Math Problem Statement
show that a = 6 and b = -13
Solution
To solve the problem, we will need to show that and for the function given the conditions:
- The graph of has a horizontal asymptote at .
- has a removable discontinuity at .
Let's break down the steps:
1. Horizontal Asymptote at :
The horizontal asymptote for a rational function depends on the degrees of the numerator and denominator :
- If the degree of is equal to the degree of , the horizontal asymptote is given by .
In this case, both the numerator and the denominator are quadratic polynomials:
The leading coefficient of the numerator is , and the leading coefficient of the denominator is 1. Since the horizontal asymptote is given as , we have:
Thus, .
2. Removable Discontinuity at :
A removable discontinuity at implies that makes both the numerator and the denominator equal to zero. This is because for a discontinuity to be removable, the function must have the form at .
Substitute into the denominator:
So, the denominator is zero at .
Next, substitute into the numerator:
Since the numerator must also equal zero at :
We already know , so substitute that into the equation:
Simplify and solve for :
Given that the answer needed to show may imply an issue. Let's correct that by checking the correct steps to solve this algebraically again:
Corrected Steps:
- Set
- Substitute back into the equation and solve for .
Since there seems to be a mismatch, you might want to confirm that the problem intended this approach. The rationale used was correct for the conditions stated, but a mismatch in the horizontal asymptote may cause reconsideration or recalculation.
Do you want further elaboration on any of these steps or need a more comprehensive check? Here are five possible follow-up questions:
- How do I derive horizontal asymptotes from any rational function?
- What defines a removable discontinuity, and how can it be removed?
- How would this change if the discontinuity wasn't removable?
- Can we generalize the approach for any quadratic in the numerator?
- What if the horizontal asymptote or discontinuity was at a different -value?
Tip: When solving algebraic problems, always ensure that each variable aligns with all given conditions. Double-check each step to prevent computational errors.
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Math Problem Analysis
Mathematical Concepts
Rational functions
Horizontal asymptotes
Removable discontinuities
Quadratic equations
Formulas
Formula for horizontal asymptote of rational functions
Theorems
-
Suitable Grade Level
Grades 11-12
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