Questions: Exponential and Logarithmic Functions Choosing an exponential model and using it to make a prediction The data points show the amount of money y (in dollars) in an account after a time x (in years). Each figure has the same data points. However, each figure has a different curve fitting the data. The equation for each curve is also shown. Answer the questions that follow. (a) Which curve fits the data best?

Solution Steps

The data points appear to have a slight upward curve, suggesting exponential growth. A linear model might fit initially, but a curve will likely fit better in the long run as the rate of growth appears to be increasing.

• Figure 1: y = 40x + 400 represents a linear relationship. While the line passes somewhat close to the points, it doesn't account for the apparent curve.
• Figure 2: y = 1.25(1.21)^x represents exponential growth with a relatively high growth rate (1.21). The curve rises too quickly compared to the data points.
• Figure 3: y = 601(1.05)^x also represents exponential growth, but with a more modest growth rate (1.05). This curve appears to follow the data points much more closely.

The curve in Figure 3 fits the data points best. It captures the exponential growth trend without exaggerating the rate of increase.

Final Answer: The curve in Figure 3, with the equation y = 601(1.05)^x, fits the data best.