# Which function is shown in the graph below?

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  approaches  as values of  are approaching  or , then the line  is a horizontal asymptote of the graph of the function.

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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