Problem 23 - Algebra and Trigonometry

Question

Use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. \(f(x)=-5 x^{4}+7 x^{2}-x+9\)

Step-by-Step Solution

Answer

The end behavior of the function is: as \(x\) approaches negative infinity, the function approaches positive infinity; as \(x\) approaches positive infinity, the function also approaches positive infinity.

Step 1: Identify the Leading Term

The leading term is the term with the highest degree in the polynomial function. For the function \(f(x)=-5 x^{4}+7 x^{2}-x+9\), the leading term is \(-5x^4\). The coefficient of the leading term is -5 and the degree is 4.

Step 2: Apply The Leading Coefficient Test

The Leading Coefficient Test states that if the degree of the polynomial is even, the ends of the graph are going in the same directions. If the leading coefficient of that term is positive, the direction is up; if it's negative, the direction is down. In this case, the degree is 4, which is even, and the leading coefficient is -5, which is negative. Therefore, the graph will go down as \(x\) goes to both positive and negative infinity.

Step 3: Formulate End Behavior

Based on the Leading Coefficient Test, we can describe the end behavior of the function using limit notation. As \(x\) approaches negative infinity (\(x \to -\infty\)), the function \(f(x)\) goes to positive infinity (\(f(x) \to +\infty\)). As \(x\) approaches positive infinity (\(x \to +\infty\)), the function \(f(x)\) goes to positive infinity (\(f(x) \to +\infty\)).

Key Concepts

End Behavior of Polynomial Functions

The end behavior of a polynomial function refers to the direction in which the graph of the function extends as the input, or the independent variable, approaches extremely large positive or negative values. In simpler terms, it deals with what happens to the values of the function as we look further and further towards the left or right side of a graph. This behavior is determined primarily by examining the leading term of the polynomial function. The leading term dictates how the polynomial behaves at these extreme ends because it grows at the fastest rate compared to the other terms as the variable gets very large or very small. Here are two key rules to consider:

• If the degree of the polynomial is even, the graph's ends tend to move in the same direction.
• If the degree of the polynomial is odd, the ends move in opposite directions.

Furthermore, whether these ends go up or down depends on the sign of the leading coefficient:

• A positive leading coefficient means the graph rises (moves up) toward both ends for even degree, or to the right for odd degree.
• A negative leading coefficient means the graph falls (moves down) toward both ends for even degree, or to the right for odd degree.

Understanding these rules helps you quickly predict how the graph will behave at the extremes without graphing it entirely.

Degree of a Polynomial

The degree of a polynomial is a crucial characteristic, as it determines the fundamental shape of the graph and the complexity of the function. The degree is the highest power of the variable present in the polynomial expression. Identifying it is simple:

• Look at each term in the polynomial.
• Notice the exponents on the variable parts.
• The largest of these exponents is the degree of the polynomial.

For instance, in the function \[f(x) = -5x^4 + 7x^2 - x + 9\], we can see that the term with the highest exponent is \(-5x^4\). Thus, the degree of this polynomial is 4.The degree determines several properties of the polynomial:

• The maximum number of turning points the graph can have is the degree minus one.
• The number of roots or solutions the polynomial can have is equal to the degree.

It also plays a crucial role in determining the end behavior, as mentioned earlier. Knowing the degree helps you set expectations for the graph's shape.

Leading Term of a Polynomial

The leading term in a polynomial is the term with the highest degree. Its role is central because it dominates the behavior of the polynomial as the variable becomes very large or very small. To identify the leading term:

• List all the terms in the polynomial.
• Find the term with the highest power of the variable.
• This is your leading term.

In our example function \[f(x) = -5x^4 + 7x^2 - x + 9\], the leading term is \(-5x^4\).The leading coefficient is the coefficient attached to this leading term, which in this case is \(-5\). This coefficient affects the end behavior of the polynomial, as the sign of this coefficient (positive or negative) determines whether the graph rises or falls as the input grows large in magnitude. So, in our function, the fact that the leading coefficient is \(-5\) (negative) means the graph will fall as it approaches positive and negative infinity. Recognizing the leading term is essential in predicting the overall shape of the polynomial graph.