Why is the nature of graph of utility function different from indifference curve?

I am new to Economics, but I have this doubt. The indifference curve and utility function both have the same equation, so their graph must also be similar, which is true I guess. Then why is it that the nature of graphs are different? I mean why is it that if the indifference curve is convex then utility function is quasi concave? How did we arrive at this conclusion?

• You can refer to qr.ae/TWvVSr for answer.
– Amit
Oct 30 '19 at 2:27
• It may be worth your time asking a separate question concerning the relationship between quasi-concave utility functions and their convex contour curves (indifference curves). Although I found this answer written by @Giskard as well. Oct 30 '19 at 19:30
• @Brennan That friggin' Giskard person is everywhere, dude... Oct 30 '19 at 20:13

You seem to be confusing some things.

The indifference curve and utility function both have the same equation

The utility function has a formula, not an equation.

$$U(x_1,x_2)$$ is a utility function.

The points $$(x_1,x_2)$$ for which $$U(x_1,x_2) = 13$$ form an indifference curve. (13 was picked as a random number)

The graph of a utility function has one more dimension than the graph of an indifference curve: in addition to the space of goods, this graph would also include the utility value, so its points would be something like $$(x_1,x_2,u)$$. Chances are you have not seen a graph of any such function, as 3D drawings are complicated. This is why maps of 2D indifference curves are used to get a feeling for the shapes of utility functions.

Utility functions and indifference curves are the same object considered from different visual/conceptual angles. Below an illustration on $$U(x_1, x_2) = x_1^{0.5} x_2^{0.5}$$ (representing Cobb-Douglas preferences), done with excel: