Graph of f(x) = 5(0.6)^x — y-intercept, behavior as x increases, and horizontal asymptote

Step-by-step explanation for f(x)=5(0.6)^x: y-intercept at (0,5), the function decreases (exponential decay) as x increases, and the horizontal asymptote is y=0.

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Given f(x) = 5(0.6)x:

  1. Y-intercept: set x = 0. f(0) = 5(0.6)0 = 5(1) = 5. So the graph crosses the y-axis at (0, 5).
  2. Behavior as x increases: the base 0.6 is between 0 and 1, so (0.6)x gets smaller as x increases. Therefore the function decreases (exponential decay) and approaches 0 as x → ∞.
  3. Asymptote: because f(x) → 0 as x → ∞, the horizontal asymptote is y = 0.

Filled statements:

The graph crosses the y-axis at the point (0, 5). As the value of x increases, the function decreases (exponential decay). The equation of the graph's asymptote is y = 0.

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