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?

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 $x \to -\infty$.