Leticia invests \( \$ 200 \) at \( 5 \% \) interest. If \( y \) represents the amount of money after \( x \) time periods, which describes the graph of the exponential function relating time and money? The initial value of the graph is 200 . The graph increases by a factor of 1.05 per 1 unit increase in time. The initial value of the graph is 200 . The graph increases by a factor of 5 per 1 unit increase in time. The initial value of the graph is 500 . The graph increases by a factor of 2 per 1 unit increase in time. The initial value of the graph is 500 . The graph increases by a factor of 1.02 per 1 unit increase in time.

To determine the correct description of the graph of the exponential function relating time and money for Leticia's investment, we need to analyze the information given. 1. **Initial Investment**: Leticia invests \( \$200 \). This means the initial value of the graph, when \( x = 0 \), is \( 200 \). 2. **Interest Rate**: The investment earns \( 5\% \) interest. In terms of growth factor, this means that for each time period (let's say each year), the amount of money increases by \( 5\% \). - A \( 5\% \) increase can be expressed as a growth factor of \( 1 + 0.05 = 1.05 \). 3. **Exponential Growth Function**: The general form of the exponential growth function can be expressed as: \[ y = a \cdot b^x \] where: - \( a \) is the initial amount (in this case, \( 200 \)), - \( b \) is the growth factor (in this case, \( 1.05 \)), - \( x \) is the number of time periods. Given this analysis, we can summarize the correct description of the graph: - The initial value of the graph is \( 200 \). - The graph increases by a factor of \( 1.05 \) per 1 unit increase in time. Now, let's evaluate the options provided: 1. **The initial value of the graph is 200. The graph increases by a factor of 1.05 per 1 unit increase in time.** (This is correct.) 2. The initial value of the graph is 200. The graph increases by a factor of 5 per 1 unit increase in time. (Incorrect, as the growth factor is \( 1.05 \), not \( 5 \).) 3. The initial value of the graph is 500. The graph increases by a factor of 2 per 1 unit increase in time. (Incorrect, as the initial value is \( 200 \) and the growth factor is \( 1.05 \).) 4. The initial value of the graph is 500. The graph increases by a factor of 1.02 per 1 unit increase in time. (Incorrect, as the initial value is \( 200 \) and the growth factor is \( 1.05 \).) Thus, the correct answer is: **The initial value of the graph is 200. The graph increases by a factor of 1.05 per 1 unit increase in time.**