Graphing Techniques

12 Aug 3pm – 5pm

HCI College Auditorium

1. DHS/I/10(b)

The diagram shows the graph of f ,y x which has a turning point at 2,2 .A

Sketch, on separate diagrams, the graphs of

(i) f 'y x , [3]

(ii) 1

fy

x , [3]

showing clearly all relevant asymptotes, intercepts and turning point(s), where possible.

2. HCI/I/12

The curves 1C and 2C have equations 2 2 22 1x a y and

2 4

1

xy

x

, where 1 2, a

respectively. Describe the geometrical shape of C1. [1]

(a) State a sequence of transformations which transforms the graph of 2 2 1x y to the

graph of 1C . [3]

(b) (i) Sketch 1C and 2C on the same diagram, stating the coordinates of any points

of intersection with the axes and the equations of any asymptotes. [6]

(ii) Show algebraically that the x-coordinates of the points of intersection of 1C

and 2C satisfy the equation 22 2 22 2 21 2 1 4x x a x a x . [2]

(iii) Deduce the number of real roots of the equation in part (ii). [1]

y

O x

2,2A

x = 2

3. NJC/I/11

The curve C has equation 2 1

f ,ax bx

xx c

where a, b and c are real constants.

Given that the line 12 xy is an asymptote of C, find the value of a and show that

2 1b c . [3]

(i) For 1c , using algebraic method, prove that the curve C cannot lie between 2

values, which are to be determined. [3]

(ii) Sketch the graph of 22 1

f ,1

x xx

x

showing clearly its asymptotes, the

coordinates of the axial intercepts, and turning point(s) (if any). [3]

Hence, state the range of x for which f x is concave downwards. [1]

(iii) Given that the line 3y kx k , where k is a real constant, passes through the

intersection of the asymptotes of C, deduce the range of k where

22 1 3 1x x kx k x

has 2 real solutions. [1]

4. Adapted from TJC/2010/II/1

A curve C is represented by the parametric equations

2 2x t t , 3 9y t t for 2t .

Sketch the curve C and show that C intersects the x-axis at the point (15, 0). [3]