Review question

# Can we sketch the curve $y=1/((x-a)^2-1)$? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R6855

## Suggestion

1. Sketch the curve $y=f(x)$ where $f(x)=\frac{1}{(x-a)^2-1} \qquad \qquad (x\neq a\pm 1),$ and $a$ is a constant.

2. The function $g(x)$ is defined by $g(x)=\frac{1}{((x-a)^2-1)((x-b)^2-1)} \qquad \qquad (x\neq a\pm 1,\,x\neq b \pm 1),$ where $a$ and $b$ are constants, and $b>a$. Sketch the curves $y=g(x)$ in the two cases $b>a+2$ and $b=a+2$, finding the values of $x$ at the stationary points.

These suggestions apply to both parts of this question.

How can we determine the vertical asymptotes of a curve? The horizontal ones?

Can the curve cross an asymptote? Does the function change sign here or not?

Are there any roots to the equation $f(x)=0$ or $g(x)=0$? In other words, what are the $x$ intercepts? What are the $y$-intercepts?

How do the curves behave as $x\rightarrow\infty$?

These curves are each symmetrical about a line $x=c$. What is this $c$ in each case?

How can we find the stationary points of the curve?

In the second part of the question, how do the two different conditions $b > a+2$ and $b = a+2$ affect the asymptotes?