How to Sketch the Curve y^2 = x^3?

[aurao2003]

Homework Statement

Hi everyone, I am not sure how to sketch this:
y^2 = X^3
I have a feeling it will form a parabola.

If I find the square root of both sides I get:
y = X^3/2
Differentiating gives me
dy/dx = 1.5X^1/2. Kind of stuck from here. Please help.

Homework Equations

The Attempt at a Solution

 

[Mentallic]

Think about the shape of the graph for y=x², then y=x³, y=x⁴ etc. only for values x>0. The shape is roughly the same, no? The only difference is that for x>1 the graph is steeper and for 0<x<1 the graph is lower for larger powers.
So y=x^3/2 will be roughly the same as y=x² for x>0
But remember that when you take the square root of both sides, you need to remember that $$\pm$$ so your final result of $$y^2=x^3$$ or $$y=\pm x^{3/2}$$ is the graph of y=x^3/2 for values of x>0 and then exactly the same thing but reflected in the x-axis for y=-x^3/2.

 

[Mark44]

Homework Statement

Hi everyone, I am not sure how to sketch this:
y^2 = X^3
I have a feeling it will form a parabola.

If I find the square root of both sides I get:
y = X^3/2

Actually, it's
$$y = \pm x^{3/2}$$
There are two branches.

Differentiating gives me
dy/dx = 1.5X^1/2. Kind of stuck from here. Please help.

The derivative also has two branches.
$$dy/dx = \pm 1.5x^{1/2}$$

Homework Equations

The Attempt at a Solution