Asked by j
transformation between f(x)=4^x and f(x)=4^x - 3
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GPT-5 mini
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The graph of f(x) = 4^x - 3 is the graph of f(x) = 4^x shifted vertically downward by 3 units.
Consequences:
- Vertical translation: down 3 units (no horizontal change).
- Horizontal asymptote: y = 0 for 4^x becomes y = -3 for 4^x - 3.
- Domain: (−∞, ∞) (unchanged).
- Range: (−3, ∞) (was (0, ∞) for 4^x).
- y-intercept: (0, 1) for 4^x becomes (0, −2) because 4^0 − 3 = −2.
- x-intercept: solve 4^x − 3 = 0 ⇒ x = log_4(3).
- End behavior: as x → ∞, f(x) → ∞; as x → −∞, f(x) → −3 from above.
Consequences:
- Vertical translation: down 3 units (no horizontal change).
- Horizontal asymptote: y = 0 for 4^x becomes y = -3 for 4^x - 3.
- Domain: (−∞, ∞) (unchanged).
- Range: (−3, ∞) (was (0, ∞) for 4^x).
- y-intercept: (0, 1) for 4^x becomes (0, −2) because 4^0 − 3 = −2.
- x-intercept: solve 4^x − 3 = 0 ⇒ x = log_4(3).
- End behavior: as x → ∞, f(x) → ∞; as x → −∞, f(x) → −3 from above.
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