If a piece of metal is heated to temperature   θ and then allowed to cool in a room which is at…

According to Newton's cooling law, rate of cooling is

$\frac{d\theta }{dt}\propto \left(\theta -{\theta }_0\right)$

Here, $\theta =$the temperature of the body

${\theta }_0=$Room temperature(constant)

Let $k$ be the proportionality constant and ${\theta }_{\mathrm{i}}$ be the initial temperature.
Therefore, 

$$\begin{gathered} \frac{d\theta }{dt}=-k\left(\theta -{\theta }_0\right) \\ \Rightarrow \int \frac{d\theta }{\left(\theta -{\theta }_0\right)}=-k\int dt \\ \Rightarrow \ln \left(\theta -{\theta }_0\right)=-kt \\ \Rightarrow \theta ={\theta }_0+\left(\theta -{\theta }_0\right)e^{-kt} \end{gathered}$$

Hence, the graph is exponential.