In a question, I was given a graph which has a cubic curve, passing through x=-2 and touches 0.

The question asks me to explain how the number of roots of the equation f(x)=k(x+2) depends on k.

Sorry,I haven't had any maths program so I'm unable to draw the graph here. So I'm not expecting any full solutions but some advice on solving such problem.

My attempt: Since the curve touches zero, obviously it has two real and equal roots which are zero.

Hence, I assume that f(x)=k=x^2

if K>0, the cup-shaped curve would be heading upwards. (If the cup-shaped curve has centre of (0,0), it intersects with the curve f(x)=k(x+2), having two equal roots... )

if K=0, the means f(x)=0, which means the number of roots are two equal and real roots.

if K<0, the cup-shaped curve would be heading downwards, so to calculate the number of roots, I should take into account all possible intersections between this cup-shaped curve with the curve f(x)=k(x+2) right?

Thanks for your advice!! Sorry if my writing is confused for you

Looking forward to hearing from you soon.