Worksheet On Inflection
Y = -7(x- 2)(x + 5)(x - 1)(x+ 4)(x - 3)(x+4)
A 6th degree polynomial with six real distinct real linear factors has 6 roots, which the cuts the x-axis six times, has 5 turning points and 4 points of inflection as shown in this graph. 2, -5, 1, -4, 3, -4 5 4
Y= (x+7)(x- 4)(x-6)(x+2)(x - 6)(x+5)
A degree 6th polynomial has 6 roots, which the cuts the x-axis six times, 5 turning points and 4 points of inflection as shown in this graph. -7, 4, 6, -2, 6, -5 5 4
The conjecture of the first form of the degree 4 (polynomial), is proven correct because the polynomial of 6th degree evidences that the turning points is n -1 and the points of inflection is n – 2. It is unnecessary if the polynomials have two or more distinct real linear factors.
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Present the information in a table as above.
Function Roots used Number of turning points Number of points of inflection Graph
Y=1(x-2)^2 (x+1)(x-3)
A degree 4th polynomial with a squared real linear factor and two distinct real linear factors has 3 roots because the squared root is a repeated root, has 3 turning points and 2 points of inflection as shown in this graph. 2, -1, 3 3 2
Y=3(x+3)^2 (x+1)(x-2)
A degree 4th polynomial with a squared real linear factor and two distinct real linear factors has 3 roots, has 3 turning points and 2 points of inflection as shown in this graph. -3, -1, 2 3 2
Y=-2(x-6)^2 (x-2)(x+4)
A degree 4th polynomial with a squared real linear factor and two distinct real linear factors has 3 roots because a repeated root is not counted twice, has 3 turning points and 2 points of inflection as shown in this graph. 6, 2, -4 3
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