Question
Question
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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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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Answers
GPT-4o mini
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**
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**