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Q8 Identify the key features of the graph below. Identify the y-intercept, the rate of decay/growth, and the common ratio. if it pass through (-2,10) and (0,3) and is an exponential growth function.
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First note: the two points (−2, 10) and (0, 3) give an exponential decay (y falls as x increases), so the statement “exponential growth” is inconsistent with these points.
Take y = a b^x.
- From (0,3): a = 3, so y = 3 b^x.
- From (−2,10): 10 = 3 b^(−2) → b^(−2) = 10/3 → b^2 = 3/10 → b = √(3/10) ≈ 0.5477.
Therefore:
- y-intercept: 3 (since y(0)=3).
- common ratio (per 1 unit increase in x): b = √(3/10) ≈ 0.5477.
- rate of decay: 1 − b ≈ 0.4523 → about 45.23% decay per unit increase in x.
Exact forms of the model: y = 3(√(3/10))^x = 3(3/10)^{x/2}.
Take y = a b^x.
- From (0,3): a = 3, so y = 3 b^x.
- From (−2,10): 10 = 3 b^(−2) → b^(−2) = 10/3 → b^2 = 3/10 → b = √(3/10) ≈ 0.5477.
Therefore:
- y-intercept: 3 (since y(0)=3).
- common ratio (per 1 unit increase in x): b = √(3/10) ≈ 0.5477.
- rate of decay: 1 − b ≈ 0.4523 → about 45.23% decay per unit increase in x.
Exact forms of the model: y = 3(√(3/10))^x = 3(3/10)^{x/2}.
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