ADVANCED MATHEMATICS FORM 5 – FUNCTION AND RELATION

VERTICAL ASSYMPTOTES(V.A)

Is the one which

HORIZONTAL ASSYMPTOTES
Is the one which

RATIONAL FUNCTION SKETCH

Horizontal assymptote (H.A)

Sketch the function

Horizontal assymptotes.

Intercepts

Sketch.

Intercepts

(y-1) x²-2(y-1)x-3(y-1)=-4x+8

(y-1)x²-2(y-1)x+4x-3(y-1)-8=0

(y-1)x²-2yx+2x+4x-3y+3-8=0

(y-1)x² +(-2y+6)x-(3y+5)=0

For real value of x

b²-4ac ≥ 0

(-2y+6)² +4(y-1) (3y+5)≥0

(4y²-24y+36)+ (12y²+8y-20)

16y² – 16y +16 ≥0

y²-y+1>0

y has no restriction: It can be any value

For the Historical A

Intercept

2xy -3y= 4x² + 8x-5

4x² +8x-2xy-5+3y

4x² (8-2y)x +(3y-5)=0

For the real value of x

b²-4ac ≥ 0

(8-2y)²-4.4(3y-5)≥0

64-32y+4y²-48y+80≥0

4y²-80y+144≥0

y²-20y +36≥0

(y-2) (y-18)≥0

Condition

(y-2)≥0          y-18≥0

(y-2)≤2,          y-18≤0

y  ≥   2,           y ≥18

y   ≤ 2,            y ≤18

Function can not lie between 2 and 18