Parametric curve x=t^2+3, y=t(t^2+3): why does (dy/dx)^2≥9 rule out certain graph shapes?

[Harmony]

A curve is given by the parametric equations x=t^2+3, y=t(t^2+3)
a)Show that the curve is symmetric about the x-axis.
b)Show that there are no parts of this curve where x<3
c)Find dy/dx in terms of t, and show that (dy/dx)^2 greater or equals to 9.

Sketch the curve by using the above results.

I have no problem in a b and c, but I am a bit confuse ith graph sketching. What is the use of result c on the graph sketching? The answer given is the curve with the black line, but how can we be sure that the graph is not the one with the red line? (refer to attachment)

 

[HallsofIvy]

A curve is given by the parametric equations x=t^2+3, y=y(t^2+3)

Is "y= y(t^2+ 3)" a typo? What is y as a function of t? At first I thought you meant just y= t^2+ 3 but then it reduces to just y= x, for x>= 3.

a)Show that the curve is symmetric about the x-axis.
b)Show that there are no parts of this curve where x<3
c)Find dy/dx in terms of t, and show that (dy/dx)^2 greater or equals to 9.

Sketch the curve by using the above results.

I have no problem in a b and c, but I am a bit confuse ith graph sketching. What is the use of result c on the graph sketching? The answer given is the curve with the black line, but how can we be sure that the graph is not the one with the red line? (refer to attachment)

Can't help until you have corrected the statement of the problem.

 

[Harmony]

Sorry, the y should be t. Correction done.