Note: Your teacher will grade your responses to questions 20–24 to ensure you receive proper credit for your answers.
Explain how to determine the end behavior of a polynomial.
To determine the end behavior of a polynomial, we look at the degrees and leading coefficients of the polynomial's terms.
1. Start by identifying the polynomial's degree, which is the highest exponent of the variable in the polynomial. For example, in a polynomial of the form f(x) = ax^n + bx^(n-1) + ... + cx + d, the degree is n.
2. Next, determine whether the leading coefficient (the coefficient of the term with the highest degree) is positive or negative. This can be found by looking at the sign in front of the term. For example, in f(x) = -3x^4 + 2x^2 + 5x - 1, the leading coefficient is -3.
3. Now, consider the end behavior of the polynomial based on the degree and leading coefficient:
- If the degree of the polynomial is even and the leading coefficient is positive, then the end behavior will be positive infinity as x approaches both positive and negative infinity. This means that the graph of the polynomial will rise on both sides.
- If the degree of the polynomial is even and the leading coefficient is negative, then the end behavior will be negative infinity as x approaches both positive and negative infinity. This means that the graph of the polynomial will fall on both sides.
- If the degree of the polynomial is odd and the leading coefficient is positive, then the end behavior will be positive infinity as x approaches positive infinity and negative infinity as x approaches negative infinity. This means that the graph of the polynomial will rise on the right side and fall on the left side.
- If the degree of the polynomial is odd and the leading coefficient is negative, then the end behavior will be negative infinity as x approaches positive infinity and positive infinity as x approaches negative infinity. This means that the graph of the polynomial will fall on the right side and rise on the left side.
By considering the degree and leading coefficient of a polynomial, we can determine its end behavior and understand how its graph behaves at the far right and far left ends of the x-axis.