Problem 62 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the statement is true or false. Justify your answer. The graph of a logistic growth function will always have an \(x\) -intercept.

Step-by-Step Solution

Answer

The statement is false. A logistic growth function does not always have an \(x\)-intercept.

Step 1: Recall the Equation of a Logistic Function

The general form of a logistic function is \(f(x) = \frac{c}{1 + ae^{-bx}}\), where \(a\), \(b\), and \(c\) are constants

Step 2: Consider the x-intercepts

A function has an x-intercept where it intersects the x-axis. This happens when \(f(x) = 0\). For a logistic function, \(f(x) = 0\) when \(x\) is equal to \(\ln(-\frac{c}{a})/b\), as long as \(a < 0\). If \(a >= 0\), the logistic function will not have an \(x\)-intercept.

Step 3: Apply to Problem

Therefore, the statement 'the graph of a logistic growth function will always have an \(x\)-intercept' is false. It will only have an x-intercept if \(a < 0\).

Key Concepts

X-Intercept

In mathematical terms, the x-intercept is the point where a function or graph crosses the x-axis of a coordinate plane. Specifically, it is the value of x where the function f(x) equals zero. For the logistic growth function, this concept is a little tricky because the existence of an x-intercept depends on certain parameters within the equation.

For instance, the logistic function might never actually touch the x-axis, which means it would not have an x-intercept. This is counterintuitive to our usual experience with many other types of functions, such as linear functions, that commonly cross the x-axis. In the context of logistic functions, whether or not there's an x-intercept hinges on the value of parameter a. If a is a non-negative number, the function will not cross the x-axis, thereby having no x-intercept. In simple terms, the presence of an x-intercept is conditional and not a guaranteed feature of logistic growth functions.

Logistic Function Equation

The logistic function equation represents a type of growth that starts rapidly and then slows down as it approaches a maximum value called the carrying capacity. The standard form of the logistic function is expressed as \(f(x) = \frac{c}{1 + ae^{-bx}}\), where the constants a, b, and c shape the curve.

Understanding the Parameters

• c: Represents the eventual maximum value or carrying capacity that the function will approach as x becomes large.
• a: Impacts the location of the curve along the x-axis. If a is negative, we may observe an x-intercept.
• b: Determines the growth rate. A larger value of b means faster initial growth.

The equation models phenomena in biology, such as population dynamics, and in other fields like economics, with the parameters being adjusted to suit the specific situation being modeled. It's a foundational concept in understanding real-world growth scenarios that don't exhibit constant growth rates.

Properties of Logistic Functions

Logistic functions have several unique characteristics that make them distinctive from other types of functions. Here's a closer look at some of their key properties:

• The function has a characteristic 'S' shape, known as a sigmoid curve.
• There is an initial exponential growth phase which slows down as it approaches the carrying capacity c.
• The growth rate is fastest at the inflection point, which is the point on the curve where the second derivative changes sign.
• The function never crosses the horizontal asymptote y = c, which represents the carrying capacity.
• It models real-world situations where growth is limited by resources or other factors.
• The values of a, b, and c have significant effects on the function's graph, shifting it horizontally, affecting the growth rate, and setting the upper limit, respectively.

The logistic function is often used to illustrate how certain values or populations grow in a confined space with limited resources. By analyzing the properties of logistic functions, students can gain insight into the behavior of complex systems in nature and other fields.