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We are given that Grace invested $200 one year ago, and the account has grown to $212 today. The growth is modeled by an exponential function of the form $ y = a(b)^x $, where:
- $ y $ is the value of the account after $ x $ years,
- $ a $ is the initial amount,
- $ b $ is the growth factor per year,
- $ x $ is the number of years since the investment.

### Step 1: Find the Exponential Equation
1. The initial value $ a $ is the amount Grace initially invested, which is $200.
   \[
   a = 200
   \]
2. The value after 1 year is $212. So, we use this information to find the growth factor $ b $. At $ x = 1 $, the equation becomes:
   \[
   212 = 200(b)^1
   \]
   Solving for $ b $:
   \[
   b = \frac{212}{200} = 1.06
   \]

Thus, the exponential model is:
\[
y = 200(1.06)^x
\]

### Step 2: Predict the Value After 6 Years
We now calculate the value of the account after 6 years. Plugging $ x = 6 $ into the equation:
\[
y = 200(1.06)^6
\]
\[
y = 200 \times 1.4185 \approx 283.70
\]

Therefore, the value of the account after 6 years will be approximately **$284**.

Would you like more details or have any questions?

Here are five related questions to further your understanding:
1. What does the growth factor $ b = 1.06 $ represent in this context?
2. How would the equation change if the account grew at a different rate?
3. What if Grace withdrew $50 after the second year? How would that affect the exponential model?
4. How does compounding influence the growth rate in this problem?
5. Could this equation be used to predict the account value over a longer time frame?

**Tip:** Exponential growth is characterized by a constant percentage increase each period, unlike linear growth, which increases by a fixed amount.

One year ago, Grace put $200 she earned from tutoring into an investment account. Today, the account's value has increased to $212, and it is expected to continue growing each year. Write an exponential equation in the form y = a(b)^x that can model the value of the investment account, y, x years after Grace started the account. To the nearest dollar, what can Grace expect the value of the account to be 6 years after starting the account if she makes no other deposits or withdrawals?

Explore how Grace's $200 investment grows exponentially to $212 in one year and predict its future value using the formula y = a(b)^x. Learn how to model the value after 6 years, demonstrating concepts of exponential growth and algebra, ideal for high school students (Grades 9-12).

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