Problem 22 - College Algebra

Question

Determine the end behavior of the functions. $$f(x)=3 x^{2}+x-2$$

Step-by-Step Solution

Answer

The function rises on both ends.

Step 1: Identify the Leading Term

To determine the end behavior of a polynomial function, focus on the leading term as its influence becomes dominant for very large or very small values of \(x\). The leading term in \(f(x) = 3x^2 + x - 2\) is \(3x^2\).

Step 2: Analyze the Degree

The degree of the leading term \(3x^2\) is 2, which is an even number. For polynomial functions with an even degree, both ends of the graph will tend to rise or fall in the same direction.

Step 3: Determine the Leading Coefficient's Influence

The leading coefficient of \(3x^2\) is 3, which is positive. For functions where the leading coefficient is positive and the degree is even, the end behavior means the function will rise on both ends.

Step 4: State the End Behavior Conclusion

Based on the analysis, as \(x\) approaches infinity (\(+\infty\)), \(f(x)\) approaches infinity (\(+\infty\)). Similarly, as \(x\) approaches negative infinity (\(-\infty\)), \(f(x)\) also approaches infinity (\(+\infty\)).

Key Concepts

Polynomial Function

A polynomial function is a mathematical expression that involves variables raised to whole number powers, along with coefficients. These terms are combined using addition, subtraction, and sometimes multiplication. Polynomial functions come in various forms, but they always rely on the arrangement of terms with different degrees or powers of a variable like \(x\).

• Polynomial Example: \(f(x) = 3x^2 + x - 2\)
• Characteristics: Smooth curves, continuous lines, no sharp turns.
• Graphing: Helps in visualizing behavior such as intercepts and turning points.

Understanding polynomial functions is key to predicting how the graph will look and behave in the long term—this is referred to as the 'end behavior'. The end behavior is greatly influenced by the leading term.

Leading Term

When evaluating a polynomial function, the leading term is the first element that you should focus on. It is the term with the highest power of the variable. For instance, in the polynomial \(f(x) = 3x^2 + x - 2\), the leading term is \(3x^2\).
The leading term primarily determines the polynomial's end behavior because its influence becomes dominant as \(x\) becomes very large or very small.

• Importance: Dictates general shape of the polynomial's graph.
• Calculation: Look for the term with the highest exponent.

Knowing the leading term is essential in predicting whether the function will rise or fall as \(x\) increases or decreases towards infinity.

Even Degree

The degree of a polynomial function is the highest power of the variable in its expression. An even degree, like 2 in \(3x^2 + x - 2\), often leads to specific end behavior patterns.

• Symmetry: Even degree functions often exhibit symmetrical behavior in the ends of their graph.
• End Behavior: Both ends of the graph tend to either rise or fall together.

For even-degree polynomials, the ends of the graph mirror each other. Either both rise upwards or both fall, depending on other factors like the leading coefficient.

Leading Coefficient

The leading coefficient is the coefficient of the leading term in a polynomial. For the function \(f(x) = 3x^2 + x - 2\), the leading coefficient is 3.
A positive leading coefficient in an even-degree polynomial indicates that both ends of the graph will rise as they move towards infinity. Conversely, a negative leading coefficient would mean both ends fall.

• Influence: Affects the direction (rise/fall) of the end behavior.
• Determination: It's the number before the variable with the highest exponent in the polynomial.

Understanding the leading coefficient helps in quick visualization of how a polynomial behaves at extreme values of \(x\).