To analyze the function \( y = -2^{-x+2} + 1 \), we can rewrite it in a more standard form:
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Rewrite the function: \[ y = -2^{-(x - 2)} + 1 \]
Here, we can see that the function is a transformation of the exponential function \( 2^x \).
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Identify the transformations:
- The \( - \) sign indicates a reflection across the horizontal axis.
- The \( +2 \) inside the exponent modifies it by shifting the graph to the right by 2 units.
- The \( +1 \) outside translates the graph vertically upwards by 1 unit.
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Domain: The domain of the function \( y = -2^{-x+2} + 1 \) is all real numbers, since there are no restrictions on the input \( x \). \[ \text{Domain: } (-\infty, \infty) \]
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Range: The exponential function \( 2^{-x} \) approaches 0 as \( x \) approaches positive infinity, and it approaches positive infinity as \( x \) approaches negative infinity. Reflecting it across the horizontal axis and shifting up by 1 means it will approach 1 as \( x \) approaches positive infinity and will decrease without bound as \( x \) approaches negative infinity: \[ \text{Range: } (-\infty, 1) \]
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Intercepts:
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Y-Intercept: To find the y-intercept, set \( x = 0 \): \[ y = -2^{-0 + 2} + 1 = -2^{-2} + 1 = -\frac{1}{4} + 1 = \frac{3}{4} \] So, the y-intercept is \( (0, \frac{3}{4}) \).
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X-Intercept: To find the x-intercept, set \( y = 0 \): \[ 0 = -2^{-(x - 2)} + 1 \implies 2^{-(x - 2)} = 1 \implies -(x - 2) = 0 \implies x - 2 = 0 \implies x = 2 \] So, the x-intercept is \( (2, 0) \).
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Asymptotes: The function approaches a horizontal asymptote as \( x \) approaches positive infinity. Since the function approaches 1 as \( x \to \infty \): \[ \text{Horizontal asymptote: } y = 1 \]
In summary:
- Domain: \( (-\infty, \infty) \)
- Range: \( (-\infty, 1) \)
- Y-Intercept: \( (0, \frac{3}{4}) \)
- X-Intercept: \( (2, 0) \)
- Horizontal Asymptote: \( y = 1 \)
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