Last year, Mitch inherited a first-edition book from his grandfather. When he first received it, he looked online and found that the value of the book was listed at \( \$ 250 \). He checked again one year later and saw that the value has increased to \( \$ 265 \). The value of the book is expected to continue to increase over the next few years. Write an exponential equation in the form \( y=a(b)^{x} \) that can model the value of Mitch's book, \( y \), in \( x \) years. Use whole numbers, decimals, or simplified fractions for the values of a and b. \( y= \)

To model the value of Mitch's book using an exponential equation of the form \( y = a(b)^x \), we'll follow these steps: 1. **Identify the Initial Value (\( a \))**: - When Mitch first received the book (consider this as \( x = 0 \) years), its value was \( \$250 \). - So, \( a = 250 \). 2. **Determine the Growth Factor (\( b \))**: - After 1 year (\( x = 1 \)), the book's value increased to \( \$265 \). - The growth factor \( b \) can be calculated using the formula: \[ b = \frac{\text{Value after 1 year}}{\text{Initial value}} = \frac{265}{250} = 1.06 \] - This means the book's value increases by 6% each year. 3. **Construct the Exponential Equation**: - Plugging \( a \) and \( b \) into the exponential form: \[ y = 250 \times (1.06)^x \] **Final Model:** \[ y = 250 \times (1.06)^x \] This equation models the value of Mitch's book \( y \) in \( x \) years, assuming the value continues to increase by 6% annually. **Answer:** \( y = 250 \times (1.06)^{\,x} \)