micah is comparing two long - term investment options. the first option is shown in the graph above. the…

micah is comparing two long - term investment options. the first option is shown in the graph above. the second option is given by the function g(x)=500(1.05)^x, where x is the number of years and g is the amount of money in the account. which option is better in the long run? a option 1 is better in the long run, because it has the higher starting value. b option 2 is better in the long run, because it has the higher starting value. c option 1 is better in the long run, because it has the higher percent rate of change. d option 2 is better in the long run, because it has the higher percent rate of change.

micah is comparing two long - term investment options. the first option is shown in the graph above. the second option is given by the function g(x)=500(1.05)^x, where x is the number of years and g is the amount of money in the account. which option is better in the long run? a option 1 is better in the long run, because it has the higher starting value. b option 2 is better in the long run, because it has the higher starting value. c option 1 is better in the long run, because it has the higher percent rate of change. d option 2 is better in the long run, because it has the higher percent rate of change.

Answer

Explanation:

Step1: Find starting - value of Option 1

From the graph, when (x = 0), the value of Option 1 is around (1000).

Step2: Find starting - value of Option 2

For the function (g(x)=500(1.05)^x), when (x = 0), (g(0)=500(1.05)^0=500) (since any non - zero number to the power of (0) is (1)). So Option 1 has a higher starting value.

Step3: Analyze the growth rate

The general form of an exponential function is (y = a(1 + r)^x), where (r) is the rate of change. For (g(x)=500(1.05)^x), the rate of change (r = 0.05) or (5%). To analyze the rate of change of Option 1 from the graph, we can see that it is also an exponential - like growth. As (x) gets larger, we can calculate the ratio of consecutive (y) - values for Option 1 at different points. Let's assume two points ((x_1,y_1)) and ((x_2,y_2)) on the graph of Option 1. By observing the graph, we can see that the rate of growth of Option 1 is less than that of Option 2. As (x) (number of years) increases, the function (g(x)=500(1.05)^x) will out - grow the function represented by the graph of Option 1 due to its higher percentage rate of change.

Answer:

D. Option 2 is better in the long run because of its higher percent rate of change.