Here's the graph of the function $f(x) = \frac{3}{2} x \left( \frac{4}{3} \right)^x - 1$.

Summary:

1. $f(1) = 1$
2. $y$-intercept: $-1$
3. Horizontal asymptote: $y = -1$

Would you like further details on any of these steps?

Here are some related questions to consider:

1. What would happen to the function if we changed the base of the exponent?
2. How does the coefficient $\frac{3}{2}$ affect the shape of the graph?
3. What if we changed the constant $-1$ to a different number?
4. How does this function differ from a basic exponential function?
5. What would the graph look like over a larger interval of $x$-values?

Tip: Horizontal asymptotes in exponential functions can reveal the long-term behavior of the function as $x \to \infty$ or $x \to -\infty$.