WMA11 Jan 2022

10. WMA11/01 Edexcel IAL P1 January 2022, Q10 (Graphs and Transformation: Sketching Graphs)

The curve C has equation​​ 

y =1x2 – 9 

(a) Sketch the graph of C.​​ 

On your sketch​​ 

• show the coordinates​​ of any points of intersection with the coordinate axes​​ 

• state clearly the equations of any asymptotes​​ 

(4)

The curve D has equation y = kx2​​ where k is a constant.​​ 

Given that C meets D at 4 distinct points,​​ 

(b) find the range of possible values for k.​​ 

(5)

SOLUTION​​ 

a- ​​ On substituting y=0, the solutions or x-intercept of curve C is​​ 

y =1x2 – 9

0=1x2-9

1x2=9

9x2=1

x2=19

x=±13

To find the horizontal asymptote, substitute the value of x as 0. As x approches to infinity, y=-9.​​ 

Thus, the equation of horizontal asymptote is​​ 

y=-9

b-​​ 

From the graph​​ below, it is observable that the quadratic graph is sketched cup down the solution or point of intersection of quadratic curve with reciprocal graph is 4. This means that the value of k has to be less than zero because than and only we may have 4 distinct​​ points.​​ 

When​​ k<0, solving the equation of quadratic curve (y=kx2) and reciprocal graphs (y=1/x2​​ -9) simultaneously. ​​ 

kx2=1x2-9 

kx4=1-9x2

kx4+9x2-1=0

kx2+9x-1=0

b2-4ac>0

a=k

b=9

c=-1

81-4k-1>0

81+4k>0

4k>-81

k>-814

-814<k<0