To visualize the situation described, we can analyze the function \( a(t) = 1,000(1.09)^t \). This function represents exponential growth due to the annual compounding interest at a rate of 9%.
Key Characteristics of the Graph:
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Initial Value: When \( t = 0 \) (the time the scholarship was awarded), the amount \( a(0) \) is: \[ a(0) = 1,000(1.09)^0 = 1,000 \] This means the graph starts at the point (0, 1000).
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Growth Rate: The function grows exponentially, so the graph will curve upward as \( t \) increases.
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Value after 10 Years: To find out how much Callie will have after 10 years, we can calculate \( a(10) \): \[ a(10) = 1,000(1.09)^{10} \] Calculating \( (1.09)^{10} \) gives approximately \( 2.3674 \) (using a calculator). Thus, \[ a(10) \approx 1,000 \times 2.3674 = 2,367.4 \]
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Shape of the Graph: The graph will be a smooth curve that rises steeply after several years, characteristic of exponential functions.
Appearance of the Graph:
- The x-axis will represent time (in years).
- The y-axis will represent the amount in the account.
- The graph will start at (0, 1000) and will rise sharply upwards, crossing points like (10, 2367.4) and even higher values as time progresses.
Conclusion:
The correct graph will start at (0, 1000) and will follow a steeply rising curve as time increases. It will not be a straight line but rather an upward-curving line reflecting the exponential growth nature of the compounded interest. If you’re looking at a set of graphs, choose the one that has this characteristic upward exponential growth starting from that initial amount.