If The Graph Of F(x):=9x²+36x+413x+5has An Oblique Asymptote At Y = 3x + K, What Is The Value Of K?
Introduction
In mathematics, an oblique asymptote is a line that a function approaches as the input (or x-value) gets arbitrarily large in the positive or negative direction. It is called "oblique" because it is not a horizontal line. In this article, we will explore the concept of oblique asymptotes and how to find the value of k in the equation y = 3x + k, given that the graph of f(x) = 9x² + 36x + 413x + 5 has an oblique asymptote at y = 3x + k.
Understanding Oblique Asymptotes
An oblique asymptote is a line that a function approaches as the input (or x-value) gets arbitrarily large in the positive or negative direction. It is called "oblique" because it is not a horizontal line. In other words, an oblique asymptote is a line that a function approaches as x goes to infinity or negative infinity.
Dividing Polynomials
To find the oblique asymptote of a function, we need to divide the polynomial by x. This is because the oblique asymptote is the quotient of the division, and the remainder is the error term. In this case, we need to divide the polynomial 9x² + 36x + 413x + 5 by x.
Performing Polynomial Division
To perform polynomial division, we need to divide the leading term of the polynomial (9x²) by the leading term of the divisor (x). This gives us 9x. We then multiply the divisor (x) by 9x and subtract the result from the polynomial.
Subtracting the Result
We subtract the result (9x² + 36x) from the polynomial (9x² + 36x + 413x + 5). This gives us 377x + 5.
Dividing the Result
We then divide the result (377x + 5) by x. This gives us 377 + 5/x.
Finding the Oblique Asymptote
The oblique asymptote is the quotient of the division, which is 377 + 5/x. However, since we are looking for the value of k in the equation y = 3x + k, we need to find the value of k that makes the equation y = 3x + k equal to the oblique asymptote 377 + 5/x.
Equating the Two Equations
We equate the two equations y = 3x + k and y = 377 + 5/x. This gives us 3x + k = 377 + 5/x.
Solving for k
We solve for k by subtracting 377 from both sides of the equation. This gives us 3x + k - 377 = 5/x.
Simplifying the Equation
We simplify the equation by combining like terms. This gives us 3x + k - 377 = 5/x.
Multiplying Both Sides by x
We multiply both sides of the equation by x to eliminate the fraction. This gives us 3x² + kx - 377x = 5.
Rearranging the Terms
We rearrange the terms to isolate k. This gives us kx = 5 + 377x - 3x².
Factoring Out x
We factor out x from the right-hand side of the equation. This gives us kx = x(377 - 3x) + 5.
Equating the Coefficients
We equate the coefficients of x on both sides of the equation. This gives us k = 377 - 3x + 5/x.
Finding the Value of k
We find the value of k by setting x = 0. This gives us k = 377 - 3(0) + 5/0.
Simplifying the Expression
We simplify the expression by evaluating the limit as x approaches 0. This gives us k = 377.
Conclusion
In this article, we explored the concept of oblique asymptotes and how to find the value of k in the equation y = 3x + k, given that the graph of f(x) = 9x² + 36x + 413x + 5 has an oblique asymptote at y = 3x + k. We performed polynomial division to find the oblique asymptote and then equated the two equations to solve for k. We found that the value of k is 377.
Final Answer
The final answer is:
Introduction
In our previous article, we explored the concept of oblique asymptotes and how to find the value of k in the equation y = 3x + k, given that the graph of f(x) = 9x² + 36x + 413x + 5 has an oblique asymptote at y = 3x + k. We performed polynomial division to find the oblique asymptote and then equated the two equations to solve for k. In this article, we will answer some frequently asked questions related to oblique asymptotes and the value of k.
Q&A
Q: What is an oblique asymptote?
A: An oblique asymptote is a line that a function approaches as the input (or x-value) gets arbitrarily large in the positive or negative direction. It is called "oblique" because it is not a horizontal line.
Q: How do I find the oblique asymptote of a function?
A: To find the oblique asymptote of a function, you need to divide the polynomial by x. This is because the oblique asymptote is the quotient of the division, and the remainder is the error term.
Q: What is the value of k in the equation y = 3x + k?
A: The value of k is 377.
Q: How do I perform polynomial division?
A: To perform polynomial division, you need to divide the leading term of the polynomial by the leading term of the divisor. You then multiply the divisor by the result and subtract the result from the polynomial.
Q: What is the remainder in polynomial division?
A: The remainder in polynomial division is the error term. It is the difference between the polynomial and the quotient.
Q: Can I have a step-by-step guide on how to perform polynomial division?
A: Yes, here is a step-by-step guide on how to perform polynomial division:
- Divide the leading term of the polynomial by the leading term of the divisor.
- Multiply the divisor by the result and subtract the result from the polynomial.
- Repeat steps 1 and 2 until the degree of the remainder is less than the degree of the divisor.
- The quotient is the result of the division, and the remainder is the error term.
Q: What is the significance of the value of k in the equation y = 3x + k?
A: The value of k represents the vertical shift of the oblique asymptote. It is the difference between the oblique asymptote and the horizontal line y = 3x.
Q: Can I have an example of how to find the value of k in the equation y = 3x + k?
A: Yes, here is an example:
Suppose we have the function f(x) = 9x² + 36x + 413x + 5 and we want to find the value of k in the equation y = 3x + k, given that the graph of f(x) has an oblique asymptote at y = 3x + k.
We perform polynomial division to find the oblique asymptote:
f(x) = 9x² + 36x + 413x + 5
x | 9x² + 36x + 413x + 5 | 9x² + 36x
377x + 5
We then equate the two equations to solve for k:
y = 3x + k y = 377 + 5/x
We solve for k by subtracting 377 from both sides of the equation:
k = 5/x - 377
We simplify the expression by evaluating the limit as x approaches 0:
k = 377
Therefore, the value of k is 377.
Conclusion
In this article, we answered some frequently asked questions related to oblique asymptotes and the value of k. We provided a step-by-step guide on how to perform polynomial division and an example of how to find the value of k in the equation y = 3x + k. We hope that this article has been helpful in understanding the concept of oblique asymptotes and the value of k.
Final Answer
The final answer is: