1. **Problem Statement:** Simplify and analyze the exponential function $$f(x) = 4(2)^x - 2$$ using exponential rules, identify the parent function, describe transformations, complete tables for asymptotes and points, and describe domain and range. 2. **Parent Function:** The base exponential function here is $$y = 2^x$$. 3. **Simplification:** The function is already simplified as $$f(x) = 4 \cdot 2^x - 2$$. 4. **Transformation Description:** - The coefficient 4 vertically stretches the graph by a factor of 4. - The subtraction of 2 shifts the graph down by 2 units. 5. **Mapping:** - The parent point $$(x, y)$$ maps to $$(x, 4y - 2)$$. 6. **Horizontal Asymptote:** - The parent function has asymptote $$y=0$$. - After transformation, the asymptote shifts down by 2, so $$y = -2$$. 7. **Table of Values:** | x | $2^x$ | $f(x) = 4 \cdot 2^x - 2$ | |---|-------|--------------------------| | -1| $\frac{1}{2}$ | $4 \times \frac{1}{2} - 2 = 2 - 2 = 0$ | | 0 | 1 | $4 \times 1 - 2 = 4 - 2 = 2$ | | 1 | 2 | $4 \times 2 - 2 = 8 - 2 = 6$ | 8. **Domain and Range:** - Domain: All real numbers, $$D = \{x \in \mathbb{R}\}$$. - Range: Since the graph is vertically stretched and shifted down, the minimum value approaches the asymptote $$y = -2$$ but never reaches it, so $$R = \{y \in \mathbb{R} : y > -2\}$$. **Final Answer:** $$f(x) = 4 \cdot 2^x - 2$$ - Parent: $$y = 2^x$$ - Transformation: vertical stretch by 4, shift down 2 - Horizontal asymptote: $$y = -2$$ - Domain: $$\mathbb{R}$$ - Range: $$y > -2$$ Tables and mapping as above.