Select The Correct Answer.What Is The \[$ Y \$\]-intercept Of \[$ F(x) = 3^{x+2} \$\]?A. \[$ (9,0) \$\] B. \[$ (0,9) \$\] C. \[$ (0,-9) \$\] D. \[$ (9,-9) \$\]

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The y-intercept of a function is a crucial concept in mathematics, particularly in algebra and calculus. It represents the point at which the graph of a function intersects the y-axis. In other words, it is the value of the function when the input (or x-value) is equal to zero. In this article, we will focus on finding the y-intercept of a given function and provide a step-by-step guide on how to select the correct answer.

What is the y-Intercept?

The y-intercept is a point on the graph of a function where the x-coordinate is zero. It is denoted by the symbol (0, y) or (0, f(0)), where y is the value of the function at x = 0. The y-intercept is an important concept in mathematics, as it provides valuable information about the behavior of a function.

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 the function at x = 0, which is the y-intercept.

Step-by-Step Guide to Finding the y-Intercept

  1. Substitute x = 0 into the function: Replace x with 0 in the given function.
  2. Evaluate the resulting expression: Simplify the expression to find the value of the function at x = 0.
  3. Identify the y-intercept: The resulting value is the y-intercept of the function.

Example: Finding the y-Intercept of f(x) = 3^(x+2)

Let's apply the step-by-step guide to find the y-intercept of the function f(x) = 3^(x+2).

Step 1: Substitute x = 0 into the function

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

Step 2: Evaluate the resulting expression

f(0) = 3^2

Step 3: Identify the y-intercept

f(0) = 9

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

Selecting the Correct Answer

Based on our calculation, the correct answer is:

  • B. (0, 9)

The other options are incorrect because they do not match the y-intercept of the function f(x) = 3^(x+2).

Conclusion

In conclusion, finding the y-intercept of a function is a straightforward process that involves substituting x = 0 into the function and evaluating the resulting expression. By following the step-by-step guide provided in this article, you can easily find the y-intercept of any function. Remember to substitute x = 0 into the function, evaluate the resulting expression, and identify the y-intercept. With practice, you will become proficient in finding the y-intercept of various functions.

Frequently Asked Questions

  • What is the y-intercept of a function?
    • The y-intercept is a point on the graph of a function where the x-coordinate is zero.
  • How do I find the y-intercept of a function?
    • Substitute x = 0 into the function and evaluate the resulting expression.
  • What is the y-intercept of f(x) = 3^(x+2)?
    • The y-intercept of f(x) = 3^(x+2) is (0, 9).

Additional Resources

For more information on finding the y-intercept of a function, you can refer to the following resources:

  • Algebra textbooks: Most algebra textbooks cover the concept of y-intercept in detail.
  • Online resources: Websites such as Khan Academy, Mathway, and Wolfram Alpha provide interactive lessons and examples on finding the y-intercept of a function.
  • Mathematical software: Software such as Mathematica and Maple can be used to visualize and analyze the graph of a function, including finding the y-intercept.
    Q&A: Finding the y-Intercept of a Function =============================================

In our previous article, we discussed the concept of y-intercept and provided a step-by-step guide on how to find the y-intercept of a function. In this article, we will address some frequently asked questions related to finding the y-intercept of a function.

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

A: The y-intercept of a function is a point on the graph of a function where the x-coordinate is zero. It is denoted by the symbol (0, y) or (0, f(0)), where y is the value of the function at 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. This will give you the value of the function at x = 0, which is the y-intercept.

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

A: The y-intercept of f(x) = 3^(x+2) is (0, 9). To find this, we substitute x = 0 into the function and evaluate the resulting expression: f(0) = 3^(0+2) = 3^2 = 9.

Q: Can I use a calculator to find the y-intercept of a function?

A: Yes, you can use a calculator to find the y-intercept of a function. Simply substitute x = 0 into the function and evaluate the resulting expression using the calculator.

Q: What if the function is a polynomial or a rational function?

A: If the function is a polynomial or a rational function, you can use the same method to find the y-intercept. Substitute x = 0 into the function and evaluate the resulting expression.

Q: Can I find the y-intercept of a function using a graphing calculator?

A: Yes, you can use a graphing calculator to find the y-intercept of a function. Graph the function and use the calculator to find the point where the graph intersects the y-axis.

Q: What if the function has a variable in the exponent?

A: If the function has a variable in the exponent, you may need to use logarithmic properties to simplify the expression and find the y-intercept.

Q: Can I find the y-intercept of a function using a computer algebra system (CAS)?

A: Yes, you can use a CAS to find the y-intercept of a function. Enter the function into the CAS and use the software to find the point where the graph intersects the y-axis.

Q: What if I'm having trouble finding the y-intercept of a function?

A: If you're having trouble finding the y-intercept of a function, try the following:

  • Check your work to make sure you're substituting x = 0 into the function correctly.
  • Simplify the expression to make it easier to evaluate.
  • Use a calculator or computer algebra system to check your work.
  • Ask a teacher or tutor for help.

Conclusion

Finding the y-intercept of a function is an important concept in mathematics, and it can be used to analyze and understand the behavior of a function. By following the step-by-step guide provided in this article, you can easily find the y-intercept of any function. Remember to substitute x = 0 into the function, evaluate the resulting expression, and identify the y-intercept. With practice, you will become proficient in finding the y-intercept of various functions.

Frequently Asked Questions

  • What is the y-intercept of a function?
    • The y-intercept is a point on the graph of a function where the x-coordinate is zero.
  • How do I find the y-intercept of a function?
    • Substitute x = 0 into the function and evaluate the resulting expression.
  • What is the y-intercept of f(x) = 3^(x+2)?
    • The y-intercept of f(x) = 3^(x+2) is (0, 9).

Additional Resources

For more information on finding the y-intercept of a function, you can refer to the following resources:

  • Algebra textbooks: Most algebra textbooks cover the concept of y-intercept in detail.
  • Online resources: Websites such as Khan Academy, Mathway, and Wolfram Alpha provide interactive lessons and examples on finding the y-intercept of a function.
  • Mathematical software: Software such as Mathematica and Maple can be used to visualize and analyze the graph of a function, including finding the y-intercept.