Problem 1 - Precalculus : Building Concepts and Connections

Question

Complete them to review topics relevant to the remaining exercises. An exponential function of the form \(f(x)=C a^{x},\) where \(C>0\) and \(a>1,\) models exponential __________.

Step-by-Step Solution

Answer

Growth

Step 1: Understand the Question

The exercise gives an exponential function \(f(x)=Ca^{x}\) with \(C>0\) and \(a>1\). The function models exponential _______. This means you need to fill in the missing word that describes a characteristic of an exponential function.

Step 2: Define the Exponential Function

Exponential functions have the form \(f(x) = Ca^{x}\). In this form, 'C' is a constant, and 'a' is the base of the exponent. When the base 'a' is greater than 1, the function represents exponential growth.

Step 3: Fill in the Blank

Based on the definition of an exponential function, when \(C>0\) and \(a>1\), it models exponential growth. So the missing word in the sentence is 'growth'.

Key Concepts

Exponential Growth

When we talk about exponential growth, we're referring to an increase whose rate becomes more rapid in proportion to the growing total number or size. In other words, as the value of 'x' increases, the function's output grows at an accelerating rate. This is often used to describe phenomena in real life that multiply at a consistent rate over time, such as population growth, compound interest, and the spread of diseases.

For example, suppose a bacteria culture doubles in size every hour. If you start with one cell, after one hour you'll have two, after two hours you'll have four, and this pattern continues, resulting in 2, 4, 8, 16, and so on. Notice how the size of the increase grows larger with each step—this is the essence of exponential growth.

Exponential Function Definition

In mathematics, an exponential function is a powerful tool for modeling scenarios where quantities grow or decay at a constant relative rate. By definition, an exponential function can be written in the form \(f(x) = Ca^{x}\), where 'C' is a positive constant known as the coefficient, and 'a' is the base of the exponent that must be a positive real number other than 1. The variable 'x' often represents time or another independent variable.

When the base 'a' is greater than 1, each increase in 'x' results in the output of the function being multiplied by 'a', leading to exponential growth. Alternatively, if 'a' is between 0 and 1, it results in exponential decay, where the function's output decreases at a rate proportional to its current value.

Exponential Function Characteristics

There are several key characteristics that define exponential functions. One, they have a constant base 'a', that's not equal to 1, and it determines the rate of growth or decay. Two, the exponent 'x' is a variable, which represents how many times 'a' will be multiplied by itself. Three, the coefficient 'C' scales the function vertically, and if it's negative, the graph of the function is reflected across the x-axis.

Additionally, the graph of an exponential function with growth (where a > 1) is a curve that starts at 'C' on the y-axis (as \(f(0) = C\)) for a positive 'C', and rises to the right. The larger the value of 'a', the steeper the curve. A key property of this function type is that the slope of the graph gets steeper and steeper as it moves to the right—demonstrating accelerating growth. If 'a' is between 0 and 1, the graph exhibits a decline, getting closer to the x-axis but never quite touching it, illustrating decay.