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Aufgabe

Lösungsschritte

1. Understand the question

   which of the following is a rational function with horizontal asymptote $y = 5$ and vertical asymptote $x = -4$?
   a $y = \\frac{5x + 4}{x + 4}$
   b $y = \\frac{5x}{x - 4}$
   c $y = \\frac{x - 5}{x + 4}$
   d $y = \\frac{-4x}{x - 5}$

2. Explanation

   Step1: Find vertical asymptote condition

   A rational function $y=\frac{f(x)}{g(x)}$ has a vertical asymptote where $g(x)=0$. We need $x=-4$ to make the denominator zero, so denominator should be $x+4$. This eliminates options B and D.

   Step2: Find horizontal asymptote condition

   For a rational function where degree of numerator = degree of denominator, horizontal asymptote is $y=\frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}$. We need this to be $y=5$.

   • For option A: $\frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}=\frac{5}{1}=5$
   • For option C: $\frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}=\frac{1}{1}=1$

   This confirms option A matches the required horizontal asymptote.

3. Final answer

   A. $y = \frac{5x + 4}{x + 4}$

Antwort

Question Analysis

OCR Text

which of the following is a rational function with horizontal asymptote $y = 5$ and vertical asymptote $x = -4$?
a $y = \\frac{5x + 4}{x + 4}$
b $y = \\frac{5x}{x - 4}$
c $y = \\frac{x - 5}{x + 4}$
d $y = \\frac{-4x}{x - 5}$

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