What Is The Y Y Y -intercept Of This Exponential Function? F ( X ) = 24 ( 2 ) X − 2 + 3 F(x) = 24(2)^{x-2} + 3 F ( X ) = 24 ( 2 ) X − 2 + 3 A. Y = − 2 Y = -2 Y = − 2 B. Y = 3 Y = 3 Y = 3 C. Y = 9 Y = 9 Y = 9 D. Y = 2 Y = 2 Y = 2

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Understanding the $y$-intercept

The $y$-intercept of a function is the point at which the graph of the function crosses the $y$-axis. In other words, it is the value of $y$ when $x$ is equal to zero. To find the $y$-intercept of an exponential function, we need to substitute $x = 0$ into the equation of the function.

The Exponential Function

The given exponential function is $f(x) = 24(2)^{x-2} + 3$. This function has a base of $2$ and a coefficient of $24$. The exponent is $x-2$, which means that the function will increase exponentially as $x$ increases.

Finding the $y$-intercept

To find the $y$-intercept, we need to substitute $x = 0$ into the equation of the function. This gives us:

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

Simplifying the Equation

Now, let's simplify the equation by evaluating the exponent:

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

Evaluating the Exponent

The exponent $-2$ means that we need to take the reciprocal of the base and raise it to the power of $2$. In this case, the base is $2$, so we have:

$f(0) = 24 \left( \frac{1}{2} \right)^2 + 3$

Simplifying the Equation

Now, let's simplify the equation by evaluating the expression inside the parentheses:

$f(0) = 24 \left( \frac{1}{4} \right) + 3$

Evaluating the Expression

Now, let's evaluate the expression by multiplying the coefficient by the fraction:

$f(0) = 6 + 3$

Finding the $y$-intercept

Now, let's add the two numbers to find the $y$-intercept:

$f(0) = 9$

Conclusion

The $y$-intercept of the exponential function $f(x) = 24(2)^{x-2} + 3$ is $y = 9$. This means that when $x$ is equal to zero, the value of $y$ is $9$.

Answer

The correct answer is:

C. $y = 9$

Discussion

The $y$-intercept of an exponential function is an important concept in mathematics. It is used to determine the value of the function when $x$ is equal to zero. In this case, the $y$-intercept is $y = 9$, which means that when $x$ is equal to zero, the value of $y$ is $9$.

Related Topics

• Exponential functions
• $y$-intercept
• Graphing functions
• Mathematics

Further Reading

• Exponential functions: A comprehensive guide
• Graphing functions: A step-by-step guide
• Mathematics: A subject that is all around us

References

• [1] Exponential functions. (n.d.). In Encyclopedia Britannica. Retrieved from https://www.britannica.com/topic/exponential-function
• [2] Graphing functions. (n.d.). In Math Open Reference. Retrieved from https://www.mathopenref.com/graphing.html
• [3] Mathematics. (n.d.). In Wikipedia. Retrieved from https://en.wikipedia.org/wiki/Mathematics
  

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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 crosses the $y$-axis. In other words, it is the value of $y$ when $x$ is equal to zero.

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

A: To find the $y$-intercept of an exponential function, you need to substitute $x = 0$ into the equation of the function.

Q: What is the formula for finding the $y$-intercept of an exponential function?

A: The formula for finding the $y$-intercept of an exponential function is:

$f(0) = a \cdot b^0 + c$

where $a$ is the coefficient, $b$ is the base, and $c$ is the constant term.

Q: What is the value of $b^0$ in the formula?

A: The value of $b^0$ is always $1$, regardless of the value of $b$.

Q: How do I simplify the equation after substituting $x = 0$?

A: After substituting $x = 0$, you need to simplify the equation by evaluating the exponent and any expressions inside parentheses.

Q: What is the final step in finding the $y$-intercept?

A: The final step in finding the $y$-intercept is to add or subtract any constant terms to get the final value of $y$.

Q: Can you give an example of finding the $y$-intercept of an exponential function?

A: Let's say we have the exponential function $f(x) = 24(2)^{x-2} + 3$. To find the $y$-intercept, we need to substitute $x = 0$ into the equation:

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

Simplifying the equation, we get:

$f(0) = 24 \left( \frac{1}{2} \right)^2 + 3$

Evaluating the expression, we get:

$f(0) = 24 \left( \frac{1}{4} \right) + 3$

Simplifying further, we get:

$f(0) = 6 + 3$

Finally, we get:

$f(0) = 9$

So, the $y$-intercept of the function is $y = 9$.

Q: What is the significance of the $y$-intercept in exponential functions?

A: The $y$-intercept is an important concept in exponential functions because it helps us understand the behavior of the function when $x$ is equal to zero. It also helps us determine the value of the function at the origin.

