https://file.aiquickdraw.com/mathbot/file/173013014468837lxzhdc.jpg

To analyze the given equation:

\[
y = 2 \cdot 4^{3 - x} - 2
\]

we need to identify the asymptote and calculate two points \((x, y)\) on the graph. Let’s go through this step-by-step.

### 1. Determine the Asymptote
In exponential functions of the form \( y = a \cdot b^{k \cdot x} + c \), the asymptote is generally \( y = c \). Here, \( c = -2 \), which means the asymptote equation is:
\[
y = -2
\]

### 2. Calculate Two Points \((x, y)\)
To graph the function, we should find two specific \((x, y)\) pairs by substituting values for \( x \).

#### When \( x = 0 \):
\[
y = 2 \cdot 4^{3 - 0} - 2 = 2 \cdot 4^3 - 2 = 2 \cdot 64 - 2 = 128 - 2 = 126
\]

So, one point is:
\[
(0, 126)
\]

#### When \( x = 1 \):
\[
y = 2 \cdot 4^{3 - 1} - 2 = 2 \cdot 4^2 - 2 = 2 \cdot 16 - 2 = 32 - 2 = 30
\]

So, another point is:
\[
(1, 30)
\]

### Summary
- **Asymptote Equation**: \( y = -2 \)
- **Points**: \( (0, 126) \) and \( (1, 30) \)

Would you like further details on how to graph this or on any other specific part of the solution?

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Here are some related questions to extend your understanding:

1. How would the graph change if the asymptote shifted to a different value?
2. What effect does changing the base (e.g., from 4 to another number) have on the graph?
3. How does changing the coefficient from 2 to another number affect the steepness of the graph?
4. How would you find additional points on the graph for increased accuracy?
5. What transformations are applied to the parent function \( y = 4^x \) to get this function?

**Tip**: In exponential functions, the asymptote gives you the horizontal line the function approaches as \( x \to \infty \) or \( x \to -\infty \).

Graph the equation shown below by first marking the asymptote and then marking two (x, y) coordinates.

\( y = 2 \cdot 4^{3 - x} - 2 \)

Asymptote Equation: _____

Learn how to graph the exponential function y = 2 * 4^(3 - x) - 2 by identifying the asymptote equation and finding specific (x, y) points. This solution provides a step-by-step guide to marking the asymptote and calculating points for accurate graphing. Suitable for high school students in Grades 10-12.

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