Find The X-intercept.${ y = \frac{6x - 18}{x + 9} }${ (x, 0)\$}

Mar 12, 2025 by ADMIN 66 views

**Finding the X-Intercept of a Rational Function**

Understanding the Concept of X-Intercept

The x-intercept of a function is the point at which the graph of the function crosses the x-axis. In other words, it is the value of x at which the function has a y-coordinate of zero. To find the x-intercept of a rational function, we need to set the numerator of the function equal to zero and solve for x.

The Given Rational Function

The given rational function is:

${ y = \frac{6x - 18}{x + 9} }$

Setting the Numerator Equal to Zero

To find the x-intercept, we need to set the numerator of the function equal to zero and solve for x. The numerator of the function is 6x - 18. Setting it equal to zero, we get:

${ 6x - 18 = 0 }$

Solving for x

To solve for x, we need to isolate x on one side of the equation. We can do this by adding 18 to both sides of the equation and then dividing both sides by 6.

${ 6x - 18 + 18 = 0 + 18 }$ ${ 6x = 18 }$ ${ \frac{6x}{6} = \frac{18}{6} }$ ${ x = 3 }$

The X-Intercept

Therefore, the x-intercept of the given rational function is x = 3.

Graphical Representation

The graph of the rational function is a hyperbola that opens to the left and right. The x-intercept is the point at which the graph crosses the x-axis. In this case, the x-intercept is at x = 3.

Real-World Applications

Finding the x-intercept of a rational function has many real-world applications. For example, in physics, the x-intercept of a function can represent the point at which a projectile lands on the ground. In economics, the x-intercept of a function can represent the point at which a company's revenue equals its costs.

Conclusion

In conclusion, finding the x-intercept of a rational function involves setting the numerator of the function equal to zero and solving for x. The x-intercept is the point at which the graph of the function crosses the x-axis. It has many real-world applications and is an important concept in mathematics.

Example Problems

Problem 1

Find the x-intercept of the rational function:

${ y = \frac{2x - 6}{x - 3} }$

Solution

To find the x-intercept, we need to set the numerator of the function equal to zero and solve for x.

${ 2x - 6 = 0 }$ ${ 2x = 6 }$ ${ \frac{2x}{2} = \frac{6}{2} }$ ${ x = 3 }$

Therefore, the x-intercept of the given rational function is x = 3.

Problem 2

Find the x-intercept of the rational function:

${ y = \frac{4x - 12}{x + 4} }$

Solution

To find the x-intercept, we need to set the numerator of the function equal to zero and solve for x.

${ 4x - 12 = 0 }$ ${ 4x = 12 }$ ${ \frac{4x}{4} = \frac{12}{4} }$ ${ x = 3 }$

Therefore, the x-intercept of the given rational function is x = 3.

Tips and Tricks

To find the x-intercept of a rational function, set the numerator of the function equal to zero and solve for x.
The x-intercept is the point at which the graph of the function crosses the x-axis.
Finding the x-intercept has many real-world applications, including physics and economics.

Common Mistakes

Not setting the numerator equal to zero when finding the x-intercept.
Not solving for x when finding the x-intercept.
Not checking if the x-intercept is a real number.

Conclusion

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

A: The x-intercept of a function is the point at which the graph of the function crosses the x-axis. In other words, it is the value of x at which the function has a y-coordinate of zero.

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

A: To find the x-intercept of a rational function, you need to set the numerator of the function equal to zero and solve for x.

Q: What is the numerator of a rational function?

A: The numerator of a rational function is the expression on top of the fraction. For example, in the rational function y = (2x - 3)/(x + 1), the numerator is 2x - 3.

Q: How do I set the numerator equal to zero?

A: To set the numerator equal to zero, you need to add or subtract a value from both sides of the equation until the numerator is equal to zero. For example, if the numerator is 2x - 3, you can add 3 to both sides of the equation to get 2x = 3.

Q: How do I solve for x?

A: To solve for x, you need to isolate x on one side of the equation. You can do this by adding or subtracting a value from both sides of the equation, or by multiplying or dividing both sides of the equation by a value.

Q: What if the numerator is a complex expression?

A: If the numerator is a complex expression, you may need to use algebraic techniques such as factoring or the quadratic formula to solve for x.

Q: What if the x-intercept is not a real number?

A: If the x-intercept is not a real number, it means that the function does not cross the x-axis at that point. In this case, the x-intercept is said to be "complex" or "imaginary".

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

A: Yes, you can find the x-intercept of a function using a graphing calculator. Simply graph the function and use the calculator's built-in features to find the x-intercept.

Q: What are some common mistakes to avoid when finding the x-intercept?

A: Some common mistakes to avoid when finding the x-intercept include:

Not setting the numerator equal to zero
Not solving for x
Not checking if the x-intercept is a real number
Not using the correct algebraic techniques to solve for x

Q: Why is finding the x-intercept important?

A: Finding the x-intercept is important because it can help you understand the behavior of a function and its graph. It can also be used to solve real-world problems in fields such as physics, engineering, and economics.

Q: Can I find the x-intercept of a function using calculus?

A: Yes, you can find the x-intercept of a function using calculus. Specifically, you can use the derivative of the function to find the x-intercept.

Q: What are some real-world applications of finding the x-intercept?

A: Some real-world applications of finding the x-intercept include:

Physics: Finding the x-intercept of a function can help you understand the motion of an object and its trajectory.
Engineering: Finding the x-intercept of a function can help you design and optimize systems and structures.
Economics: Finding the x-intercept of a function can help you understand the behavior of economic systems and make predictions about future trends.

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

A: Yes, you can find the x-intercept of a function using a computer algebra system (CAS). Simply enter the function into the CAS and use its built-in features to find the x-intercept.

Q: What are some tips for finding the x-intercept of a function?

A: Some tips for finding the x-intercept of a function include:

Use algebraic techniques such as factoring and the quadratic formula to solve for x.
Check if the x-intercept is a real number.
Use a graphing calculator or computer algebra system (CAS) to find the x-intercept.
Understand the behavior of the function and its graph.
Use the x-intercept to solve real-world problems in fields such as physics, engineering, and economics.