Problem 15 - College Algebra

Question

For the following exercises, determine whether the equation represents exponential growth, exponential decay, or neither. Explain. $$ y=220(1.06)^{x} $$

Step-by-Step Solution

Answer

Exponential growth (rate = 6%).

Step 1: Identify the General Form of an Exponential Equation

The general form of an exponential equation can be written as \( y = a(1 + r)^x \) or \( y = a(1 - r)^x \), where \( a \) is the initial value, \( r \) is the rate of change, and \( x \) is the exponent.

Step 2: Compare Given Equation with General Form

The given equation is \( y = 220(1.06)^x \). It resembles the form \( y = a(1 + r)^x \) because of the structure \( 1.06 \).

Step 3: Determine the Rate of Change

Compare \( 1.06 \) with \( 1 + r \). This indicates that \( r = 0.06 \), which represents a 6% increase.

Step 4: Determine Type of Exponential Function

Because \( r \) is positive (6% increase), the equation represents exponential growth.

Key Concepts

Exponential Equation

An exponential equation is a type of mathematical expression where a constant base is raised to a variable exponent. This form can be seen in a variety of real-world scenarios, such as population growth, radioactive decay, and interest calculations. The general form of an exponential equation is often presented as \( y = a(1 + r)^x \) or \( y = a(1 - r)^x \). Here, \( a \) represents the initial value, \( r \) is the rate of change, and \( x \) is the exponent.

The crucial aspect of exponential equations is the multiplication effect, where small increases in the exponent can result in significant changes in the overall value. This is because the base is raised to a higher power each time. In many cases, these equations model processes that proceed at a rate related to their current value, leading to accelerating increases (or decreases) over time.

Rate of Change

The rate of change in an exponential equation indicates how quickly the quantity grows or decays. This rate is expressed within the formula as a percentage and is symbolized by \( r \) in the equation \( y = a(1 + r)^x \) or \( y = a(1 - r)^x \).

In the context of the original exercise, the given exponential equation is \( y = 220(1.06)^x \). Here, 1.06 represents \( 1 + r \), meaning \( r = 0.06 \). This reflects a 6% growth rate. Positive values of \( r \) (greater than zero) lead to an increase or exponential growth, while negative values (below zero) lead to exponential decay.

• A positive \( r \) means the function is growing over time.
• A negative \( r \) would indicate decay or decreasing values.

This rate of change helps describe how fast the initial value is multiplying itself in each step along the timeline.

Initial Value

The initial value in exponential equations serves as the starting point or the baseline from which the growth or decay begins. It is represented as \( a \) in the general exponential formula \( y = a(1 + r)^x \) or \( y = a(1 - r)^x \).

In our specific example, the equation is \( y = 220(1.06)^x \). Here, \( a = 220 \). This means the situation modeled by the equation starts with a base value of 220.

• The initial value is crucial as it sets the stage for the exponential process. Every calculation of future outcomes is anchored by this starting point.
• Changes to the initial value directly affect the entire scaling of the curve associated with the equation.

Understanding the initial value allows one to accurately interpret real-world situations, such as predicting future population changes from a known starting population.