Question

Trace the curve y

2 = (x − 1)(x − 2)(x − 3).

Answer 1

Solution

$y=(x-1)(x-2)(x-3)-2$

• As it is easily discoverable,

$(1,-2) ,(2,-2),(3,-2),(0,-8)$

• For the next calculations it is easy if this is expanded as follows

$y=x^3-6x^2+11x-8$

• To estimate where would the graph (y) tend as x tends towards +/- infinity, set x to either +/-infinity.

$y\to+\infty$ as $x\to+\infty$

$y\to -\infty$ as $x \to -\infty$

• To evaluate the maxima and minima, take the derivative & find the roots

$y'=3x^2-12x+11=0$

• This gives

$x=2.57$ or $x=1.42$

• Then the corresponding coordinates are

$(2.57,-2.38)$ and $(1.42,-1.62)$

• With all these coordinates, the graph can be traced as follows