To analyze the function \( y = -2^{-x+2} + 1 \), we can break it down and identify its domain, range, intercepts, and asymptotes.
Step 1: Identify the Domain
The domain of the function \( y = -2^{-x+2} + 1 \) is all real numbers. Therefore:
- Domain: \( (-\infty, \infty) \)
Step 2: Identify the Range
The expression \( -2^{-x+2} \) will vary based on the exponent. As \( x \) increases, \( -2^{-x+2} \) approaches \( 0 \) from below since \( 2^{-x+2} \) becomes very small. The function will never actually reach \( 0 \) but can get arbitrarily close to it. The vertical shift upwards by \( 1 \) means that the values of the function approach \( 1 \) from below. However, when \( x \) is very large and \( -x+2 \) becomes negative, \( 2^{-x+2} \) gets larger, and thus \( -2^{-x+2} \) approaches \( -\infty \). Hence:
- Range: \( (-\infty, 1) \)
Step 3: Identify the Intercepts
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:
- Y-Intercept: \( (0, \frac{3}{4}) \)
X-Intercept: To find the x-intercept, set \( y = 0 \): \[ 0 = -2^{-x+2} + 1 \] \[ 2^{-x+2} = 1 \] Taking the log base 2 of both sides: \[ -x + 2 = 0 \implies x = 2 \] Thus, the x-intercept is:
- X-Intercept: \( (2, 0) \)
Step 4: Identify the Asymptotes
The function has a horizontal asymptote because as \( x \to \infty \), \( y \to 1 \).
- Horizontal Asymptote: \( y = 1 \)
Summary:
- Domain: \( (-\infty, \infty) \)
- Range: \( (-\infty, 1) \)
- Y-Intercept: \( (0, \frac{3}{4}) \)
- X-Intercept: \( (2, 0) \)
- Horizontal Asymptote: \( y = 1 \)