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

Understanding Exponential Functions

Exponential functions are a type of mathematical function that describes a relationship between two quantities, where one quantity is a constant power of the other. The general form of an exponential function is:

$f(x) = a \cdot b^{x-c} + d$

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

The Given Exponential Function

The given exponential function is:

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

This function has the following components:

• $a = 24$
• $b = 2$
• $c = 2$
• $d = 3$

Finding the $y$-Intercept

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

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

Simplifying the Equation

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

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

We know that $2^{-2} = \frac{1}{2^2} = \frac{1}{4}$.

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

Simplifying Further

Now, let's simplify the equation further by multiplying $24$ and $\frac{1}{4}$.

$f(0) = 6 + 3$

The Final Answer

Now, let's add $6$ and $3$ to get the final answer.

$f(0) = 9$

Therefore, the $y$-intercept of the given exponential function is $y = 9$.

Conclusion

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In the previous article, we discussed how to find the $y$-intercept of an exponential function. In this article, we will answer some frequently asked questions related to finding the $y$-intercept of exponential functions.

Q: What is the $y$-intercept of the exponential function $f(x) = 5(3)^{x-1} + 2$?

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

$f(0) = 5(3)^{0-1} + 2$

We know that $3^{-1} = \frac{1}{3}$.

$f(0) = 5 \cdot \frac{1}{3} + 2$

Now, let's simplify the equation further by multiplying $5$ and $\frac{1}{3}$.

$f(0) = \frac{5}{3} + 2$

To add $\frac{5}{3}$ and $2$, we need to find a common denominator, which is $3$.

$f(0) = \frac{5}{3} + \frac{6}{3}$

Now, let's add the fractions.

$f(0) = \frac{11}{3}$

Therefore, the $y$-intercept of the given exponential function is $y = \frac{11}{3}$.

Q: What is the $y$-intercept of the exponential function $f(x) = 2(4)^{x-3} - 1$?

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

$f(0) = 2(4)^{0-3} - 1$

We know that $4^{-3} = \frac{1}{4^3} = \frac{1}{64}$.

$f(0) = 2 \cdot \frac{1}{64} - 1$

Now, let's simplify the equation further by multiplying $2$ and $\frac{1}{64}$.

$f(0) = \frac{1}{32} - 1$

To subtract $1$ from $\frac{1}{32}$, we need to find a common denominator, which is $32$.

$f(0) = \frac{1}{32} - \frac{32}{32}$

Now, let's subtract the fractions.

$f(0) = -\frac{31}{32}$

Therefore, the $y$-intercept of the given exponential function is $y = -\frac{31}{32}$.

Q: What is the $y$-intercept of the exponential function $f(x) = 3(2)^{x-2} + 4$?

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

$f(0) = 3(2)^{0-2} + 4$

We know that $2^{-2} = \frac{1}{2^2} = \frac{1}{4}$.

$f(0) = 3 \cdot \frac{1}{4} + 4$

Now, let's simplify the equation further by multiplying $3$ and $\frac{1}{4}$.

$f(0) = \frac{3}{4} + 4$

To add $\frac{3}{4}$ and $4$, we need to find a common denominator, which is $4$.

$f(0) = \frac{3}{4} + \frac{16}{4}$

Now, let's add the fractions.

$f(0) = \frac{19}{4}$

Therefore, the $y$-intercept of the given exponential function is $y = \frac{19}{4}$.

Conclusion

In this article, we answered some frequently asked questions related to finding the $y$-intercept of exponential functions. We used the given functions as examples and substituted $x = 0$ into the equations to find the $y$-intercepts. We simplified the equations step by step and arrived at the final answers.