Question

The diagram shows part of the graph of y = 2** +3, and its horizontal asymptote.
The graph passes through the points (0, a) and (b,3.5).

Answer 1

It seems like there might be a typo or missing information in the equation you've provided. The expression \( y = 2** + 3 \) is not clear. If you're referring to a function like \( y = 2^x + 3 \) (an exponential function), then I can assist you with that.

Assuming the function is \( y = 2^x + 3 \), let's discuss its properties, including the horizontal asymptote.

1. **Horizontal Asymptote**: For \( y = 2^x + 3 \), as \( x \) approaches negative infinity, \( 2^x \) approaches 0. Therefore, the horizontal asymptote is \( y = 3 \).

2. **Points on the Graph**:
- To find the value \( a \) when \( x = 0 \):
\[
y = 2^0 + 3 = 1 + 3 = 4.
\]
So, the point is \( (0, 4) \) and hence \( a = 4 \).

- To find the \( x \)-coordinate \( b \) when \( y = 3.5 \):
\[
3.5 = 2^x + 3.
\]
Subtracting 3 from both sides gives:
\[
0.5 = 2^x.
\]
Taking the logarithm base 2:
\[
x = \log_2(0.5) = -1.
\]
Therefore, the point is \( (-1, 3.5) \), meaning \( b = -1 \).

So, the answers are:
- \( a = 4 \)
- \( b = -1 \)

If this was not the function you intended to refer to, please provide the correct function or additional context!