Question
Which description best matches the function shown: a) Degree four, with negative leading coefficient. b) Even degree with positive leading coefficient. c) Degree three, with negative leading coefficient. d) Odd degree with positive leading coefficient.
Answer
4.7
(306 Votes)
Jude
Professional · Tutor for 6 years
Answer
To determine which description best matches the function shown in the graph, we need to analyze the given points and the behavior of the function.Step 1: Analyze the behavior of the function.- The function initially decreases, then increases, followed by another decrease and increase. This indicates that the function has at least two turning points, which suggests that the function is of at least degree three.Step 2: Consider the given points.- The function passes through the points (-1, 0), (0, 1), (1, 0), and (2, 0). The fact that the function has three x-intercepts (also known as zeros or roots) suggests that the function could be of degree three or higher.Step 3: Determine the degree of the function.- Since the function has at least two turning points and three x-intercepts, the minimum degree it could be is three. However, the function could also be of a higher degree, such as four, if it has a double root at one of the x-intercepts.Step 4: Determine the leading coefficient.- The function initially decreases as it approaches from the left, which suggests that the leading coefficient is negative. If the leading coefficient were positive, the function would initially increase as it approaches from the left.Step 5: Determine the parity of the degree (even or odd).- The function does not have the same behavior as x approaches positive and negative infinity. If the function had an even degree with a negative leading coefficient, it would go down on both ends. However, since the function initially decreases and then increases, it suggests that the function has an odd degree.Step 6: Combine the information.- The function has an odd degree and a negative leading coefficient. This rules out options b) and d), which suggest an even degree and a positive leading coefficient, respectively.Step 7: Choose between the remaining options.- Option a) suggests a degree four function with a negative leading coefficient. However, a degree four polynomial would typically have up to three turning points, and the behavior at infinity would be the same on both ends, which is not consistent with the description of the graph.- Option c) suggests a degree three function with a negative leading coefficient. This matches the behavior of the function, which has two turning points and different end behaviors.Final Answer:The description that best matches the function shown is c) Degree three, with negative leading coefficient.