Question

the curve c has equation y = (x-2)(x-a)/(x-1)(x-3) where a is a constant not equal to 1,2 or 3
(i) write down the equation of the asymptotes of c
(ii) show that c meets the asymptote parallel to the x-axis at the point where x = (2a-3)/(a-2)
(iii) show that the x-coordinates of any stationary points on C satisfy
(a-2)x^2 + (6 - 4a)x + (5a-6) = 0
and hence find the set of values of a for which c has stationary points
(iv) sketch the graph of c for
(a) a > 3
(b) 2 < a < 3

Answer 1

Answer:

The graph of y f(x) is transformed into the graph of

y f(-

x) by a reflection in the y-axis.

Hence the new curve has equation y 2 tan (-

x).

However, since tan(-

x) -

tan x, the equation of the new

curve can be written as y 2 -

tan x.

(b) Recall that a translation of -

transforms the graph of

y f(x) into the graph of y f(x -

a) b.

After translation through the vector -

, the curve

y 2 tan x has equation y 2 tan

x -

3

5

or y 7 tan

x -

3

.

(c) The graph of y f(x) is transformed into the graph of

y f

x

c

by a stretch of scale factor c in the x-direction.

Hence y 2 tan x is transformed into y 2 tan

0

x

.5

or y 2 tan 2x.