Questions: Graph all asymptotes of the rational function. f(x) = (-6x^2 - 7x + 1) / (2x + 3)

Transcript text: Graph all asymptotes of the rational function. \[ f(x)=\frac{-6 x^{2}-7 x+1}{2 x+3} \]

Solution

Solution Steps

Step 1: Identify the Vertical Asymptote

The vertical asymptote occurs where the denominator is zero. Set the denominator equal to zero and solve for $x$: \[ 2x + 3 = 0 \] \[ x = -\frac{3}{2} \]

Step 2: Identify the Horizontal Asymptote

For rational functions where the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote. Instead, there is an oblique asymptote.

Step 3: Identify the Oblique Asymptote

Since the degree of the numerator is one more than the degree of the denominator, perform polynomial long division to find the oblique asymptote: \[ \frac{-6x^2 - 7x + 1}{2x + 3} \] The result of the division is: \[ y = -3x + \frac{1}{2} \] The oblique asymptote is $y = -3x + \frac{1}{2}$.

Final Answer

The vertical asymptote is at $x = -\frac{3}{2}$ and the oblique asymptote is $y = -3x + \frac{1}{2}$.

{"axisType": 3, "coordSystem": {"xmin": -5, "xmax": 5, "ymin": -10, "ymax": 10}, "commands": ["y = -3x + (1/2)", "x = -3/2"], "latex_expressions": ["$y = -3x + \\frac{1}{2}$", "$x = -\\frac{3}{2}$"]}

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