Review question

# Can we sketch $y=ax/(x^2+x+1)$ when $a$ is positive? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R7129

## Suggestion

If $y=\frac{ax}{x^2+x+1} \qquad (a>0)$ show that for real values of $x$ $-a\leq y \leq \frac{1}{3}a.$

Could we turn this into a quadratic in $x$?

What condition now do we need to ensure the roots for $x$ are real?

Sketch the graph of $y$ in the case $a=6$.

Are there any asymptotes? How does the curve behave as $\vert x\vert \rightarrow \infty$? What about roots, intercepts and stationary points?

How could we use the first part in our sketch?

By drawing on the same diagram a certain straight line show that the equation $x^3-1=6x$ has three real roots, two negative and one positive.

Can we factorise $x^3-1$?