Rational functions and graphs: แบบฝึกหัด

Consider the rational function $f(x) = \frac{x^2 - x - 2}{x-3}$. Which of the following statements about its asymptotes is correct?

Determine the set of values for the constant \( c \) such that the graph of \( y = \frac{x^2-2x+c}{x-1} \) has no stationary points.

Consider the graph of \( y = \frac{2x+1}{x-1} \). Which of the following describes the horizontal asymptote of the transformed graph \( y = \left| \frac{2x+1}{x-1} \right| \)?

The graph of \(y = \frac{4x+1}{x+1}\) intersects the line \(y = x+1\). Find the coordinates of all intersection points.

Consider the rational function \(f(x) = \frac{x^2 - 4x + 3}{x - 2}\). Find the equations of its vertical and oblique asymptotes.

State the equations of the vertical and horizontal asymptotes for the rational function \(f(x) = \frac{3x+1}{x-2}\).

Determine the equation of the oblique asymptote for the rational function \(f(x) = \frac{x^2+x-1}{x-1}\).

The function \(g(x) = \frac{ax+b}{cx+d}\) passes through the point \((0,1))\), has a vertical asymptote at \(x=2\) and a horizontal asymptote at \(y=3\). Determine the values of \(a, b, c, d\).

(a) Find the equations of the vertical and horizontal asymptotes of the rational function \(y = \frac{1}{x^2-4}\).

(b) Find the coordinates of the points where the graph of \(y = \frac{1}{x^2-4}\) intersects the coordinate axes.

(c) Sketch the graph of \(y = \frac{1}{x^2-4}\), showing clearly the asymptotes and intercepts.

Let \(f(x) = \frac{x}{x^2-1}\).

(a) Find the equations of the asymptotes and the coordinates of the axial intercepts for \(y = f(x)\). Sketch the graph of \(y = f(x)\).

(b) Using your graph from part (a), sketch the graph of \(y^2 = f(x)\). Show clearly any asymptotes and restricted domains for \(x\).