Problem 42 - College Algebra and Calculus: An Applied Approach

Question

Sketch the graph of the function and determine whether the function is even, odd, or neither. \(f(t)=-t^{4}\)

Step-by-Step Solution

Answer

The plot of the function \(f(t) = -t^{4}\) shows that the graph is reflected symmetrically around the y-axis. The function is determined as even because it satisfies the condition \(f(-t) = f(t)\) for every value of \(t\).

Step 1: Understanding the given function

Here, we are given a function, \(f(t) = -t^{4}\), which is a degree 4 polynomial. The negative sign changes the orientation of the graph, but does not affect whether the function is even or odd.

Step 2: Plotting the function

To graph the function, we can make a table of values for \(t\) and \(f(t)\) and plot the corresponding points. Because the degree of the polynomial is even and the leading coefficient is negative, the ends of the graph point downward.

Step 3: Determining the nature

To determine if a function is even, we replace \(t\) with \(-t\) and simplify the expression. If it is identical to the original function, then it's even. When we insert \(-t\) in place of \(t\) in the given function, we get \(-(-t)^4 = -t^4\), which mirrors the expression \(f(t)\). Hence, our function is even.

Key Concepts

Degree of polynomial

The **degree of a polynomial** is a fundamental concept that tells us the highest power of the variable in a polynomial expression. For example, the function \(f(t) = -t^4\) is a polynomial, and the degree of this polynomial is 4. This is because the highest exponent in the expression is 4. The degree of a polynomial can provide us with valuable information about the polynomial's characteristics, such as:

• The number of roots or solutions it can have.
• The general shape and behavior of its graph.
• The number of turning points it may exhibit.

For the polynomial \(f(t) = -t^4\), being of the fourth degree, it indicates that the graph will have up to three turning points. Also, since the degree is even, the graph has a general symmetrical shape similar at both ends.

Graphing polynomials

Graphing polynomials is a visual way to understand the behavior and properties of a polynomial function. When graphing the polynomial \(f(t) = -t^4\), we can follow these steps:

• Create a table of values to calculate corresponding \(f(t)\) values for selected \(t\) values.
• Plot the points on a coordinate plane based on these values.
• Connect the points smoothly, taking into account the polynomial's degree and leading coefficients to determine the end behavior.

For \(f(t) = -t^4\), since the leading coefficient is negative, the ends of the graph will point downwards. This is contrary to a positive leading coefficient where the ends would point upwards. Additionally, the symmetrical nature of the graph can be observed due to its even degree. This means points equidistant from the y-axis will have the same function value, reflecting across this axis.

Polynomial functions

A **polynomial function** is a mathematical expression involving a sum of powers of variables, each multiplied by coefficients. These functions can be classified based on their degree and coefficients. For example, \(f(t) = -t^4\) is a specific type of polynomial function known as a monomial because it contains a single term. Key characteristics of polynomial functions include:

• The possibility to classify them as even or odd based on symmetry properties.
• They can exhibit a wide range of shapes depending on their coefficients and degree.
• They have smooth, continuous graphs without breaks or holes.

In determining if a polynomial is even or odd, you can replace \(t\) with \(-t\) in the function. If the resulting expression equates to the original, it's even; if it equates to the negative, it's odd. For \(f(t) = -t^4\), substituting \(-t\) gives \(-(-t)^4 = -t^4\), confirming it as an even function, and showcasing the usefulness of polynomial functions in mathematical analysis.