Exploring the End Behavior of Polynomials

As Emma studies the graph of a polynomial, she notices that as x approaches infinity, the graph climbs higher and higher. She realizes that this suggests a positive end behavior. What can we determine about the leading coefficient and the degree of this polynomial?

Question: What is the end behavior of a polynomial function with a leading coefficient greater than 0 and an even degree?

Answer: As x approaches positive or negative infinity, the polynomial approaches positive infinity.

Jake is analyzing a polynomial that has an odd degree and a negative leading coefficient. He recalls the important traits of the end behavior of this polynomial. What would Jake expect the graph to do as x approaches positive and negative infinity?

Question: What is the end behavior of a polynomial function with a leading coefficient less than 0 and an odd degree?

Answer: As x approaches positive infinity, the polynomial approaches negative infinity, and as x approaches negative infinity, the polynomial approaches positive infinity.

In a math competition, Mia grapples with drawing the graph of a degree 4 polynomial with a leading coefficient of -3. She remembers the significance of end behavior in her sketch.

Question: What can Mia conclude about the end behavior of her degree 4 polynomial?

Answer: As x approaches positive or negative infinity, the polynomial approaches negative infinity.

At the end of the school year, a group of seniors is celebrating their graduation. Aaron creates a polynomial that goes through the x-axis at multiple points. He writes a polynomial function of degree 5 with a positive leading coefficient. How does the function behave at extreme x values?

Question: What is the end behavior of a degree 5 polynomial function with a positive leading coefficient as x approaches positive and negative infinity?

Answer: As x approaches positive infinity, the polynomial approaches positive infinity; as x approaches negative infinity, the polynomial approaches negative infinity.

During her final project, Leah compiles data onto a polynomial graph. She encounters another polynomial with a leading coefficient of 2 and an odd degree of 3. What does this tell her about the behavior of the graph at the ends?

Question: What is the end behavior of a polynomial function with an odd degree and a positive leading coefficient?

Answer: As x approaches positive infinity, the polynomial approaches positive infinity, and as x approaches negative infinity, the polynomial approaches negative infinity.

Carter's math tutor gave him a polynomial function with a degree of 6, and he is curious about its end behavior. He learns that the leading coefficient is -1. What is the expected behavior of this polynomial when x is at extreme values?

Question: What is the end behavior of a degree 6 polynomial function with a leading coefficient of -1?

Answer: As x approaches positive infinity, the polynomial approaches negative infinity; as x approaches negative infinity, the polynomial also approaches negative infinity.

During math practice, Jenna wrote down a polynomial function which had a degree of 7 and a negative leading coefficient. She wanted to summarize what she learned about its end behavior. Can you help her?

Question: What is the end behavior for a degree 7 polynomial with a negative leading coefficient?

Answer: As x approaches positive infinity, the polynomial approaches negative infinity, and as x approaches negative infinity, the polynomial approaches positive infinity.

Tom was graphing polynomial functions for his math class and noted that one polynomial had an even degree and a positive leading coefficient. He wondered about its behavior as he extended the graph further left and right. What conclusion can he make?

Question: What is the end behavior of a polynomial function with a positive leading coefficient and an even degree?

Answer: As x approaches positive infinity, the polynomial approaches positive infinity; as x approaches negative infinity, the polynomial also approaches positive infinity.

During a group study session, Rachel explained to her classmates how the end behavior of polynomials changes based on degree and leading coefficient. She wrote down an example of a polynomial of degree 9 and confirmed its leading coefficient is 4. Can you predict its end behavior?

Question: What is the end behavior of a polynomial function with a degree of 9 and a positive leading coefficient?

Answer: As x approaches positive infinity, the polynomial approaches positive infinity; as x approaches negative infinity, the polynomial also approaches negative infinity.

As the school year flourished, David helped his friends with understanding end behavior in polynomials. He examined a polynomial of degree 8 with a leading coefficient of -2. What can David say about the end behavior of this function?

Question: What is the end behavior of a degree 8 polynomial function with a negative leading coefficient?

Answer: As x approaches positive infinity, the polynomial approaches negative infinity; as x approaches negative infinity, the polynomial also approaches negative infinity.