Use the graph to answer the question.

Question

The graph shows an exponentially decreasing line passing through (0, 1) and (1, 0.5)

Write the exponential equation shown in the graph.

Answer 1

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:

1. We know from the first point that: \[ y = 1 \cdot b^0 \Rightarrow y = 1 \]

2. 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:

\[ y = 0.5^x \]