponent, we can rewrite it as f(x) = 24(1/2^2) + 3.

Q: How do I evaluate the expression after simplifying the exponent?

A: After simplifying the exponent, we can evaluate the expression by multiplying the numerator and denominator by 4, which gives us f(x) = 6 + 3.

Q: Can you explain the concept of exponential growth and decay?

A: Exponential growth and decay are two important concepts in mathematics that describe how quantities change over time. Exponential growth occurs when a quantity increases at a rate proportional to its current value, while exponential decay occurs when a quantity decreases at a rate proportional to its current value.

Q: How do I apply the concept of exponential growth and decay to real-world problems?

A: Exponential growth and decay have many real-world applications, such as modeling population growth, chemical reactions, and financial investments. To apply these concepts, we need to understand the underlying mathematical models and use them to make predictions and decisions.

Q: Can you provide some examples of real-world applications of exponential functions?

A: Some examples of real-world applications of exponential functions include:

  • Modeling population growth and decline
  • Calculating compound interest and investment returns
  • Analyzing chemical reactions and decay rates
  • Predicting the spread of diseases and epidemics
  • Modeling the growth and decay of financial markets

Q: How do I use technology to graph and analyze exponential functions?

A: There are many software programs and online tools available that can help you graph and analyze exponential functions, such as graphing calculators, computer algebra systems, and online graphing tools.

Q: Can you provide some tips for graphing and analyzing exponential functions?

A: Here are some tips for graphing and analyzing exponential functions:

  • Use a graphing calculator or computer algebra system to graph the function and analyze its behavior.
  • Identify the key features of the function, such as the y-intercept, asymptotes, and inflection points.
  • Use the graph to make predictions and decisions about the behavior of the function.
  • Analyze the function's behavior over different intervals and domains.
  • Use the function to model real-world problems and make predictions about future behavior.

Conclusion

In conclusion, exponential functions and the y-intercept are important concepts in mathematics that have many real-world applications. By understanding these concepts and using them to model and analyze real-world problems, we can make predictions and decisions about the behavior of complex systems.