To draw the graph of $g(x) = 3(4)^{x-2} - 3$, let's analyze the function and understand how it differs from the base function shown.

1. Identify the Parent Function: The base function here is $h(x) = 4^x$, an exponential growth function.

2. Transformations:

   • The factor of $3$ outside the base function $4^{x-2}$ represents a vertical stretch by a factor of $3$.
   • The term $x - 2$ in the exponent indicates a horizontal shift to the right by $2$ units.
   • The $-3$ at the end of the equation represents a vertical shift downward by $3$ units.

3. Constructing Key Points: To accurately sketch $g(x) = 3(4)^{x-2} - 3$, we can evaluate a few points around the center of the transformation, at $x = 2$.

   • When $x = 2$: $g(2) = 3(4)^{2-2} - 3 = 3(4^0) - 3 = 3(1) - 3 = 0$ So, $(2, 0)$ is a point on the graph.

   • When $x = 3$: $g(3) = 3(4)^{3-2} - 3 = 3(4^1) - 3 = 3(4) - 3 = 12 - 3 = 9$ So, $(3, 9)$ is a point on the graph.

   • When $x = 1$: $g(1) = 3(4)^{1-2} - 3 = 3(4^{-1}) - 3 = 3\left(\frac{1}{4}\right) - 3 = \frac{3}{4} - 3 = -2.25$ So, $(1, -2.25)$ is a point on the graph.

4. Sketching the Graph:

   • The graph of $g(x)$ will look similar to the base function $f(x) = 4^x$, but it will be shifted to the right by 2 units, stretched vertically by a factor of 3, and shifted downward by 3 units.

If you'd like, I can plot this function step-by-step or clarify any specific part of the transformation process.

Would you like details on the graphing process or have any questions?

Follow-up Questions:

1. What is the general formula for transforming exponential functions?
2. How does changing the base of an exponential function (e.g., from 4 to another number) affect its growth rate?
3. How would you graphically represent a horizontal shift in an exponential function?
4. What are some real-world examples of exponential growth functions?
5. How does a vertical shift affect the asymptote of an exponential function?

Tip:

When sketching transformed graphs, start by identifying the base function and then apply shifts, stretches, or reflections one by one to see the impact of each transformation.