Graphs of Equations and Compound Interest: Foundations for Calculus

Graphs of Equations

Introduction to Equations in Two Variables

Equations such as $y = 2x + 1$ and $x^2 + y^2 = 25$ are called equations in two variables. These equations relate two quantities and their solutions are ordered pairs that satisfy the equation.

• Ordered Pair: A solution to an equation in two variables is an ordered pair $(x, y)$ that makes the equation true.

• Graph of an Equation: The graph is a visual representation of all ordered pairs that are solutions to the equation.

• Example: For $x + y = 5$, some solutions are $(1, 4)$, $(2, 3)$, $(3, 2)$, $(4, 1)$, and $(5, 0)$.

Definition: The graph of an equation is a drawing that represents all ordered pairs that are solutions of the equation.

Creating Graphs

Graphs can be created by plotting points that satisfy the equation and looking for patterns. Technology can assist in graphing more complex equations.

• Mathematical Models: Equations can be used to model real-world phenomena and predict outcomes.

• Applications: Graphs are used in finance, science, engineering, and many other fields to visualize relationships between variables.

Compound Interest

Compound Interest Formula

Compound interest is a common application of mathematical equations in finance. When money is invested at a certain interest rate and compounded over time, the amount grows according to a specific formula.

• Principal ($P$): The initial amount of money invested.

• Interest Rate ($r$): The annual interest rate, expressed as a decimal.

• Number of Compounding Periods ($n$): The number of times interest is compounded per year.

• Time ($t$): The number of years the money is invested.

• Amount ($A$): The total amount in the account after $t$ years.

The formula for compound interest is:

$A = P \left(1 + \frac{r}{n}\right)^{nt}$

Theorem 1: Compound Interest

If an amount $P$ is invested at interest rate $r$, compounded $n$ times a year, after $t$ years it will grow to an amount $A$ given by:

$A = P \left(1 + \frac{r}{n}\right)^{nt}$

Examples of Compound Interest

• Example 6: Suppose $1000 is invested in the Fibonacci Investment Fund at 5%, compounded annually. How much is in the account at the end of the second year?

• Example 7: Jim invests $1000 in the Green View Fund at 2.75%, compounded quarterly. How much is in the account at the end of 3 years?

Solution Method: For each example, substitute the values for $P$, $r$, $n$, and $t$ into the compound interest formula to find $A$.

Example Calculation

• Example 6 Solution: $P = 1000$, $r = 0.05$, $n = 1$, $t = 2$ $A = 1000 \left(1 + \frac{0.05}{1}\right)^{1 \times 2}$ $A = 1000 (1.05)^2 = 1000 \times 1.1025 = 1102.50$ After 2 years, the account will have $1102.50.

• Example 7 Solution: $P = 1000$, $r = 0.0275$, $n = 4$, $t = 3$ $A = 1000 \left(1 + \frac{0.0275}{4}\right)^{4 \times 3}$ $A = 1000 (1.006875)^{12}$ $A \approx 1000 \times 1.0851 = 1085.10$ After 3 years, the account will have approximately $1085.10.

Comparison Table: Simple vs. Compound Interest

Type	Formula	Growth Pattern
Simple Interest	$A = P(1 + rt)$	Linear
Compound Interest	$A = P \left(1 + \frac{r}{n}\right)^{nt}$	Exponential

Additional info: Compound interest is a foundational concept in calculus, as it introduces exponential growth, which is later analyzed using derivatives and integrals.