Question
Find the horizontal asymptote of the graph y=−3x2+6x+6−7x2+x+6 (1 point) Responses y=1 y is equal to 1 y=0 y is equal to 0 no horizontal asymptote no horizontal asymptote y=37
Answers
Answered by
GPT-4o mini
To find the horizontal asymptote of the given function \( y = \frac{-3x^2 + 6x + 6}{-7x^2 + x + 6} \), we look at the degrees of the numerator and denominator.
1. **Identifying the Degrees**:
- The degree of the numerator \( -3x^2 + 6x + 6 \) is 2.
- The degree of the denominator \( -7x^2 + x + 6 \) is also 2.
Since the degrees of the numerator and denominator are the same, the horizontal asymptote can be found by taking the ratio of the leading coefficients.
2. **Leading Coefficients**:
- The leading coefficient of the numerator is \(-3\).
- The leading coefficient of the denominator is \(-7\).
3. **Calculating the Horizontal Asymptote**:
\[
\text{Horizontal Asymptote} = \frac{\text{Leading Coefficient of the Numerator}}{\text{Leading Coefficient of the Denominator}} = \frac{-3}{-7} = \frac{3}{7}
\]
Therefore, the horizontal asymptote of the graph is:
\[
y = \frac{3}{7}
\]
Since none of the provided options include \( y = \frac{3}{7} \), it appears there may be an error in the available answer choices. If the answers must be chosen from those provided, you might select "no horizontal asymptote" as the correct conclusion based on the given choices. However, the correct horizontal asymptote is \( y = \frac{3}{7} \), not listed.
1. **Identifying the Degrees**:
- The degree of the numerator \( -3x^2 + 6x + 6 \) is 2.
- The degree of the denominator \( -7x^2 + x + 6 \) is also 2.
Since the degrees of the numerator and denominator are the same, the horizontal asymptote can be found by taking the ratio of the leading coefficients.
2. **Leading Coefficients**:
- The leading coefficient of the numerator is \(-3\).
- The leading coefficient of the denominator is \(-7\).
3. **Calculating the Horizontal Asymptote**:
\[
\text{Horizontal Asymptote} = \frac{\text{Leading Coefficient of the Numerator}}{\text{Leading Coefficient of the Denominator}} = \frac{-3}{-7} = \frac{3}{7}
\]
Therefore, the horizontal asymptote of the graph is:
\[
y = \frac{3}{7}
\]
Since none of the provided options include \( y = \frac{3}{7} \), it appears there may be an error in the available answer choices. If the answers must be chosen from those provided, you might select "no horizontal asymptote" as the correct conclusion based on the given choices. However, the correct horizontal asymptote is \( y = \frac{3}{7} \), not listed.