Problem 8 - College Algebra with Corequisite Support

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 an exponential function; it's a quadratic function.

Step 1: Identify the Form of the Function

First, let's identify the function given: \( h(t) = -4.9 t^2 + 18 t + 40 \). An exponential function generally has the form \( f(t) = a \cdot b^t \), where \(a\) and \(b\) are constants and \(b > 0\). In an exponential function, the variable is in the exponent. Let's compare this form with the given function.

Step 2: Examine for Exponential Characteristics

The function \( h(t) = -4.9 t^2 + 18 t + 40 \) is a quadratic function because the highest power of \(t\) is 2 (i.e., \(t^2\)). In exponential functions, the variable \(t\) would be in the exponent, such as \(a \cdot b^t\). Since in this function the variable is not in the exponent but rather in the base raised to a power of 2, it does not fit the characteristics of an exponential function.

Step 3: Conclusion on Exponential Representation

Since the function \( h(t) = -4.9 t^2 + 18 t + 40 \) does not have the variable in the exponent, it does not represent an exponential function. Instead, it is a polynomial function of degree 2, specifically a quadratic function representing a parabolic path of a projectile.

Key Concepts

Exponential Characteristics

Exponential functions are a unique category of functions with distinct characteristics. In an exponential function, the variable is found in the exponent of the equation. The general form of an exponential function is \( f(t) = a \cdot b^t \), where \(a\) and \(b\) are constants, and \(b > 0\). The base \(b\) determines the growth or decay rate of the function.
For any exponential function, you'll notice:

• The graph has a constant percentage rate of change.
• The function's value will never be zero. It gets closer and closer to zero but never quite reaches it, which is known as an asymptote.
• In growth, as \(t\) increases, \(f(t)\) rapidly increases.
• In decay, as \(t\) increases, \(f(t)\) rapidly decreases.

Understanding these characteristics can help differentiate exponential functions from other types, such as polynomial and quadratic functions, where the variable is not in the exponent.

Quadratic Functions

Quadratic functions are a subtype of polynomial functions, typically having the form \( f(t) = at^2 + bt + c \), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). The characteristic feature here is the squared term \(t^2\), which signifies that the function will produce a parabolic graph, usually either a U-shaped or an inverted U-shaped curve.
Some important properties of quadratic functions include:

• The highest degree (power) of the variable is 2.
• The graph of a quadratic function is a parabola.
• Quadratics have a vertex, which is the highest or lowest point on the graph depending on the sign of \(a\).
• They have a line of symmetry which passes through the vertex.

Quadratic functions are used extensively in physics to model the path of projectiles, as seen in the function \(h(t) = -4.9t^2 + 18t + 40\). This represents the height of a projectile over time, manifesting its parabolic, quadratic nature.

Polynomial Functions

Polynomial functions encompass a broad category of functions that include variables raised to whole number powers and coefficients. The general form is \( P(t) = a_n t^n + a_{n-1} t^{n-1} + \ldots + a_1 t + a_0 \), where each \(a_i\) signifies a coefficient and \(n\) is a non-negative integer representing the degree of the polynomial.
Key aspects of polynomial functions include:

• The degree of the polynomial is the highest power of the variable in the expression, which influences the graph's shape and behavior.
• Polynomials of degree two are known as quadratic functions.
• Higher-degree polynomials can have more complex graphs with multiple turning points.
• They are continuous and smooth, with no sharp corners or breaks.

The function \(h(t) = -4.9t^2 + 18t + 40\) is a polynomial function of degree 2, highlighting its quadratic properties as it describes a projectile's path. Understanding polynomial functions enables the analysis of a variety of practical scenarios where such curves are applicable.