Question

draw the graph of (f(x) = \frac{1}{4} \cdot (4)^{x-2} + 2)

Explanation:

Step1: Identify the horizontal asymptote

$$y = 2$$

Step2: Simplify the function expression

$$f(x) = 4^{-1} \cdot 4^{x-2} + 2 = 4^{x-3} + 2$$

Step3: Calculate key coordinate points

$$f(2) = 4^{-1} + 2 = 2.25$$

$$f(3) = 4^{0} + 2 = 3$$

$$f(4) = 4^{1} + 2 = 6$$

$$f(5) = 4^{2} + 2 = 18$$

Step4: Plot points and draw curve

Plot \((2, 2.25)\), \((3, 3)\), \((4, 6)\), \((5, 18)\) and sketch the curve approaching \(y = 2\).

Answer:

The graph of \(f(x) = \frac{1}{4} \cdot (4)^{x-2} + 2\) is an exponential curve with a horizontal asymptote at \(y = 2\), passing through the key points \((2, 2.25)\), \((3, 3)\), \((4, 6)\), and \((5, 18)\).