To identify the correct graph for the exponential equation \( y = (2.5)^x \), keep the following characteristics in mind:
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Base Greater than 1: Since the base is 2.5, which is greater than 1, the graph will show exponential growth. This means that as \( x \) increases, \( y \) will increase rapidly.
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y-intercept: The graph will intersect the y-axis at \( y = 1 \) when \( x = 0 \), since \( (2.5)^0 = 1 \).
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Behavior as x approaches negative infinity: As \( x \) approaches negative infinity, \( y \) will approach 0 but never actually reach it (the graph will not cross the x-axis).
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Curvature: The graph will be increasing and curving upwards, getting steeper as \( x \) increases.
With these characteristics in mind, look for a graph that:
- Starts close to zero for negative \( x \) values.
- Passes through the point \( (0, 1) \).
- Increases rapidly for positive \( x \) values.
If provided with multiple graph options, select the one that matches these behaviors.