Problem 13 - A Graphical Approach to Precalculus with Limits

Question

Describe the end behavior of the graph of each function. Do not use a calculator. $$P(x)=2.74 x^{4}-3 x^{2}+x-2$$

Step-by-Step Solution

Answer

As \(x \to \infty\) or \(x \to -\infty\), \(P(x) \to \infty\). The graph rises on both ends.

Step 1: Identify the Leading Term

The leading term of a polynomial determines the end behavior of its graph. In the polynomial \(P(x) = 2.74x^4 - 3x^2 + x - 2\), the leading term is \(2.74x^4\).

Step 2: Determine the Degree and Leading Coefficient

The degree of the polynomial is 4, which is the highest power of \(x\) in the polynomial. The leading coefficient, which is the coefficient of the leading term, is 2.74.

Step 3: Analyze the End Behavior for Even Degree and Positive Leading Coefficient

For polynomials with an even degree and a positive leading coefficient, as \(x\) approaches infinity or negative infinity, the polynomial \(P(x)\) approaches positive infinity. Thus, the graph will rise to the left and rise to the right.

Key Concepts

Leading Term

The leading term of a polynomial is an essential component as it directly influences the end behavior of the polynomial's graph. In a polynomial, terms are arranged in descending order of their powers, which means the term with the highest power of the variable, typically denoted as \(a_n x^n\), is the leading term.

• If you look at the polynomial \(P(x) = 2.74x^4 - 3x^2 + x - 2\), the leading term is \(2.74x^4\).
• The leading term usually captures the most significant behavior as \(x\) becomes very large or very small.

Understanding this term helps us gauge how the polynomial will behave in extreme cases and is crucial for analyzing its graph.

Degree of Polynomial

The degree of a polynomial is determined by the highest power of the variable present in the polynomial expression. The degree plays a crucial role in dictating various characteristics of the polynomial's graph.

• For the polynomial \(P(x) = 2.74x^4 - 3x^2 + x - 2\), the highest power of \(x\) is 4, so the degree is 4.
• A polynomial with an even degree like this one suggests that the graph's end behavior will have both arms in the same vertical direction.
• A higher degree can also imply more complex graph shapes, with possibly more turning points.

Knowing the degree assists us in predicting how the graph behaves beyond just the immediate surroundings of any given \(x\) values.

Leading Coefficient

The leading coefficient of a polynomial is the coefficient of its leading term. It significantly determines the scale and direction of the polynomial's end behavior, particularly because of its sign (positive or negative).

• In our polynomial \(P(x) = 2.74x^4 - 3x^2 + x - 2\), the leading coefficient is \(2.74\).
• The positive sign of this coefficient reveals that regardless of the even or odd degree, as \(x\) moves toward positive or negative infinity, the polynomial will tend toward positive infinity.
• This results in the graph rising both to the left and right since the degree is even and the leading coefficient is positive.

Recognizing the leading coefficient helps in accurately predicting the overall "shape" of the polynomial's graph as \(x\) strays far from zero.