The diagram shows part of the graph of y = 2^-x +3, and its horizontal asymptote.

Question

The graph passes through the points (0, a) and (b,3.5).

Answer 1

To analyze the function \( y = 2^{-x} + 3 \) and find the points \( (0, a) \) and \( (b, 3.5) \), let's start with some computations:

1. Find \( a \) when \( x = 0 \): \[ y = 2^{-0} + 3 = 1 + 3 = 4 \] So the point \( (0, a) \) corresponds to \( a = 4 \). Thus, the point is \( (0, 4) \).

2. Find \( b \) when \( y = 3.5 \). We need to solve the equation: \[ 3.5 = 2^{-x} + 3 \] Subtract 3 from both sides: \[ 0.5 = 2^{-x} \] Now, rewrite \( 0.5 \) as \( 2^{-1} \): \[ 2^{-x} = 2^{-1} \] Since the bases are the same, we can equate the exponents: \[ -x = -1 \] Therefore, \( x = 1 \). The point is \( (b, 3.5) \) corresponds to \( b = 1 \). Thus, the point is \( (1, 3.5) \).

In summary, the points identified are:

• \( (0, 4) \)
• \( (1, 3.5) \)