What Is The $y$-intercept Of $f(x)=3^{x+2}$?A. \$(9,0)$[/tex\] B. $(0,9)$ C. $(0,-9)$ D. \$(9,-9)$[/tex\]

Introduction

In mathematics, the $y$-intercept of a function is the point at which the graph of the function intersects the $y$-axis. It is a crucial concept in understanding the behavior of a function and is used extensively in various mathematical and scientific applications. In this article, we will explore the concept of the $y$-intercept and determine the $y$-intercept of the function $f(x)=3^{x+2}$.

What is the $y$-Intercept?

The $y$-intercept of a function is the point at which the graph of the function intersects the $y$-axis. It is the value of $y$ when $x=0$. In other words, it is the point on the graph where the $x$-coordinate is zero. The $y$-intercept is denoted by the symbol $y$-intercept or $f(0)$.

Finding the $y$-Intercept of a Function

To find the $y$-intercept of a function, we need to substitute $x=0$ into the function and evaluate the resulting expression. This will give us the value of $y$ when $x=0$, which is the $y$-intercept of the function.

Determining the $y$-Intercept of $f(x)=3^{x+2}$

To determine the $y$-intercept of the function $f(x)=3^{x+2}$, we need to substitute $x=0$ into the function and evaluate the resulting expression.

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

Using the properties of exponents, we can simplify the expression as follows:

$$f(0)=3^2$$

$$f(0)=9$$

Therefore, the $y$-intercept of the function $f(x)=3^{x+2}$ is $(0,9)$.

Conclusion

In conclusion, the $y$-intercept of a function is the point at which the graph of the function intersects the $y$-axis. It is a crucial concept in understanding the behavior of a function and is used extensively in various mathematical and scientific applications. In this article, we determined the $y$-intercept of the function $f(x)=3^{x+2}$ to be $(0,9)$.

Answer

The correct answer is:

• B. $(0,9)$

Additional Information

• The $y$-intercept of a function is denoted by the symbol $y$-intercept or $f(0)$.
• To find the $y$-intercept of a function, we need to substitute $x=0$ into the function and evaluate the resulting expression.
• The $y$-intercept of the function $f(x)=3^{x+2}$ is $(0,9)$.

References

• [1] Khan Academy. (n.d.). Y-intercept. Retrieved from https://www.khanacademy.org/math/algebra/x2f1f4/x2f1f4_1f1f1f1
• [2] Math Open Reference. (n.d.). Y-intercept. Retrieved from https://www.mathopenref.com/yintercept.html

Related Topics

• Finding the $y$-Intercept of a Linear Function
• Finding the $y$-Intercept of a Quadratic Function
• Finding the $y$-Intercept of an Exponential Function

Finding the $y$-Intercept of a Linear Function

To find the $y$-intercept of a linear function, we need to substitute $x=0$ into the function and evaluate the resulting expression. The $y$-intercept of a linear function is the value of $y$ when $x=0$.

Finding the $y$-Intercept of a Quadratic Function

To find the $y$-intercept of a quadratic function, we need to substitute $x=0$ into the function and evaluate the resulting expression. The $y$-intercept of a quadratic function is the value of $y$ when $x=0$.

Finding the $y$-Intercept of an Exponential Function

Frequently Asked Questions

Q: What is the $y$-intercept of a function?

A: The $y$-intercept of a function is the point at which the graph of the function intersects the $y$-axis. It is the value of $y$ when $x=0$.

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

A: To find the $y$-intercept of a function, you need to substitute $x=0$ into the function and evaluate the resulting expression.

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

A: The $y$-intercept of the function $f(x)=3^{x+2}$ is $(0,9)$.

Q: Why is the $y$-intercept important?

A: The $y$-intercept is an important concept in understanding the behavior of a function. It is used extensively in various mathematical and scientific applications.

Q: Can you give an example of how to find the $y$-intercept of a linear function?

A: Yes, let's consider the linear function $f(x)=2x+1$. To find the $y$-intercept, we need to substitute $x=0$ into the function:

$$f(0)=2(0)+1$$

$$f(0)=1$$

Therefore, the $y$-intercept of the function $f(x)=2x+1$ is $(0,1)$.

Q: Can you give an example of how to find the $y$-intercept of a quadratic function?

A: Yes, let's consider the quadratic function $f(x)=x^2+2x+1$. To find the $y$-intercept, we need to substitute $x=0$ into the function:

$$f(0)=(0)^2+2(0)+1$$

$$f(0)=1$$

Therefore, the $y$-intercept of the function $f(x)=x^2+2x+1$ is $(0,1)$.

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

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

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

$$f(0)=3^2$$

$$f(0)=9$$

Therefore, the $y$-intercept of the function $f(x)=3^{x+2}$ is $(0,9)$.

Additional Resources

• Khan Academy: Y-Intercept
• Math Open Reference: Y-Intercept
• Wolfram Alpha: Y-Intercept

Related Topics

• Finding the $y$-Intercept of a Linear Function
• Finding the $y$-Intercept of a Quadratic Function
• Finding the $y$-Intercept of an Exponential Function

Finding the $y$-Intercept of a Linear Function

To find the $y$-intercept of a linear function, we need to substitute $x=0$ into the function and evaluate the resulting expression. The $y$-intercept of a linear function is the value of $y$ when $x=0$.

Finding the $y$-Intercept of a Quadratic Function

To find the $y$-intercept of a quadratic function, we need to substitute $x=0$ into the function and evaluate the resulting expression. The $y$-intercept of a quadratic function is the value of $y$ when $x=0$.

Finding the $y$-Intercept of an Exponential Function

To find the $y$-intercept of an exponential function, we need to substitute $x=0$ into the function and evaluate the resulting expression. The $y$-intercept of an exponential function is the value of $y$ when $x=0$.