Practice section 3 of 6

Sketch the graph of the following curves, indicating clearly axial intercept(s), asymptote(s) and stationary point(s), if any:

(i) $y=\dfrac1{3-x}$.

(ii) $y=\dfrac{2x+1}{x-2}$.

(iii) $y=\dfrac{x^2}{x-1}$.

(iv) $y=\dfrac{x^2+2x-3}{x+2}$.

Sketch the graphs of $y=x-1$ and $y=|x^2-3|$ on the same axes and determine the coordinates of the point of intersections of these two graphs.

The curve $C$ has an equation $y=\dfrac{ax^2+bx+a}{x^2-b}$ where $a$ and $b$ are constants.

(i) Given that two of the asymptotes of $C$ are $x=2$ and $y=3$, determine the values of $a$ and $b$.

(ii) Sketch the curve $C$, giving the coordinates of the point(s) where the curve crosses the axes, stationary points and the equations of all asymptotes.

The curve $C$ has equation $y=\dfrac{x^2-2x+2}{x+1}$.

(i) Determine the asymptotes of $C$.

(ii) Sketch $C$, showing clearly any axial intercepts, turning points and asymptotes.

(iii) Deduce the number of real roots of the equation $x^4-2x^3+2x^2-x-1=0$.

The curve $C$ has equation $y=\dfrac{x^2+3x-1}{x-2}$, $x\in\mathbb R$, $x\ne2$.

(i) Using an algebraic method, find the range of values that $y$ can take.

(ii) Sketch $C$, stating the equations of the asymptotes, axial intercepts and the coordinates of the turning points, if any.

(iii) State the range of values that $m$ can take such that the equation $\dfrac{x^2+3x-1}{x-2}=m(x-2)+7$ has 2 distinct solutions, justifying your answer.

Sketch each graph, clearly indicating asymptotes, axial intercepts and coordinates of any turning points.

(a) $y=2x^3-5x^2+x-2$.

(b) $y=4x^2e^x$.

(c) $y=\ln x/(2x)$.

(d) $y=(x^2+2x-8)/(x^2+2x-3)$.

(e) $y=|2x-x^2|$.

(f) $y=(ax+1)/(x-b)$, where (i)−1<a<0,0<b<1; or (ii)a<−1,b>1. What happens if ab+1=0?