To define the exponential function $f(x) = a^x$ that passes through the given points, we can use the point $(x_0, y_0)$ on the graph to find the value of $a$. Let's break it down for both parts:

Part 1:

We are given the point $(3, 27)$.

The general form of the exponential function is $f(x) = a^x$. Since $f(3) = 27$, we can substitute these values into the equation:

$f(3) = a^3 = 27$

Now, solve for $a$:

$a^3 = 27 \quad \Rightarrow \quad a = \sqrt[3]{27} = 3$

Thus, the exponential function is:

$f(x) = 3^x$

Part 2:

We are given the point $(-2, 36)$.

Again, using the general form $f(x) = a^x$ and substituting $f(-2) = 36$, we get:

$f(-2) = a^{-2} = 36$

Now solve for $a$:

$a^{-2} = 36 \quad \Rightarrow \quad \frac{1}{a^2} = 36 \quad \Rightarrow \quad a^2 = \frac{1}{36} \quad \Rightarrow \quad a = \frac{1}{6}$

Thus, the exponential function is:

$f(x) = \left( \frac{1}{6} \right)^x$

Final Answers:

1. $f(x) = 3^x$ for the point $(3, 27)$.
2. $f(x) = \left( \frac{1}{6} \right)^x$ for the point $(-2, 36)$.

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Do you want details on how I solved for $a$ in part (b)? Here are 5 related questions:

1. What are the characteristics of exponential functions?
2. How do we determine if a function is exponential from a set of points?
3. What happens when the base of an exponential function is a fraction?
4. How can we generalize exponential growth and decay functions?
5. How do transformations affect the graph of exponential functions?

Tip: For exponential functions, the base $a$ determines the rate of growth or decay: $a > 1$ implies growth, while $0 < a < 1$ implies decay.