https://file.aiquickdraw.com/mathbot/file/17304010690508xc86ari.jpg

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.

The graph of f(x) = 4^x is shown. Draw the graph g(x) = 3(4)^(x-2) - 3.

Learn how to graph the transformed exponential function g(x) = 3(4)^(x-2) - 3, using shifts, stretches, and translations from the base function f(x) = 4^x. This solution covers key concepts in exponential functions and transformations, suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation