If a piece of metal is heated to temperature θ and then allowed to...

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

$\dfrac{d \theta}{d t} ∝ \left(\theta - {\theta}_{0}\right)$

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

${\theta}_{0} =$Room temperature(constant)

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

$\dfrac{d \theta}{d t} = - k\left(\theta - {\theta}_{0}\right) ⇒ ∫ \dfrac{d \theta}{\left(\theta - {\theta}_{0}\right)} = - k ∫ d t ⇒ \ln \left(\theta - {\theta}_{0}\right) = - kt ⇒ \theta = {\theta}_{0} + \left(\theta - {\theta}_{0}\right){e}^{- kt}$

Hence, the graph is exponential.