Problem 1 - Algebra 1

Question

The function \(y=a b^{x}\) is a(n) \(\underline{?}\) function.

Step-by-Step Solution

Answer

The function \(y=a b^{x}\) is an exponential function.

Step 1: Identify the formula

The formula given in the exercise is \(y=a b^{x}\). This is a classic format used in mathematics to express a specific type of function. It is composed of three variables: 'y', 'a', and 'x', and one constant 'b'. 'a' and 'b' are coefficients while 'x' and 'y' are variables.

Step 2: Match the formula with the function type

The formula \(y=a b^{x}\) matches the generalized form of an exponential function. Exponential functions are defined as those where the variable 'x' is the exponent. In this case, 'x' is indeed the exponent of 'b'.

Step 3: Formulate the answer

So, the function \(y=a b^{x}\) is an exponential function, as it takes the form of a constant raised to a variable power. Hence, the complete sentence is 'The function \(y=a b^{x}\) is an exponential function.'

Key Concepts

Exponential Function Formula

The exponential function is one of the most pivotal formulas in mathematics, and it can be identified by its distinct structure: \( y = a b^{x} \). At its core, this formula represents an exponential relationship between two variables, 'y' and 'x'. Let's unpack this formula further.

In the expression, \( a \) and \( b \) are constants where \( a \) serves as the initial value and \( b \) is the base of the exponential function. It's crucial that \( b \) is a positive number, but not equal to 1, to ensure the exponential nature is preserved. The variable \( x \) is the exponent, which dictates the rate at which the function grows or decays.

Understanding the Components

• \( y \): This is typically the dependent variable, representing the output or the value of the function.
• \( a \): The coefficient \( a \) is known as the initial amount in many real-life applications such as compound interest or population growth.
• \( b^{x} \): The exponential part, where \( b \) is raised to the power of \( x \), showcases the change in growth or decay over time or across values of \( x \).

In practice, when \( x \) is positive, and \( b \) is greater than 1, the function represents exponential growth. Alternatively, when \( b \) is between 0 and 1, it indicates exponential decay.

Variables in Mathematics

Variables are the alphabets of the mathematical language and play an integral role in formulating relationships, functions, and equations. They are symbols used to represent unknown or changeable values. In the exponential function \( y = a b^{x} \), both \( x \) and \( y \) are variables, each serving a unique purpose within the function.

Role of Variables

• \( x \): Often called the independent variable or input, \( x \) is the value that users can manipulate or choose. In exponential functions, it's set as the power to which the base is raised.
• \( y \): This is the dependent variable or output, which changes in response to different values of \( x \) according to the specific function or equation.

Variables allow for the representation of general relationships in various mathematical contexts. They are foundational in algebra and enable the exploration of functions such as exponentials across many disciplines including physics, finance, and biology.

Exponential Growth and Decay

Exponential growth and decay are two phenomena well-captured by the exponential function \( y = a b^{x} \). They describe processes that increase or decrease at rates proportional to their current size.

Exponential Growth

Exponential growth occurs when the base \( b \) of the exponential function is greater than 1. It reflects a situation where the quantity grows by a constant percentage over equal increments of time. Common examples include population growth, where the number of individuals increases by a certain percentage each year, or compound interest, where money in a bank account grows due to the repeated addition of interest.

Exponential Decay

Conversely, exponential decay describes a decrease over time, with the base \( b \) between 0 and 1. This pattern is found in processes like radioactive decay, where unstable atoms disintegrate over time, or depreciation, where the value of an asset decreases over its useful life.

Understanding the principles behind exponential growth and decay not only assists in the study of mathematics but also allows for the prediction and modeling of real-world scenarios. Simpler examples for the classroom might involve tracking the amount of a drug in the bloodstream over time or the depreciation of a car's value.