Determine whether the following statement makes sense or does not make sense, and explain your reasoning:
When I'm trying to determine end behavior, it's the coefficient of the leading term of a polynomial function that I should inspect.
A) This makes sense because the coefficient of the leading term tells us how the polynomial behaves on the left side of the graph and the right side of the graph.
B) This does not make sense because although the leading term tells us how the polynomial behaves on the right and left sides of the graph, it does not tell the complete story. We also need to know the degree of the polynomial.
A) is correct. "End behavior" is the right and left sides of the graph.
B) seems to be jibberish
I always thought that the "leading term" (highest exponent) and the degree were the same thing, anyway
B) This does not make sense because although the leading term tells us how the polynomial behaves on the right and left sides of the graph, it does not tell the complete story. We also need to know the degree of the polynomial.
When determining the end behavior of a polynomial function, both the coefficient of the leading term and the degree of the polynomial are important. The leading term is the term with the highest power, and its coefficient indicates the direction of the graph as x approaches positive or negative infinity. If the coefficient is positive, the graph will have an upward trend on both sides. If the coefficient is negative, the graph will have a downward trend on both sides.
However, the degree of the polynomial also plays a significant role. A polynomial can have an even or odd degree, which affects the behavior of the graph.
If the degree of the polynomial is even, such as 2, 4, or 6, the end behavior will be the same on both sides of the graph. If the leading term has a positive coefficient, the graph will increase without bound as x approaches positive or negative infinity. If the leading term has a negative coefficient, the graph will decrease without bound as x approaches positive or negative infinity.
On the other hand, if the degree of the polynomial is odd, such as 1, 3, or 5, the end behavior will be different on the left and right sides of the graph. If the leading term has a positive coefficient, the graph will increase without bound as x approaches positive infinity and decrease without bound as x approaches negative infinity. If the leading term has a negative coefficient, the graph will decrease without bound as x approaches positive infinity and increase without bound as x approaches negative infinity.
Therefore, the statement does not make sense because in addition to the coefficient of the leading term, the degree of the polynomial also needs to be taken into consideration when determining the end behavior of a polynomial function.