Step-by-step solution:

Notice from the given figure that the graph of the function passes through points:

(0,9),(1,5)

The given graph resembles the **Exponential Decay Function**:

There is a **Horizontal Asymptote** at y=1.

If **f(x)** approaches **c** as values are approaching minus infinity or plus infinity, then the line **y=c** is a horizontal asymptote of the graph of the function.

The general form of the exponential function is:

y=a\cdot b^x+k

Here **k** is the horizontal asymptote, a is greater than **0** and **b** is the decay factor.

**k **is the horizontal asymptote,

It means,

k=1

{y}=ab^{x}+\textcolor{primary}{k}

Substitute **k=1** into the above equation:

{y}=ab^{x}+\textcolor{primary}{1}

Substitute **(x, y)=(0,9)** into the equation:

{9}=ab^{0}+{1}

After simplifying:

{9}=ab^{0}+{1}\implies9=a+1\implies a=9-1=8

The value of **a** is** 8**:

Substitute **(x, y)=(1,5)** into the equation:

\textcolor{secondary}{5}=ab^{1}+1

Substitute the **a=8** into the above:

5=(8)b+1\implies 8b=5-1\implies b=\frac{4}{8}=\frac{1}{2}

The value of **b **is** 1/2**:

Substitute the **a=8**, **b=1/2** into the above:

y=8\cdot(\frac{1}{2})^x+1

The equation of the above graph is:

y=8\cdot(\frac{1}{2})^x+1

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