Exponential: \( y=4 \cdot(2)^{x-2}+1 \)

The function   y = 4 · 2^(x – 2) + 1 is an exponential function obtained by transforming the basic exponential f(x) = 2^x. Here’s a breakdown of its features and transformations: 1. Base Function:   • The original exponential function is f(x) = 2^x. 2. Horizontal Shift:   • The exponent (x – 2) means the graph is shifted to the right by 2 units. 3. Vertical Scaling:   • Multiplying 2^(x – 2) by 4 stretches the graph vertically by a factor of 4. 4. Vertical Shift:   • Finally, adding 1 moves the entire graph upward by 1 unit. 5. Domain and Range:   • Domain: All real numbers (–∞, ∞) since exponential functions are defined for every x.   • Range: The basic function 2^x has a range of (0, ∞). After the vertical transformations, the range becomes (1, ∞) because the horizontal asymptote shifts to y = 1. 6. Key Points:   • Horizontal Asymptote: y = 1 (due to the +1 shift)   • x = 2 gives y = 4 · 2^(0) + 1 = 4 · 1 + 1 = 5. So, the point (2, 5) lies on the graph.   • For x = 0: y = 4 · 2^(0 – 2) + 1 = 4 · 2^(–2) + 1 = 4 · (1/4) + 1 = 1 + 1 = 2. Thus, the y-intercept is (0, 2). Overall, the graph of y = 4 · 2^(x–2) + 1 is an exponentially increasing curve that has been shifted right 2 units and up 1 unit, and scaled vertically by a factor of 4.