In Exercises 19–24, use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. $f\left(x\right)=-11x^4-6x^2+x+3$

Verified step by step guidance

1. Identify the leading term of the polynomial function. The leading term is the term with the highest power of x, which in this case is $-11x^4$.
2. Determine the degree of the polynomial, which is the exponent of the leading term. Here, the degree is 4, an even number.
3. Look at the leading coefficient, which is the coefficient of the leading term. In this function, the leading coefficient is -11, a negative number.
4. Apply the Leading Coefficient Test: For an even degree polynomial, if the leading coefficient is negative, the end behavior of the graph is that both ends go down as x approaches positive and negative infinity.
5. Summarize the end behavior: As $x\to \infty$, $f\left(x\right)\to -\infty$, and as $x\to -\infty$, $f\left(x\right)\to -\infty$.

Key Concepts

Leading Coefficient Test

The Leading Coefficient Test uses the degree and leading coefficient of a polynomial to determine its end behavior. For large positive or negative x-values, the sign and degree dictate whether the graph rises or falls on each end. This test simplifies predicting the graph's behavior without plotting points.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the expression. It indicates the general shape and number of turning points of the graph. Even-degree polynomials have similar end behaviors on both sides, while odd-degree polynomials have opposite end behaviors.

Leading Coefficient

The leading coefficient is the coefficient of the term with the highest degree in a polynomial. Its sign (positive or negative) affects the direction of the graph's ends. A positive leading coefficient causes the graph to rise on the right end, while a negative one causes it to fall.