Problem 24 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

\(y=(0.2)^x\)

Step-by-Step Solution

Answer

The function value decreases as \(x\) increases with \(y = 0.2\) for \(x = 1\), \(y = 0.04\) for \(x = 2\) and \(y = 0.008\) for \(x = 3\).

Step 1: Substitute \(x=1\) in the equation

Let's start with \(x = 1\). Substitute \(x = 1\) in the equation to yield \(y = (0.2)^1 = 0.2\).

Step 2: Substitute \(x=2\) in the equation

Let's continue with \(x = 2\). Substitute \(x = 2\) in the equation to yield \(y = (0.2)^2 = 0.04\).

Step 3: Substitute \(x=3\) in the equation

Let's consider \(x = 3\). Substitute \(x = 3\) in the equation to yield \(y = (0.2)^3 = 0.008\).

Key Concepts

Substitution Method

The substitution method is a fundamental technique often used to solve equations, where one variable is replaced by its value or another expression. It's a straightforward 'plug and play' method. In mathematical problems, especially when dealing with functions, we frequently use substitution to evaluate the function for specific values of its variables.

For instance, given an exponential function like \(y = (0.2)^x\), we can find the value of 'y' for any given 'x' by substituting the value of 'x' into the expression. When we put \(x = 1\) into our function, we replace every 'x' with 1, leading to \(y = (0.2)^1 = 0.2\). It simplifies the identification of the function's behavior at particular points and is crucial for understanding the overall trend of the function.

Exponential Decay

Exponential decay refers to the decrease in quantity over time, following a consistent pattern described by an exponential function. In such functions, like \(y = (0.2)^x\), the base of the exponential, in this case, 0.2, is between 0 and 1. This tells us that the function represents a decay process.

The variable 'x' often represents time or sequence of events, and 'y' the remaining quantity. As 'x' increases, 'y' decreases rapidly at first and then more slowly, approaching zero but never actually reaching it. This behavior is typical in radioactive decay, population decline, and depreciation of assets. Understanding exponential decay is crucial for professionals in various fields such as physics, finance, and environmental science.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, operations, and sometimes grouping symbols that represent a particular value or relationship. An expression such as \(y = (0.2)^x\) is an exponential expression, with 'x' being the exponent and 0.2 the base. They are the building blocks of mathematical equations and functions.

• Numbers, like 0.2, are constants that provide specific values.
• Variables like 'x' and 'y' represent unknown values or values that can change.
• Operations include addition, subtraction, multiplication, and division, as well as exponentiation in this case.

Expressions are a concise way to communicate mathematical ideas and serve as a shorthand for computations that might be too lengthy or complex to describe in words alone. Students learn to manipulate these expressions using various methods including the substitution method, in order to solve equations and understand the behavior of functions like exponential decay.