Rational functions and graphs: แบบฝึกหัด

Consider the rational function \(y = \frac{x^2 + 2x + 5}{x^2 + 2x + k}\). For what range of values of \(k\) does the graph of the function have exactly two vertical asymptotes and no intersections with the x-axis?

The graph of \(y = \frac{x^2 + ax + b}{x - 1}\) has a turning point at \((0, -2)\). Find the values of \(a\) and \(b\) and identify the equation of the oblique asymptote.

A curve has the equation \(y = \frac{ax^2 + bx + c}{x + d}\). The asymptotes of the curve are \(x = 2\) and \(y = 2x + 1\). The curve passes through the origin. Find the values of \(a, b, c, d\) and determine the value of \(a+b+c+d\).

The equation of a curve is \(y = \frac{2x^2 + kx + 2}{x-1}\). It is given that the curve has no stationary points. Find the set of possible values for the constant \(k\).

The curve \(C\) has equation \(y = \frac{2x^2 + 5x + 11}{x^2 + x - 2}\). Find the set of values of \(y\) for which there are no real values of \(x\) on the curve.

Sketch the graph of \(y = \frac{x^2 - 2x - 3}{x - 2}\), clearly indicating the coordinates of any turning points and the equations of all asymptotes.

The graph of the rational function \(y = \frac{2x^2 + x - 1}{x - 3}\) has an oblique asymptote. Find the equation of this asymptote in the form \(y = mx + c\).

Sketch the graph of \(y = \frac{x^2 - 4}{x^2 - 1}\), clearly indicating the equations of all asymptotes and the coordinates of any intercepts with the axes.