Problem 12 - Algebra 2

Question

State whether each equation or function is linear. Write yes or no. If no, explain your reasoning. \(f(x)=7 x^{5}+x-1\)

Step-by-Step Solution

Answer

No, because it is of degree 5, not 1.

Step 1: Identify the polynomial degree

A linear function or equation is typically of the form \( f(x) = mx + b \) where the greatest power of \( x \) is 1. For the given function \( f(x) = 7x^5 + x - 1 \), identify the degrees of each term. The highest degree term here is \( 7x^5 \), indicating degree 5.

Step 2: Determine linearity based on polynomial degree

Since linear functions have a degree of 1, check the degree of the polynomial in the given function. The function \( f(x) = 7x^5 + x - 1 \) has a highest degree of 5, which is not equal to 1.

Step 3: Conclude if the function is linear

A function with degree higher than 1 is not linear. Since the highest degree in \( f(x) = 7x^5 + x - 1 \) is 5, this function is not linear.

Key Concepts

Polynomial Degree

The degree of a polynomial is a key factor in understanding the nature of the function. The polynomial degree is defined as the highest power of the variable present in the expression. For example, for the polynomial \( f(x) = 7x^5 + x - 1 \), the term \( 7x^5 \) has the highest power of \( x \), which is 5. Hence, the degree of this polynomial is 5.

When examining polynomials, these key points can guide:

• Only consider terms with non-zero coefficients.
• The degree is indicative of the curve's behavior; higher degrees often result in more complex shapes.
• In a polynomial, linearity is observed when the highest degree is 1.

Understanding polynomial degree is critical when identifying the type of polynomial you are working with and predicting the general shape of its graph.

Function Linearity

A linear function is characterized by its constant rate of change, forming a straight line when graphed. The standard form is \( f(x) = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept. In this standard form, the degree of the polynomial is 1, meaning the highest exponent of the variable is 1.

Recognizing a linear function involves checking:

• If the equation can be expressed as a first-degree polynomial.
• The graph of the function will be a straight line.
• The change in the dependent variable is proportional to the change in the independent variable.

In the given function \( f(x) = 7x^5 + x - 1 \), since the highest degree is 5, it does not meet the criteria for linearity, confirming that it is not a linear function.

Non-linear Functions

Non-linear functions are abundant and encompass any polynomial that is not of degree 1. These functions can produce a variety of complex curves and shapes instead of a straight line. The function \( f(x) = 7x^5 + x - 1 \), for instance, is a fifth-degree polynomial function, indicating its non-linear nature.

Non-linear functions often have these characteristics:

• They exhibit variable rates of change.
• When graphed, they can form curves, waves, or other complex shapes.
• Higher degree polynomials can have multiple turning points, reflecting complex behavior.

Understanding non-linear functions and their behaviors unlock insights into complex systems and phenomena, which are vital in fields like physics, economics, and various engineering disciplines.