For a fixed positive constant

MATHEMATICS · Multi-part · AP · AP Precalculus · Polynomial and Rational Functions · Polynomial and Rational Functions

Problem

k

, the straight line

y=kx

intersects the curve

y=\dfrac{1}{x}

in the first quadrant. (i) Solve simultaneously to obtain the coordinates of the point of intersection in terms of

k

. (3) (ii) Show that the product of the

x

– and

y

–coordinates of this intersection point is always

1

, irrespective of the value of

k

. (2) (iii) Taking

k=0.5,\,1,

and

2

in turn, calculate the corresponding points of intersection. Plot these three points together with the line segments of

y=kx

that meet the curve. (3) (iv) Discuss, with reference to your diagram, how increasing

k

shifts the intersection point along the curve, and use this observation to refine the accuracy of your sketch of the reciprocal function in the first quadrant. (2)

Related theory

Straight Lines
Rational Functions
Functions

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