1. **State the problem:** We are given the function $$f(x) = \frac{2}{3} \cdot 3^{3x + 12}$$ and want to understand its behavior and graph. 2. **Recall the exponential function form:** The general form is $$f(x) = a \cdot b^{cx + d}$$ where $a$ is a coefficient, $b$ is the base of the exponential, and $cx + d$ is the exponent. 3. **Identify components:** Here, $a = \frac{2}{3}$, $b = 3$, $c = 3$, and $d = 12$. 4. **Key properties:** - Since $b=3 > 1$, the function is an increasing exponential. - The coefficient $\frac{2}{3}$ scales the output vertically. 5. **Find the y-intercept:** Set $x=0$: $$f(0) = \frac{2}{3} \cdot 3^{3 \cdot 0 + 12} = \frac{2}{3} \cdot 3^{12}$$ 6. **Simplify $3^{12}$:** $$3^{12} = 531441$$ 7. **Calculate $f(0)$:** $$f(0) = \frac{2}{3} \times 531441 = 2 \times 177147 = 354294$$ 8. **Horizontal asymptote:** Since the base is greater than 1 and the exponent grows without bound, as $x \to -\infty$, $f(x) \to 0$. 9. **Summary:** The graph is an increasing exponential starting near zero for large negative $x$, passing through $(0, 354294)$, and increasing rapidly for positive $x$.