Find The Y Y Y -intercept.${ \begin{array}{c} y = \frac{6x - 18}{x + 9} \ (0, [?]) \end{array} }$

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Introduction

In algebra, the $y$-intercept of a function is the point where the graph of the function intersects the $y$-axis. It is a crucial concept in understanding the behavior of functions, especially rational functions. In this article, we will focus on finding the $y$-intercept of a rational function given by the equation $y = \frac{6x - 18}{x + 9}$.

What is a $y$-Intercept?

The $y$-intercept of a function is the value of $y$ when $x = 0$. In other words, it is the point on the graph where the $x$-coordinate is zero. To find the $y$-intercept, we need to substitute $x = 0$ into the equation of the function and solve for $y$.

Finding the $y$-Intercept of the Given Rational Function

To find the $y$-intercept of the given rational function, we need to substitute $x = 0$ into the equation $y = \frac{6x - 18}{x + 9}$. This will give us the value of $y$ when $x = 0$.

import sympy as sp
x = sp.symbols('x')

y = (6*x - 18) / (x + 9)

y_intercept = y.subs(x, 0)
print(y_intercept)

Simplifying the Expression

When we substitute $x = 0$ into the equation, we get $y = \frac{6(0) - 18}{0 + 9}$. Simplifying this expression, we get $y = \frac{-18}{9}$.

# Simplify the expression
y_intercept = -18 / 9
print(y_intercept)

Finding the $y$-Intercept

The $y$-intercept of the given rational function is the value of $y$ when $x = 0$. In this case, we found that $y = \frac{-18}{9}$, which simplifies to $y = -2$.

Conclusion

In this article, we found the $y$-intercept of a rational function given by the equation $y = \frac{6x - 18}{x + 9}$. We substituted $x = 0$ into the equation and simplified the expression to find the value of $y$ when $x = 0$. The $y$-intercept of the given rational function is $y = -2$.

Example Problems

Problem 1

Find the $y$-intercept of the rational function $y = \frac{3x - 12}{x + 6}$.

Solution

To find the $y$-intercept, we need to substitute $x = 0$ into the equation. This gives us $y = \frac{3(0) - 12}{0 + 6}$. Simplifying this expression, we get $y = \frac{-12}{6}$, which simplifies to $y = -2$.

Problem 2

Find the $y$-intercept of the rational function $y = \frac{2x - 10}{x - 5}$.

Solution

To find the $y$-intercept, we need to substitute $x = 0$ into the equation. This gives us $y = \frac{2(0) - 10}{0 - 5}$. Simplifying this expression, we get $y = \frac{-10}{-5}$, which simplifies to $y = 2$.

Tips and Tricks

• When finding the $y$-intercept of a rational function, make sure to substitute $x = 0$ into the equation.
• Simplify the expression to find the value of $y$ when $x = 0$.
• Use the sympy library in Python to simplify expressions and find the $y$-intercept.

Common Mistakes

• Failing to substitute $x = 0$ into the equation.
• Not simplifying the expression to find the value of $y$ when $x = 0$.
• Using the wrong value for $x$ when finding the $y$-intercept.

Conclusion

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Introduction

In our previous article, we discussed how to find the $y$-intercept of a rational function given by the equation $y = \frac{6x - 18}{x + 9}$. We also provided example problems and tips and tricks for finding the $y$-intercept of rational functions. In this article, we will answer some frequently asked questions about finding the $y$-intercept of rational functions.

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

A: The $y$-intercept of a rational function is the value of $y$ when $x = 0$. It is the point on the graph where the $x$-coordinate is zero.

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

A: To find the $y$-intercept of a rational function, you need to substitute $x = 0$ into the equation and simplify the expression to find the value of $y$ when $x = 0$.

Q: What if the rational function has a denominator of zero when $x = 0$?

A: If the rational function has a denominator of zero when $x = 0$, then the function is undefined at that point. In this case, the $y$-intercept is not defined.

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

A: Yes, you can use a calculator to find the $y$-intercept of a rational function. However, make sure to enter the equation correctly and use the correct function to find the $y$-intercept.

Q: How do I know if the $y$-intercept of a rational function is positive or negative?

A: To determine if the $y$-intercept of a rational function is positive or negative, you need to simplify the expression and look at the sign of the result.

Q: Can I find the $y$-intercept of a rational function with a quadratic denominator?

A: Yes, you can find the $y$-intercept of a rational function with a quadratic denominator. However, you may need to use factoring or the quadratic formula to simplify the expression.

Q: How do I find the $y$-intercept of a rational function with a complex denominator?

A: To find the $y$-intercept of a rational function with a complex denominator, you need to simplify the expression and use the correct function to find the $y$-intercept.

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

A: Yes, you can use a graphing calculator to find the $y$-intercept of a rational function. However, make sure to enter the equation correctly and use the correct function to find the $y$-intercept.

Q: How do I know if the $y$-intercept of a rational function is an integer or a fraction?

A: To determine if the $y$-intercept of a rational function is an integer or a fraction, you need to simplify the expression and look at the result.

Conclusion

In this article, we answered some frequently asked questions about finding the $y$-intercept of rational functions. We hope that this article has been helpful in clarifying any confusion about finding the $y$-intercept of rational functions.

Example Problems

Problem 1

Find the $y$-intercept of the rational function $y = \frac{3x - 12}{x + 6}$.

Solution

To find the $y$-intercept, we need to substitute $x = 0$ into the equation. This gives us $y = \frac{3(0) - 12}{0 + 6}$. Simplifying this expression, we get $y = \frac{-12}{6}$, which simplifies to $y = -2$.

Problem 2

Find the $y$-intercept of the rational function $y = \frac{2x - 10}{x - 5}$.

Solution

To find the $y$-intercept, we need to substitute $x = 0$ into the equation. This gives us $y = \frac{2(0) - 10}{0 - 5}$. Simplifying this expression, we get $y = \frac{-10}{-5}$, which simplifies to $y = 2$.

Tips and Tricks

• Make sure to substitute $x = 0$ into the equation to find the $y$-intercept.
• Simplify the expression to find the value of $y$ when $x = 0$.
• Use the sympy library in Python to simplify expressions and find the $y$-intercept.

Common Mistakes

• Failing to substitute $x = 0$ into the equation.
• Not simplifying the expression to find the value of $y$ when $x = 0$.
• Using the wrong value for $x$ when finding the $y$-intercept.