Question

the table represents a savings account balance each year for 3 years.

\

$$\begin{tabular}{|l|l|l|l|l|} \\hline years & 0 & 1 & 2 & 3 \\\\ \\hline balance & \\$7,000 & \\$7,175 & \\$7,354.38 & \\$7,538.23 \\\\ \\hline \\end{tabular}$$

part a: is the function linear or exponential? what is the interest rate? show all necessary steps. (12 points)

part b: write an equation to represent the investment account balance over time. identify the parts of the equation. (6 points)

Explanation:

Determine function type

Using the Compound Interest Formula knowledge point

$$
LATEXBLOCK0
$$

The constant ratio indicates an exponential function.

Calculate interest rate

Using the Compound Interest Formula knowledge point

$$
LATEXBLOCK1
$$

Formulate the equation

Using the Compound Interest Formula knowledge point

$$ A(t) = 7000(1.025)^t $$

Identify equation parts

Using the Compound Interest Formula knowledge point

• \(A(t)\): Balance after \(t\) years.
• \(7000\): Initial principal balance.
• \(1.025\): Growth factor.
• \(t\): Time in years.

Answer:

Part A

• Function Type: Exponential, because the balance increases by a constant multiplier of \(1.025\) each year rather than a constant addition.
• Interest Rate: \(2.5\%\) per year.

Part B

• Equation: \(A(t) = 7000(1.025)^t\)
• Parts of the Equation:
• \(A(t)\) represents the account balance after \(t\) years.
• \(7000\) represents the initial deposit (principal amount).
• \(1.025\) represents the growth factor (where \(1 + 0.025\) accounts for the \(2.5\%\) annual interest rate).
• \(t\) represents the time elapsed in years.