A principal of $$ \$ 2000 $$ was invested at $$ 6.25 \% $$ interest, compounded annually. Let $$ t $$ be the number of years since the start of the investment. Let $$ y $$ be the value of the investment, in dollars. Write an exponential function showing the relationship between $$ y $$ and $$ t $$.

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To write an exponential function that shows the relationship between the value of the investment $$ y $$ and the time in years $$ t $$, we can use the formula for compound interest:

$$
y = P(1 + r)^t
$$

where:
- $$ P $$ is the principal amount (initial investment),
- $$ r $$ is the annual interest rate (as a decimal),
- $$ t $$ is the number of years,
- $$ y $$ is the amount of money accumulated after $$ t $$ years, including interest.

Given:
- $$ P = 2000 $$
- $$ r = 6.25\% = 0.0625 $$

Substituting these values into the formula, we get:

$$
y = 2000(1 + 0.0625)^t
$$

This simplifies to:

$$
y = 2000(1.0625)^t
$$

Thus, the exponential function showing the relationship between $$ y $$ and $$ t $$ is:

$$
y = 2000(1.0625)^t
$$
The exponential function is $$ y = 2000(1.0625)^t $$.

A principal of was invested at interest, compounded annually.
Let be the number of years since the start of the investment. Let be the value of the investment, in dollars.
Write an exponential function showing the relationship between and .

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A principal of was invested at interest, compounded annually. Let be the number of years since the start of the investment. Let be the value of the investment, in dollars. Write an exponential function showing the relationship between and .