Find The \[$ Y \$\]-intercept Of The Function.$\[ y = \frac{3x + 12}{x - 6} \\]The \[$ Y \$\]-intercept Is \[$(0, [?])\$\].

$$

Introduction

The $y$-intercept of a function is the point at which the graph of the function intersects the $y$-axis. In other words, it is the value of $y$ when $x$ is equal to zero. To find the $y$-intercept of a function, we need to substitute $x = 0$ into the equation of the function and solve for $y$. In this article, we will find the $y$-intercept of the function $y = \frac{3x + 12}{x - 6}$.

Understanding the Function

The given function is a rational function, which is a function that can be expressed as the ratio of two polynomials. In this case, the function is $y = \frac{3x + 12}{x - 6}$. To find the $y$-intercept, we need to substitute $x = 0$ into this equation.

Finding the $y$-intercept

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

$$y = \frac{3(0) + 12}{0 - 6}$$

Simplifying the equation, we get:

$$y = \frac{12}{-6}$$

$$y = -2$$

Therefore, the $y$-intercept of the function is $(0, -2)$.

Importance of the $y$-intercept

The $y$-intercept is an important concept in mathematics, particularly in algebra and calculus. It is used to determine the behavior of a function at a particular point. In this case, the $y$-intercept of the function $y = \frac{3x + 12}{x - 6}$ is $-2$, which means that the function intersects the $y$-axis at the point $(0, -2)$.

Real-World Applications

The concept of the $y$-intercept has many real-world applications. For example, in economics, the $y$-intercept of a demand curve represents the minimum price that consumers are willing to pay for a product. In physics, the $y$-intercept of a velocity-time graph represents the initial velocity of an object.

Conclusion

In conclusion, the $y$-intercept of a function is the point at which the graph of the function intersects the $y$-axis. To find the $y$-intercept, we need to substitute $x = 0$ into the equation of the function and solve for $y$. In this article, we found the $y$-intercept of the function $y = \frac{3x + 12}{x - 6}$ to be $(0, -2)$. The concept of the $y$-intercept has many real-world applications and is an important concept in mathematics.

Additional Examples

Here are a few more examples of finding the $y$-intercept of a function:

• Example 1: Find the $y$-intercept of the function $y = 2x + 5$.
  • To find the $y$-intercept, we substitute $x = 0$ into the equation of the function.

  • $$y = 2(0) + 5$$

  • $$y = 5$$

  • Therefore, the $y$-intercept of the function is $(0, 5)$.
• Example 2: Find the $y$-intercept of the function $y = \frac{x - 2}{x + 1}$.
  • To find the $y$-intercept, we substitute $x = 0$ into the equation of the function.

  • $$y = \frac{0 - 2}{0 + 1}$$

  • $$y = -2$$

  • Therefore, the $y$-intercept of the function is $(0, -2)$.

Tips and Tricks

Here are a few tips and tricks for finding the $y$-intercept of a function:

• Tip 1: Make sure to substitute $x = 0$ into the equation of the function.
• Tip 2: Simplify the equation as much as possible before solving for $y$.
• Tip 3: Check your work by plugging the value of $y$ back into the equation of the function.

Final Thoughts

In conclusion, the $y$-intercept of a function is an important concept in mathematics. It is used to determine the behavior of a function at a particular point. To find the $y$-intercept, we need to substitute $x = 0$ into the equation of the function and solve for $y$. In this article, we found the $y$-intercept of the function $y = \frac{3x + 12}{x - 6}$ to be $(0, -2)$. The concept of the $y$-intercept has many real-world applications and is an important concept in mathematics.

$$

Introduction

In our previous article, we discussed how to find the $y$-intercept of a function. In this article, we will answer some frequently asked questions about finding the $y$-intercept of a function.

Q&A

Q1: What is the $y$-intercept of a function?

A1: 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 $x$ is equal to zero.

Q2: How do I find the $y$-intercept of a function?

A2: To find the $y$-intercept of a function, you need to substitute $x = 0$ into the equation of the function and solve for $y$.

Q3: What is the difference between the $y$-intercept and the $x$-intercept?

A3: The $y$-intercept is the point at which the graph of the function intersects the $y$-axis, while the $x$-intercept is the point at which the graph of the function intersects the $x$-axis.

Q4: Can I find the $y$-intercept of a function using a graphing calculator?

A4: Yes, you can find the $y$-intercept of a function using a graphing calculator. Simply enter the equation of the function into the calculator and use the "intersect" or "zero" function to find the $y$-intercept.

Q5: What is the significance of the $y$-intercept in real-world applications?

A5: The $y$-intercept has many real-world applications. For example, in economics, the $y$-intercept of a demand curve represents the minimum price that consumers are willing to pay for a product. In physics, the $y$-intercept of a velocity-time graph represents the initial velocity of an object.

Q6: Can I find the $y$-intercept of a function with a negative exponent?

A6: Yes, you can find the $y$-intercept of a function with a negative exponent. To do this, you need to substitute $x = 0$ into the equation of the function and solve for $y$. For example, if the function is $y = \frac{1}{x}$, the $y$-intercept is $(0, \infty)$.

Q7: What is the $y$-intercept of a function with a zero denominator?

A7: If the function has a zero denominator, the $y$-intercept is undefined. For example, if the function is $y = \frac{1}{x}$, the $y$-intercept is undefined because the denominator is zero when $x = 0$.

Q8: Can I find the $y$-intercept of a function with a complex number?

A8: Yes, you can find the $y$-intercept of a function with a complex number. To do this, you need to substitute $x = 0$ into the equation of the function and solve for $y$. For example, if the function is $y = 2 + 3i$, the $y$-intercept is $(0, 2 + 3i)$.

Conclusion

In conclusion, finding the $y$-intercept of a function is an important concept in mathematics. It is used to determine the behavior of a function at a particular point. We have answered some frequently asked questions about finding the $y$-intercept of a function, including how to find the $y$-intercept, the difference between the $y$-intercept and the $x$-intercept, and the significance of the $y$-intercept in real-world applications.

Additional Resources

Here are some additional resources for finding the $y$-intercept of a function:

• Online Graphing Calculator: A online graphing calculator that can be used to find the $y$-intercept of a function.
• Mathematics Textbook: A mathematics textbook that covers the concept of the $y$-intercept of a function.
• Mathematics Website: A mathematics website that provides tutorials and examples on finding the $y$-intercept of a function.

Final Thoughts

In conclusion, finding the $y$-intercept of a function is an important concept in mathematics. It is used to determine the behavior of a function at a particular point. We have answered some frequently asked questions about finding the $y$-intercept of a function, including how to find the $y$-intercept, the difference between the $y$-intercept and the $x$-intercept, and the significance of the $y$-intercept in real-world applications.