Understanding the shape of a Marginal Cost Curve
You are missing the average cost curve in the same diagram. 
Basic algebra gives us the following.  

Let's find the minimum of the $AC = C/Q$. We have

$$\frac {\partial AC}{\partial Q} = \frac {MC\cdot Q - C}{Q^2}$$

For this to be equal to zero, we must have $MC \cdot Q = C \implies MC = AC$. So when $AC$ is at its minimum, it equals $MC$. But we also want the second order condition, and we want the second derivative to be positive, evaluated at the critical point.

$$\frac {\partial^2 AC}{\partial Q^2} &lt;0 \implies ...MC'\cdot Q^3-2MC\cdot Q^2 + 2QC &gt;0$$

At the critical point $MC = AC = C/Q$. Inserting this we get the condition 

$$MC'\cdot Q^3-2(C/Q)\cdot Q^2 + 2QC &gt;0 \implies MC'&gt;0$$

So at the point where $MC = AC$ we want MC to be rising. But this implies that it will cross the $AC$ curve from below. In turn this implies that for quantities lower than this point, marginal cost curve will be below the average cost curve, which means that if this left part of $MC$ curve was a supply curve the firm would have losses.  

Therefore the supply curve is only an upwards-sloping part of the marginal cost curve. You can now superimpose the $AC$ curve that is consistent with all these.