Problem 8 - College Algebra

Question

For the following exercises, identify whether the statement represents an exponential function. Explain. The height of a projectile at time \(t\) is represented by the function \(h(t)=-4.9 t^{2}+18 t+40\).

Step-by-Step Solution

Answer

The function is not exponential; it's a quadratic function.

Step 1: Identify the Standard Form of an Exponential Function

An exponential function is typically of the form \( f(t) = a \, b^t \) where \(a\) is a constant, \(b\) is the base of the exponential, and \(t\) is the variable exponent. It involves a constant raised to a variable power.

Step 2: Analyze the Given Function

The given function is \( h(t) = -4.9t^2 + 18t + 40 \). This function is a quadratic function because it is of the form \( at^2 + bt + c \), where the highest degree of \(t\) is 2, not a variable exponent.

Step 3: Compare with Exponential Function Characteristics

In an exponential function, the variable \(t\) would be in the exponent, such as in \( b^t \). However, in the given quadratic function, \(t\) is not in the exponent; instead, it is part of the terms \(t^2\) and \(t\), which are polynomial characteristics.

Step 4: Conclusion: Determine the Type of Function

The function \( h(t) = -4.9t^2 + 18t + 40 \) is not an exponential function because it is a quadratic function, which is characterized by \(t\) being raised to a power (2 and 1), rather than being in the exponent.

Key Concepts

Exponential Functions

Exponential functions are truly fascinating and are commonly encountered in various fields, from finance to biology. The general form of an exponential function is given by \( f(t) = a \cdot b^t \), where:

• \(a\) is a constant which usually represents the initial amount or y-intercept.
• \(b\) is the base of the exponential and determines the growth (if \(b > 1\)) or decay (if \(0 < b < 1\)).
• \(t\) is the exponent and is commonly a variable, representing time or another continuous parameter.

This form signifies rapid changes because the variable is an exponent, making values rise or fall very quickly. You won't see terms like \(t^2\) or \(t\) in this function form. The hallmark of exponential functions is the exponent itself housing the variable, leading to a unique and predictable pattern of growth (exponential growth) or shrinkage (exponential decay). Whenever you're faced with a function, looking out for a variable in the exponent is a key step to identifying it as an exponential function.

Polynomial Characteristics

Polynomials are straightforward in their structure, different from exponential functions because they have variables raised to fixed powers. Quadratic functions, such as the one discussed \( h(t) = -4.9t^2 + 18t + 40 \), belong to the broader family of polynomial functions.

• Polynomial functions can be expressed in the form \(a_nt^n + a_{n-1}t^{n-1} + \, ... \, + a_1t + a_0\).
• Each term consists of a coefficient \(a\) multiplied by the variable \(t\) raised to an integer power.
• The highest power, in the case of polynomials like quadratics, defines its degree. For example, a quadratic function has a degree of 2, due to the term \(t^2\).

These characteristics help to distinguish polynomials from exponential functions, where the key factor is the variable being an exponent itself. In contrasting polynomial characteristics, keep an eye out for integer exponents arranged from the highest to the lowest, which is the typical layout.

Function Analysis

Function analysis is the process of examining a function to understand its behavior and characteristics. Analyzing a function like \( h(t) = -4.9t^2 + 18t + 40 \), you can learn a great deal about it by identifying its type and structure, which are critical for further predictions and understanding.

• Start by determining the function's form: quadratic, exponential, or otherwise. In our exercise, recognizing it as a quadratic function is essential.
• Examine coefficients and terms: they can tell you about the parabola’s direction (upward if positive, downward if negative) and the roots or zero points.
• Understand the impacts of each part of the function: the discriminant in quadratics, \(b^2 - 4ac\), tells us about the nature of the roots.

The examination of the function provides insights into graph behavior, such as opening direction and intersections with axes. Be curious about each component of the function since they all add layers to understanding how the function "behaves" over its domain. Function analysis is a vital tool for decoding a function’s essential features and simulating its progression over time or across inputs.