Question

Graph the function.

\[f(x)=8\cdot\left(\dfrac{5}{4}\right)^x\]

Answer 1

f(x) = 8·(5/4)^x is an exponential growth function (base 5/4 > 1).

Key features
- Domain: all real numbers
- Range: y > 0
- Horizontal asymptote: y = 0
- y-intercept: f(0) = 8 → point (0, 8)
- No x-intercepts
- Growth factor: each time x increases by 1, y is multiplied by 5/4 = 1.25 (so it grows 25% per unit)

Useful points to plot
- x = -2: f(-2) = 8·(5/4)^{-2} = 8·(4/5)^2 = 8·(16/25) = 128/25 = 5.12
- x = -1: f(-1) = 8·(4/5) = 32/5 = 6.4
- x = 0: f(0) = 8
- x = 1: f(1) = 8·5/4 = 10
- x = 2: f(2) = 8·(5/4)^2 = 8·25/16 = 200/16 = 12.5

How to draw it
1. Draw axes and mark the points above.
2. Sketch a smooth curve through the points that approaches y = 0 as x → −∞ and rises to the right as x → ∞.
3. Label the horizontal asymptote y = 0 and the y-intercept (0, 8).

If you want, I can provide a plotted image (PNG) of this graph.