Equation for the image of a vector-valued function

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So I've already found the normal vector at (s,t)=(-1,0) to be (-1,0,-2) and found the tangent plane at (1,0,-1) to be (feel free to correct me if either of those end up being wrong and important to my question).

The question reads "Find an equation for the image of in the form .

What I initially did was to say that , , and , which then just gives the equation , but this seems suspiciously simplistic. Is there some other way to define F that doesn't require that z be defined explicitly in terms of x and y, or is what I've written really all that there is to the problem?