this is the graph of an exponential function.\nwrite an equation for the function in the form ( f(x) =…

Explanation:

Step1: Identify the y-intercept (x=0)

When ( x = 0 ), the function ( f(x)=a(b)^x ) becomes ( f(0)=a(b)^0=a(1)=a ). From the graph, when ( x = 0 ), ( y=-3 ), so ( a=-3 ).

Step2: Find another point to determine ( b )

Looking at the graph, when ( x = 1 ), ( y=-1 ). We know ( a = -3 ), so substitute into ( f(x)=a(b)^x ): ( -1=-3(b)^1 )

Step3: Solve for ( b )

From ( -1 = -3b ), divide both sides by -3: ( b=\frac{-1}{-3}=\frac{1}{3}\approx0.333 )

Step4: Write the equation

Substitute ( a=-3 ) and ( b = \frac{1}{3} ) into ( f(x)=a(b)^x ), we get ( f(x)=-3\left(\frac{1}{3}\right)^x )

Answer:

( f(x) = -3\left(\frac{1}{3}\right)^x ) (or equivalent forms like ( f(x) = -3(3)^{-x} ))