Final answer:
Emily's investment function, assuming compound interest, is A = 100(1 + 0.12)^n. To calculate the average rate of change between two points, subtract the function's value at the initial point from its value at the final point, and divide by the length of the time interval between those points.
Step-by-step explanation:
Emily is interested in putting $100 into an account that compounds interest annually at a rate of 12%. To track the growth of her investment, we can create a function that models compound interest.
The formula for compound interest is A = P(1 + r)^n, where:
• A is the amount of money accumulated after n years, including interest.
• P is the principal amount (the initial amount of money).
• r is the annual interest rate (in decimal form).
• n is the number of years the money is invested or borrowed.
Using this formula, Emily's investment function would be written as A = 100(1 + 0.12)^n.
To find the average rate of change between two points on a function's curve, which is akin to finding the slope of the secant line that connects the two points, you simply subtract the initial value of the function from the final value and divide by the difference in the x-values (amount of time) associated with these points. Here is the formula:
Average Rate of Change = (f(x2) - f(x1)) / (x2 - x1)
For example, if you wanted to find the average rate of change of Emily's account balance between years 1 and 3, you would calculate the amount of money at year 3, subtract the amount at year 1, and then divide by the difference in time, which is 3-1 = 2 years.