Question

Use a computer to investigate the family of surfaces \(z = (a{x^2} + b{y^2}){e^{ - {x^2} - {y^2}}}\)how does the shape of the group depend on the numbers a and b?

Short answer

The shape of the given graph is with two “ocne” shape. As a and b get larger together, the graph reaches higher with two“cone” shape. As a or b increases alone, this increases the size of one of the“cones”

Step-by-step solution

Analyze the family of surfaces

The graph of \(z = (a{x^2} + b{y^2}){e^{ - {x^2} - {y^2}}}\)

Let us choose a few values for a and b to see what happen

Clearly, as a and b get larger together, the graph exhibits asymptotic behaviour near the origin, and reaches higher with two“cone” shape.

Look at the end behavior of the graph

As a or b increases alone, this increases the size of one of the “cones”

Therefore, the shape of the given graph is with two “cone”shape. As a and b get larger together, the graph reaches higher with two“cone”shape .As a or b increases alone, this increases the size of one of the “cones”.