Use an end behavior diagram, as shown below, to describe the end behavior of the graph of each polynomial function. ƒ(x)=7+2x-5x²-10x⁴

Verified step by step guidance

1. Identify the degree of the polynomial function. The given function is \(f(x) = 7 + 2x - 5x^2 - 10x^4\). The degree is the highest power of \(x\), which is 4 in this case.
2. Determine the leading term of the polynomial, which is the term with the highest degree. Here, the leading term is \(-10x^4\).
3. Analyze the leading coefficient and the degree to understand the end behavior. The leading coefficient is \(-10\), which is negative, and the degree 4 is even.
4. Recall the end behavior rules for polynomials: For even degree and negative leading coefficient, as \(x \to \infty\), \(f(x) \to -\infty\), and as \(x \to -\infty\), \(f(x) \to -\infty\).
5. Use this information to draw or describe the end behavior diagram, showing both ends of the graph going downwards toward negative infinity.

Key Concepts

Polynomial Degree and Leading Term

The degree of a polynomial is the highest power of the variable in the expression, and the leading term is the term with this highest power. The degree and leading coefficient determine the general shape and end behavior of the polynomial's graph.

End Behavior of Polynomial Functions

End behavior describes how the values of a polynomial function behave as x approaches positive or negative infinity. It depends primarily on the degree and leading coefficient: even-degree polynomials with positive leading coefficients rise on both ends, while odd-degree polynomials have opposite end behaviors.

Using End Behavior Diagrams

End behavior diagrams visually represent the direction of the graph's ends, typically using arrows pointing up or down on the left and right sides. These diagrams help quickly identify how the polynomial behaves for large positive and negative x-values.