Problem

Graph \(f(x)~=~16\cdot~0.5^{x}\) on the grid. State the initial value, retained factor and decay percent, verified points and ratio, horizontal asymptote, domain, and range.

Blank coordinate grid tailored to plotting f(x)=16·0.5ˣ, without a completed curve or derived features.

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What this problem is really about

Exponential decay multiplies by the same retained fraction rather than subtracting a fixed amount. We’ll identify the initial output, translate the retained factor into a percent decrease, and plot several points whose consecutive ratios check the rule. Extending that pattern in both directions explains the smooth decreasing curve, its never-reached horizontal asymptote, and its positive-output range.

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initial = f(0)=16; retained factor = 0.5; decay percent = 50% per +1 input; points = (0,16),(1,8),(2,4); ratio check = 8/16=4/8=0.5; asymptote = y=0; domain = (-∞,∞); range = (0,∞)

initial 0.5; factor 16; points (0,0.5),(1,8),(2,128)

retained factor 0.5; decay percent 0.5%; points 16,15.92,15.8404

asymptote x=0; domain x>0

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Graph \(f(x)~=~16\cdot~0.5^{x}\) on the grid. State the initial value, retained factor and decay percent, verified points and ratio, horizontal asymptote, domain, and range.

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