Which Of The Following Functions Has Vertical Asymptotes At \[$ X = 6 \$\] And \[$ X = -3 \$\], And X-intercepts At \[$(-4, 0)\$\] And \[$(0, 0)\$\]?A. \[$ Y = \frac{x^2 - 3x - 18}{3x^2 + 12x} \$\]B. \[$ Y =
Introduction
In mathematics, functions are used to describe the relationship between variables. One of the key concepts in function analysis is the presence of vertical asymptotes and x-intercepts. Vertical asymptotes are points where the function approaches infinity or negative infinity, while x-intercepts are points where the function crosses the x-axis. In this article, we will analyze two functions and determine which one has vertical asymptotes at x = 6 and x = -3, and x-intercepts at (-4, 0) and (0, 0).
Vertical Asymptotes
Vertical asymptotes occur when the denominator of a rational function is equal to zero. This is because division by zero is undefined, and the function approaches infinity or negative infinity as the denominator approaches zero. In the case of the given functions, we need to find the values of x that make the denominator equal to zero.
Function A: y = (x^2 - 3x - 18) / (3x^2 + 12x)
To find the vertical asymptotes of Function A, we need to set the denominator equal to zero and solve for x.
3x^2 + 12x = 0
We can factor out a 3x from the equation:
3x(x + 4) = 0
This gives us two possible values for x:
x = 0
x = -4
However, we are looking for vertical asymptotes at x = 6 and x = -3. Therefore, Function A does not have vertical asymptotes at these points.
Function B: y = (x^2 - 3x - 18) / (x - 6)(x + 3)
To find the vertical asymptotes of Function B, we need to set the denominator equal to zero and solve for x.
(x - 6)(x + 3) = 0
This gives us two possible values for x:
x = 6
x = -3
Therefore, Function B has vertical asymptotes at x = 6 and x = -3.
X-Intercepts
X-intercepts occur when the function crosses the x-axis, which means that the y-coordinate is equal to zero. In the case of the given functions, we need to find the values of x that make the function equal to zero.
Function A: y = (x^2 - 3x - 18) / (3x^2 + 12x)
To find the x-intercepts of Function A, we need to set the numerator equal to zero and solve for x.
x^2 - 3x - 18 = 0
We can factor the equation:
(x - 6)(x + 3) = 0
This gives us two possible values for x:
x = 6
x = -3
However, we are looking for x-intercepts at (-4, 0) and (0, 0). Therefore, Function A does not have x-intercepts at these points.
Function B: y = (x^2 - 3x - 18) / (x - 6)(x + 3)
To find the x-intercepts of Function B, we need to set the numerator equal to zero and solve for x.
x^2 - 3x - 18 = 0
We can factor the equation:
(x - 6)(x + 3) = 0
This gives us two possible values for x:
x = 6
x = -3
However, we are looking for x-intercepts at (-4, 0) and (0, 0). To find the x-intercept at (0, 0), we need to set x equal to zero and solve for y.
y = (0^2 - 3(0) - 18) / ((0 - 6)(0 + 3))
y = -18 / (-18)
y = 1
However, this is not equal to zero, so Function B does not have an x-intercept at (0, 0). To find the x-intercept at (-4, 0), we need to set x equal to -4 and solve for y.
y = ((-4)^2 - 3(-4) - 18) / ((-4 - 6)(-4 + 3))
y = (16 + 12 - 18) / (-10 * -1)
y = 10 / 10
y = 1
However, this is not equal to zero, so Function B does not have an x-intercept at (-4, 0).
Conclusion
In conclusion, Function B has vertical asymptotes at x = 6 and x = -3, but it does not have x-intercepts at (-4, 0) and (0, 0). Therefore, the correct answer is Function B.
References
- [1] "Functions" by Khan Academy
- [2] "Vertical Asymptotes" by Math Open Reference
- [3] "X-Intercepts" by Purplemath
Vertical Asymptotes and X-Intercepts: A Q&A Guide =====================================================
Introduction
In our previous article, we analyzed two functions and determined which one has vertical asymptotes at x = 6 and x = -3, and x-intercepts at (-4, 0) and (0, 0). In this article, we will provide a Q&A guide to help you better understand the concepts of vertical asymptotes and x-intercepts.
Q: What are vertical asymptotes?
A: Vertical asymptotes are points where the function approaches infinity or negative infinity. They occur when the denominator of a rational function is equal to zero.
Q: How do I find the vertical asymptotes of a function?
A: To find the vertical asymptotes of a function, you need to set the denominator equal to zero and solve for x.
Q: What are x-intercepts?
A: X-intercepts are points where the function crosses the x-axis, which means that the y-coordinate is equal to zero.
Q: How do I find the x-intercepts of a function?
A: To find the x-intercepts of a function, you need to set the numerator equal to zero and solve for x.
Q: What is the difference between a vertical asymptote and an x-intercept?
A: A vertical asymptote is a point where the function approaches infinity or negative infinity, while an x-intercept is a point where the function crosses the x-axis.
Q: Can a function have both vertical asymptotes and x-intercepts?
A: Yes, a function can have both vertical asymptotes and x-intercepts. However, the x-intercepts must occur at points where the function is not undefined.
Q: How do I determine if a function has a vertical asymptote or an x-intercept?
A: To determine if a function has a vertical asymptote or an x-intercept, you need to analyze the function and its graph. If the function approaches infinity or negative infinity at a point, it is a vertical asymptote. If the function crosses the x-axis at a point, it is an x-intercept.
Q: Can a function have multiple vertical asymptotes or x-intercepts?
A: Yes, a function can have multiple vertical asymptotes or x-intercepts. However, the points must be distinct and not coincident.
Q: How do I graph a function with vertical asymptotes and x-intercepts?
A: To graph a function with vertical asymptotes and x-intercepts, you need to plot the points where the function is undefined (vertical asymptotes) and where the function crosses the x-axis (x-intercepts).
Q: What are some common mistakes to avoid when working with vertical asymptotes and x-intercepts?
A: Some common mistakes to avoid when working with vertical asymptotes and x-intercepts include:
- Confusing vertical asymptotes with x-intercepts
- Failing to check for undefined points
- Not considering the domain of the function
- Not analyzing the graph of the function
Conclusion
In conclusion, vertical asymptotes and x-intercepts are important concepts in mathematics that can help you better understand the behavior of functions. By following the tips and guidelines in this article, you can improve your understanding of these concepts and become a more confident and proficient math student.
References
- [1] "Functions" by Khan Academy
- [2] "Vertical Asymptotes" by Math Open Reference
- [3] "X-Intercepts" by Purplemath