Question

The diagram shows part of the graph of y = 2^-x +3, and its horizontal asymptote.
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:
- \( (0, 4) \)
- \( (1, 3.5) \)