1. **State the problem:** We need to find the rule of the exponential function passing through the points $(0,1)$ and $(1,4)$. 2. **Recall the general form of an exponential function:** $$y = a^x$$ where $a$ is the base of the exponential function. 3. **Use the point $(0,1)$ to find the constant:** Substitute $x=0$ and $y=1$ into the equation: $$1 = a^0$$ Since any number to the zero power is 1, this confirms the form is correct but does not determine $a$. 4. **Use the point $(1,4)$ to find $a$:** Substitute $x=1$ and $y=4$: $$4 = a^1$$ So, $$a = 4$$ 5. **Write the function rule:** $$y = 4^x$$ 6. **Summary:** The exponential function passing through $(0,1)$ and $(1,4)$ is $$\boxed{y = 4^x}$$