Find thefollowing informationof the given function then use them to sketch the graph.

Question

b) 𝑦 = (𝑥−𝑥²) / (𝑥²−3𝑥+2)

i) Domain

ii) y-intercept & x-intercept

iii)Symmetry

iv) Asymptotes (HA/VA)

v) Intervals of increasing/decreasing

vi)Local extrema

vii)Intervals of concavity (CU/CD)

viii) Inflection point

Answer 1

y = x(1-x) / (x-1)(x-2)

= -x/(x-2) = -x(x-2)^(-1)

domain is all reals except 2, R - {2}, (-infinity, 2) u (2, infinity)

x-intercept(0,0)

y-intercept(0,0)

vertical asym. x=2

horizontal asym. -1

Per quotient rule

y' = {(x-2)(-1) - (-x)} / (x-2)^2 =

2/(x-2)^2

= 2(x-2)^(-2)

Per product rule:

(-x)(-1)(x-2)^(-2) + -(x-2)^(-1)

= (x-2)^(-2) { x - (x-2)} = 2/(x-2)^2

derivative is zero NOWHERE!!!

therefore the function is increasing EVERYWHERE in it's domain,

because y'>0

y'' = (-4)(x-2)^(-3) which is negative for x<2 and positive for x>2 so no flex points.

for x<2, it is concave up; for x>2 it is concave down