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Find the horizontal and vertical asymptotes for the graph of y equals Start Fraction 2 x plus 4 over x minus 6 End Fraction.
(1 point)
Responses

x = 6, y = 2
x = 6, y = 2

x equals negative 2, x = 6
Image with alt text: x equals negative 2 , x = 6

y equals negative Start Fraction 2 over 3 End Fraction, y = 2
Image with alt text: y equals negative Start Fraction 2 over 3 End Fraction , y = 2

x equals negative 2, y equals negative Start Fraction 2 over 3 End Fraction
Image with alt text: x equals negative 2 , Image with alt text: y equals negative Start Fraction 2 over 3 End Fraction
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1 answer

To find the horizontal and vertical asymptotes for the function \( y = \frac{2x + 4}{x - 6} \), we will analyze the function step by step.

  1. Vertical Asymptotes: Vertical asymptotes occur where the denominator equals zero, as long as the numerator is not also zero at that same point. \[ x - 6 = 0 \quad \Rightarrow \quad x = 6 \] Thus, there is a vertical asymptote at \( x = 6 \).

  2. Horizontal Asymptotes: Horizontal asymptotes are found by analyzing the behavior of the function as \( x \) approaches infinity or negative infinity. For rational functions, the horizontal asymptote can be determined by comparing the degrees of the numerator and the denominator:

    • The degree of the numerator (2x + 4) is 1.
    • The degree of the denominator (x - 6) is also 1.

    Since the degrees of the numerator and denominator are equal, the horizontal asymptote is given by the ratio of the leading coefficients: \[ \text{Horizontal Asymptote at } y = \frac{\text{leading coefficient of the numerator}}{\text{leading coefficient of the denominator}} = \frac{2}{1} = 2 \]

In conclusion, the asymptotes are:

  • Vertical asymptote: \( x = 6 \)
  • Horizontal asymptote: \( y = 2 \)

So, the correct response is: x = 6, y = 2