Q: Can you give some examples of exponential functions with different $y$-intercepts?

A: Here are a few examples of exponential functions with different $y$-intercepts:

• $f(x) = 2^x + 1$ has a $y$-intercept of $y = 1$
• $f(x) = 3 \cdot 2^x - 2$ has a $y$-intercept of $y = -2$
• $f(x) = 4 \cdot 3^x + 2$ has a $y$-intercept of $y = 2$

Q: How do I graph an exponential function with a given $y$-intercept?

A: To graph an exponential function with a given $y$-intercept, you need to use a graphing calculator or software. You can also use a table of values to plot the function.

Q: Can you give some tips for graphing exponential functions?

A: Here are a few tips for graphing exponential functions:

• Use a graphing calculator or software to graph the function.
• Use a table of values to plot the function.
• Pay attention to the $y$-intercept and the asymptote.
• Use different colors or line styles to distinguish between different functions.

Q: How do I determine the domain and range of an exponential function?

A: To determine the domain and range of an exponential function, you need to consider the values of $x$ and $y$ that make the function undefined or undefined. For example, if the function has a base of $2$, the domain is all real numbers, but the range is only positive real numbers.

Q: Can you give some examples of exponential functions with different domains and ranges?

A: Here are a few examples of exponential functions with different domains and ranges:

• $f(x) = 2^x$ has a domain of all real numbers and a range of all positive real numbers.
• $f(x) = 3^x$ has a domain of all real numbers and a range of all positive real numbers.
• $f(x) = -2^x$ has a domain of all real numbers and a range of all negative real numbers.

Q: How do I solve exponential equations?

A: To solve exponential equations, you need to use logarithms to isolate the variable. For example, if you have the equation $2^x = 8$, you can take the logarithm of both sides to get:

$x = \log_2(8)$

Simplifying the equation, you get:

$x = 3$

So, the solution to the equation is $x = 3$.

Q: Can you give some examples of exponential equations?

A: Here are a few examples of exponential equations:

• $2^x = 8$
• $3^x = 27$
• $4^x = 64$

Q: How do I solve exponential inequalities?

A: To solve exponential inequalities, you need to use logarithms to isolate the variable. For example, if you have the inequality $2^x > 8$, you can take the logarithm of both sides to get:

$x > \log_2(8)$

Simplifying the equation, you get:

$x > 3$

So, the solution to the inequality is $x > 3$.

Q: Can you give some examples of exponential inequalities?

A: Here are a few examples of exponential inequalities:

• $2^x > 8$
• $3^x > 27$
• $4^x > 64$

Q: How do I graph exponential functions with different bases?

A: To graph exponential functions with different bases, you need to use a graphing calculator or software. You can also use a table of values to plot the function.

Q: Can you give some tips for graphing exponential functions with different bases?

A: Here are a few tips for graphing exponential functions with different bases:

• Use a graphing calculator or software to graph the function.
• Use a table of values to plot the function.
• Pay attention to the $y$-intercept and the asymptote.
• Use different colors or line styles to distinguish between different functions.

Q: How do I determine the equation of an exponential function from a graph?

A: To determine the equation of an exponential function from a graph, you need to identify the base and the $y$-intercept. For example, if the graph has a $y$-intercept of $y = 1$ and a base of $2$, the equation of the function is $f(x) = 2^x + 1$.

Q: Can you give some examples of exponential functions with different equations?

A: Here are a few examples of exponential functions with different equations:

• $f(x) = 2^x + 1$
• $f(x) = 3 \cdot 2^x - 2$
• $f(x) = 4 \cdot 3^x + 2$

Q: How do I use exponential functions in real-world applications?

A: Exponential functions have many real-world applications, including:

• Modeling population growth
• Modeling chemical reactions
• Modeling financial growth
• Modeling electrical circuits

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

A: Here are a few examples of exponential functions in real-world applications:

• The population of a city grows exponentially, with a base of $1.05$ and a $y$-intercept of $1000$.
• The amount of a chemical in a reaction grows exponentially, with a base of $2$ and a $y$-intercept of $10$.
• The value of a stock grows exponentially, with a base of $1.05$ and a $y$-intercept of $100$.
• The current in an electrical circuit grows exponentially, with a base of $2$ and a $y$-intercept of $10$.

Q: How do I use logarithms to solve exponential equations?

A: To use logarithms to solve exponential equations, you need to take the logarithm of both sides of the equation. For example, if you have the equation $2^x = 8$, you can take the logarithm of both sides to get:

$x = \log_2(8)$

Simplifying the equation, you get:

$x = 3$

So, the solution to the equation is $x = 3$.

Q: Can you give some examples of using logarithms to solve exponential equations?

A: Here are a few examples of using logarithms to solve exponential equations:

• $2^