Question

this is the graph of an exponential function.
write an equation for the function in the form ( f(x) = a(b)^x ).
use whole numbers, decimals, or simplified fractions for the values of ( a ) and ( b ).
( f(x) = )

Explanation:

Step1: Identify the y - intercept

The general form of an exponential function is \(f(x)=a(b)^{x}\). The y - intercept occurs when \(x = 0\). From the graph, when \(x = 0\), \(y=- 6\). Substitute \(x = 0\) and \(y=-6\) into the function:
\(f(0)=a(b)^{0}\). Since \(b^{0}=1\) for any non - zero \(b\), we have \(-6=a\times1\), so \(a=-6\).

Step2: Find the value of \(b\)

We need another point on the graph. Let's take the point \((1,-3)\) (we can check from the graph: when \(x = 1\), \(y=-3\)). Now we know \(a=-6\), \(x = 1\), and \(y=-3\). Substitute these values into the function \(f(x)=a(b)^{x}\):
\(-3=-6\times(b)^{1}\).
To solve for \(b\), divide both sides of the equation by \(-6\):
\(b=\frac{-3}{-6}=\frac{1}{2}=0.5\)

Step3: Write the function

Now that we have \(a=-6\) and \(b = 0.5\), the exponential function is \(f(x)=-6(0.5)^{x}\)

Answer:

\(f(x)=-6(0.5)^{x}\)