To graph the function \( y = 15(2)^x \), we can identify characteristics of the function including the y-intercept.
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Find the y-intercept: The y-intercept occurs when \( x = 0 \). Substitute \( x = 0 \) into the function:
\[ y = 15(2)^0 = 15(1) = 15 \]
Therefore, the y-intercept is \( (0, 15) \).
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Graph the function: To sketch the graph of \( y = 15(2)^x \), we calculate a few values of \( y \) for various \( x \):
- For \( x = -2 \): \[ y = 15(2)^{-2} = 15 \cdot \frac{1}{4} = 3.75 \]
- For \( x = -1 \): \[ y = 15(2)^{-1} = 15 \cdot \frac{1}{2} = 7.5 \]
- For \( x = 0 \): \[ y = 15(2)^0 = 15 \]
- For \( x = 1 \): \[ y = 15(2)^1 = 30 \]
- For \( x = 2 \): \[ y = 15(2)^2 = 60 \]
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Plot these points: The points \( (-2, 3.75) \), \( (-1, 7.5) \), \( (0, 15) \), \( (1, 30) \), \( (2, 60) \) can be plotted on a graph.
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Shape of the graph: Since \( y = 15(2)^x \) is an exponential function with a base greater than 1, the graph will rise sharply as \( x \) increases, and approach \( 0 \) as \( x \) decreases.
Thus, the y-intercept of the function \( y = 15(2)^x \) is:
\[ \text{Y-intercept: } (0, 15) \]