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Chapter 3: Problem 47

Identify the asymptotes. $$ h(x)=\frac{-3 x^{2}+4 x-5}{x+6} $$

Short Answer

Expert verified
Vertical asymptote: \( x = -6 \), Oblique asymptote: \( y = -3x + 22 \). No horizontal asymptote.

Step by step solution

01

- Identify Vertical Asymptotes

Vertical asymptotes occur where the denominator of the function equals zero. Set the denominator equal to zero and solve for x: \[ x + 6 = 0 \] \[ x = -6 \] So, there is a vertical asymptote at \( x = -6 \).
02

- Identify Horizontal Asymptotes

For rational functions, the horizontal asymptote is determined by comparing the degrees of the numerator and denominator. Here, the degree of the numerator is 2 and the degree of the denominator is 1. Since the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.
03

- Identify Oblique Asymptotes

When the degree of the numerator is exactly one more than the degree of the denominator, the function has an oblique asymptote. Perform polynomial long division of the numerator by the denominator to find the oblique asymptote: \[ \frac{-3x^2 + 4x - 5}{x + 6} \] - The quotient is the equation of the oblique asymptote. Perform the division: Quotient: \( -3x + 22 \) (The remainder can be ignored for the asymptote.) Thus, the oblique asymptote is \( y = -3x + 22 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vertical Asymptotes
Vertical asymptotes occur where the function's denominator equals zero because division by zero is undefined. To find these values, we set the denominator equal to zero and solve for the variable. For the function:

We set the denominator equal to zero:



When the denominator is zero, the function heads towards infinity or negative infinity, indicating a vertical asymptote at this value.


For example, if our function is:
Horizontal Asymptotes
Horizontal asymptotes indicate the behavior of a function as it heads to infinity or negative infinity. To find horizontal asymptotes, we compare the degrees of the numerator and denominator:

  • If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0.

  • If the degrees are equal, the horizontal asymptote is y = (leading coefficient of numerator)/(leading coefficient of denominator).


In our example, the numerator's degree is 2 while the denominator's degree is 1. Since the numerator's degree is higher, there is no horizontal asymptote.
Oblique Asymptotes
Oblique asymptotes, or slant asymptotes, occur when the degree of the numerator is exactly one more than the degree of the denominator. To find it, we use polynomial long division:

  • Divide the numerator by the denominator.
  • The quotient (ignoring the remainder) is the equation of the oblique asymptote.



For the function, we perform division:


Here, the quotient is:

We ignore the remainder for finding the equation of the oblique asymptote:
Thus, the oblique asymptote is y = -3x + 22.
Polynomial Long Division
Polynomial long division is a method to divide polynomials similar to arithmetic long division. It is helpful for finding oblique asymptotes:




  • Divide the first term of the numerator by the first term of the denominator.
  • Multiply the result by the polynomial divisor.
  • Subtract this result from the original polynomial numerator.
  • Repeat with the new polynomial.


For our function:


Polynomial long division shows us that the quotient, ignoring the remainder, gives us oblique asymptote equations like y = -3x + 22 in this case.

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Most popular questions from this chapter

Write an informal definition of a horizontal asymptote of a rational function.

Given \(y=f(x)\) a. Divide the numerator by the denominator to write \(f(x)\) in the form \(f(x)=\) quotient \(+\frac{\text { remainder }}{\text { divisor }}\). b. Use transformations of \(y=\frac{1}{x}\) to graph the function. $$ f(x)=\frac{2 x+7}{x+3} $$

Determine if the statement is true or false. If -3 is a lower bound for the real zeros of \(f(x)\), then -4 is also a lower bound.

Use a variation model to solve for the unknown value. The cost to carpet a rectangular room varies jointly as the length of the room and the width of the room. A 10-yd by 15 -yd room costs \(\$ 3870\) to carpet. What is the cost to carpet a room that is 18 yd by 24 yd?

Graph the function. $$ g(x)=\frac{x^{3}+3 x^{2}-x-3}{x^{2}-2 x} $$
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