A polynomial function has a root of negative 6 with multiplicity 3 and a root of 2 with multiplicity 4 If the function has a negative leading coefficient and is of odd degree, which could be the graph of the function?
Answer
A
Image with description A Cartesian coordinate plane with horizontal x-axis and vertical y-axis, both grid-lined and ranges from negative 8 to 8 with an increment of 1. The curve begins at point arrow near "negative 7 comma negative 8" and extends upward from the lower-left quadrant. It rises sharply, crossing the x-axis at "negative 6 comma 0", and reaches a local maximum near "negative 3 point 5 comma 6". From there, it moves downward through the second quadrant, crossing the y-axis at "0 comma 2", and enters the first quadrant. It reaches a local minimum tangent to the x-axis at "2 comma 0", and then curves steeply upwards, terminating with an upward-pointing arrow near "5 comma 8".
B
Image with description On a coordinate plane with x and y axes marked from negative 8 to 8, a polynomial graph is drawn. It starts from quadrant 2, passes through (negative 8 comma 8), crosses the x axis at (negative 6 comma 0), moves down and curves up at (negative 3 point 5 comma negative 3), and crosses the y axis at (0 comma negative 1). It then touches the x axis at (2 comma 0) and moves to quadrant 4 passing through (6 comma negative 8).
C
Image with description A Cartesian coordinate plane with horizontal x-axis and vertical y-axis, both grid-lined and ranges from negative 8 to 8 with an increment of 1. The curve begins in the third quadrant near "negative 8 comma negative 1 point 5" with a downward-pointing arrow. Moving right, it rises into the second quadrant to a local maximum tangent to the x-axis at "negative 6 comma 0". It then decreases gradually to a local minimum in the third quadrant near "negative 1 comma negative 3". From there, it curves upward, crossing the y-axis at "0 comma negative 2 point 8", passing through the x-axis at "2 comma 0, and rising steeply into the first quadrant, where it terminates with an upward-pointing arrow near "4 comma 8".
D
Image with description A Cartesian coordinate plane with horizontal x-axis and vertical y-axis, both grid-lined and ranges from negative 8 to 8 with an increment of 1. The enters from the upper-left quadrant near "negative 8 comma 1" with an arrowhead pointing upward and to the left. It drops slightly to a local minimum tangent to the x-axis at "negative 6 comma 0", then rises gradually through the second quadrant, crossing the y-axis at a local maximum near "0 comma 1 point 5". From there, it enters the first quadrant, crossing the x-axis at " 2 comma 0", and curves steeply downward into the fourth quadrant where it terminates with a downward-pointing arrowhead near "5 comma negative 8".
Explanation
A root with odd multiplicity 3 at x = -6 means the graph crosses the x-axis there, while the even multiplicity 4 at x = 2 means it touches the axis and turns around. Since the degree is odd and the leading coefficient is negative, the graph rises on the left and falls on the right. Choice B has both the correct behavior at the roots and the correct end behavior.
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