To find the exponential equation for the graph described, we typically use the form of the exponential decay function:
\[ y = a \cdot b^x \]
where:
- \( a \) is the initial value (the value of \( y \) when \( x = 0 \)),
- \( b \) is the base of the exponential function, which determines the rate of decay.
From the information provided:
- The graph passes through the point (0, 1). This means when \( x = 0 \), \( y = 1 \). Thus, \( a = 1 \).
- The graph also passes through the point (1, 0.5). When \( x = 1 \), \( y = 0.5 \).
Substituting these values into the equation:
-
We know from the first point that: \[ y = 1 \cdot b^0 \Rightarrow y = 1 \]
-
From the second point (1, 0.5): \[ 0.5 = 1 \cdot b^1 \Rightarrow 0.5 = b \]
Now, substituting \( a \) and \( b \) back into the equation gives us:
\[ y = 1 \cdot 0.5^x \]
Therefore, the exponential equation shown in the graph is:
\[ y = 0.5^x \]