Problem 33 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the Leading Coefficient Test to describe the right-hand and left-hand behavior of the graph of the polynomial function. Use a graphing utility to verify your results. \(f(x)=\frac{6 x^{5}-2 x^{4}+4 x^{2}-5 x}{3}\)

Step-by-Step Solution

Answer

The polynomial function \(f(x)=\frac{6 x^{5}-2 x^{4}+4 x^{2}-5 x}{3}\) will fall to the left and rise to the right. This was confirmed by using a graphing utility to visually display the end behavior of the function.

Step 1: Identify leading term and its coefficient

The polynomial function is \(f(x)=\frac{6 x^{5}-2 x^{4}+4 x^{2}-5 x}{3}\). The leading term, having the highest degree, is \(6x^5\), and its leading coefficient is 6.

Step 2: Determine degree and leading coefficient

The highest exponent in the polynomial function, which is the degree of the polynomial, is 5 which is an odd number. The leading coefficient is positive, which is 6.

Step 3: Apply leading coefficient test

The leading coefficient test states that if the degree is odd and the leading coefficient is positive, the graph will fall to the left and rise to the right. Therefore, as x approaches negative infinity, \(f(x)\) will approach positive infinity, and as x approaches positive infinity, \(f(x)\) will also approach positive infinity.

Step 4: Verify results using a graphing utility

By plotting the function on a graphing tool, the behavior of the function can be visually confirmed. The graph of the function \(f(x)=\frac{6 x^{5}-2 x^{4}+4 x^{2}-5 x}{3}\) will show that as x moves to the left, the y-values are decreasing (falling right to left). And as x moves the right, the y-values are increasing (rising left to right), thus validating our results.

Key Concepts

Polynomial Function

A polynomial function is an expression made up of variables and coefficients, where the variables are raised to whole number powers and combined using addition, subtraction, and multiplication. Polynomials are considered to be very smooth functions because their graphs are continuous with no breaks or sharp corners. Some key components of a polynomial include:

• Terms: Parts of the polynomial separated by addition or subtraction.
• Degree: The highest power of the variable in the polynomial.
• Coefficients: The numbers in front of each term.

In our example, the polynomial function is given by \[ f(x) = \frac{6x^{5} - 2x^{4} + 4x^{2} - 5x}{3} \]This consists of four terms, with the leading term being \(6x^5\). The overall expression is a fraction, which can be handled by considering each term of the polynomial is divided by 3. This does not change its basic behavior as a polynomial.

Graphing Utility

A graphing utility is a tool, often a software or calculator, that helps us visualize mathematical functions and data. Using graphing utilities can greatly aid in understanding the behavior and nature of functions, especially for complex polynomials. When plotting functions:

• You can observe where the graph intersects the axes.
• Calculate important points like maxima, minima, and inflection points.
• Test your predictions about the function's behavior, such as end behavior.

In the exercise, a graphing utility helps confirm the behavior predicted by the Leading Coefficient Test. By plotting \( f(x) = \frac{6x^{5} - 2x^{4} + 4x^{2} - 5x}{3} \), it's easy to see the graph falling to the left and rising to the right, aligning perfectly with the theoretical prediction.

Degree of a Polynomial

The degree of a polynomial is an important descriptor that tells us about the function's characteristics. It is defined as the highest power of the variable in the polynomial expression. The degree provides insights into the general shape of a polynomial's graph and the number of roots it might have. Key points about polynomial degree include:

• A polynomial of degree \( n \) can have up to \( n \) roots and \( n-1 \) turning points.
• The degree of the polynomial also indicates the curve's end behavior—whether both ends go in the same or opposite directions.

In our function \( f(x) = \frac{6x^{5} - 2x^{4} + 4x^{2} - 5x}{3} \), the highest power is 5. This tells us it is a degree 5 polynomial, suggesting that graphically, it might have up to 5 intersections with the x-axis and 4 turning points.

Leading Coefficient

The leading coefficient of a polynomial is the coefficient of the term with the highest degree. This coefficient plays a critical role in determining the graph's end behavior. The leading coefficient, in combination with the degree of the polynomial, can predict how the function behaves as the input values become extremely large or small. Here's how it works:

• If the degree is odd and the leading coefficient is positive, the graph will fall to the left and rise to the right.
• If the degree is odd and the leading coefficient is negative, the graph will rise to the left and fall to the right.
• If the degree is even, regardless of whether the coefficient is positive or negative, both ends of the graph will go in the same direction—up for a positive leading coefficient and down for a negative one.

In our polynomial, \( f(x) = \frac{6x^{5} - 2x^{4} + 4x^{2} - 5x}{3} \), the leading term is \( 6x^5 \), making the leading coefficient 6. Since the degree is odd, and the leading coefficient is positive, as predicted by the Leading Coefficient Test, the graph falls to the left and rises to the right.