Question
Find the horizontal asymptote of the graph y=4x2−8x+9/3x2+6x−2 (1 point) Responses y=0 y=0 y=43 y is equal to 4 thirds y=52 y is equal to 5 halves There is no horizontal asymptote
Answers
Answered by
GPT-4o mini
To find the horizontal asymptote of the rational function
\[
y = \frac{4x^2 - 8x + 9}{3x^2 + 6x - 2},
\]
we look at the leading terms of the numerator and the denominator as \(x\) approaches infinity.
The leading term of the numerator \(4x^2 - 8x + 9\) is \(4x^2\), and the leading term of the denominator \(3x^2 + 6x - 2\) is \(3x^2\).
As \(x \to \infty\), the behavior of the function can be approximated by:
\[
y \approx \frac{4x^2}{3x^2} = \frac{4}{3}.
\]
Thus, the horizontal asymptote is
\[
y = \frac{4}{3}.
\]
So, the correct response is \(y\) is equal to \(\frac{4}{3}\).
\[
y = \frac{4x^2 - 8x + 9}{3x^2 + 6x - 2},
\]
we look at the leading terms of the numerator and the denominator as \(x\) approaches infinity.
The leading term of the numerator \(4x^2 - 8x + 9\) is \(4x^2\), and the leading term of the denominator \(3x^2 + 6x - 2\) is \(3x^2\).
As \(x \to \infty\), the behavior of the function can be approximated by:
\[
y \approx \frac{4x^2}{3x^2} = \frac{4}{3}.
\]
Thus, the horizontal asymptote is
\[
y = \frac{4}{3}.
\]
So, the correct response is \(y\) is equal to \(\frac{4}{3}\).