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: Find the value of \( a \)

The exponential function is in the form \( f(x) = a(b)^x \). When \( x = 0 \), \( f(0)=a(b)^0=a(1)=a \). From the graph, when \( x = 0 \), \( y=-2 \). So, \( a=-2 \).

Step2: Find the value of \( b \)

We need another point to find \( b \). Let's take a point on the graph. Let's see when \( x = 1 \), what's \( y \)? Wait, maybe take \( x = 1 \), but let's check the behavior. Wait, the function is decreasing, so \( b \) should be greater than 0 and less than 1? Wait, no, wait the graph: when \( x = 0 \), \( y=-2 \); let's see when \( x = 1 \), what's the \( y \)-value? Wait, the graph at \( x = 1 \), let's check the grid. Wait, maybe another approach. Wait, let's assume a point. Wait, when \( x = 1 \), let's see the graph. Wait, maybe the function is \( f(x)=-2(2)^x \)? Wait, no, let's test. Wait, if \( a=-2 \), and let's take \( x = 1 \), what's \( y \)? Wait, maybe I made a mistake. Wait, let's check the graph again. Wait, when \( x = 0 \), \( y=-2 \). Let's take \( x = 1 \), the graph at \( x = 1 \) is at \( y=-4 \)? Wait, no, the graph at \( x = 1 \) is going down to \( -7 \)? Wait, no, maybe I misread. Wait, the graph: when \( x = 0 \), \( y=-2 \); when \( x = 1 \), let's see the grid. The vertical axis is \( y \), horizontal is \( x \). So at \( x = 0 \), \( y=-2 \); at \( x = 1 \), the point is at \( y=-4 \)? Wait, no, the green line at \( x = 1 \) is at \( y=-4 \)? Wait, no, the graph is a curve. Wait, maybe the base \( b = 2 \). Let's test: if \( f(x)=-2(2)^x \), then when \( x = 0 \), \( f(0)=-2(1)=-2 \), which matches. When \( x = 1 \), \( f(1)=-2(2)=-4 \), does that match the graph? Let's see the graph at \( x = 1 \), the \( y \)-value is -4? Wait, the graph at \( x = 1 \) is going down, so maybe \( f(1)=-4 \), which would mean \( b = 2 \). So let's confirm. So \( a=-2 \), \( b = 2 \). So the function is \( f(x)=-2(2)^x \). Wait, but let's check another point. When \( x=-1 \), \( f(-1)=-2(2)^{-1}=-2(\frac{1}{2})=-1 \). Does the graph at \( x=-1 \) have \( y=-1 \)? Yes! From the graph, at \( x=-1 \), \( y=-1 \). So that matches. So \( a=-2 \), \( b = 2 \).

Answer:

\( f(x)=-2(2)^x \)