# Aim: To find out where the tangent lines at the average of any two roots intersect the curve again in cubic functions. Functions with Three Roots

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Introduction

Zeros of Cubic Functions Chun Yu Lai Aim: To find out where the tangent lines at the average of any two roots intersect the curve again in cubic functions. Functions with Three Roots *The first example is the function taken from the task sheet. There will be detailed explanation on my working steps, and the rest of the investigation will follow up the same format. Example 1- (The first derivative of the original function will allow me to find out the slope value, "m" at any given x value) Roots = -3, -1.5, 1.5 (I found out the roots of the function by using the graph calculator) 1. Roots -3 & -1.5 (Two roots are taken at a time to find out where the tangent line at the average of these two roots intersects the original cubic function again. Same steps will be taken, but with other combination pair of roots to prove that my observations apply to any random pair of roots) Average of the two roots = (Their average value, "-2.25" is the x value of the original function, and the task is simply asking for where the tangent line intersects the cubic function when the tangent point is at "x=-2.25". ...read more.

Middle

Example 2- Roots = 7, 3, -4 1. Roots 7 & 3 Average of the two roots = Plug the point, (5, -36) into the function Intersection = (-4,0) 2. Roots 7 & -4 Average of the two roots = Plug the point, (1.5, 45.375) into the function Intersection = (3,0) 3. Roots 3 & -4 Average of the two roots = Plug the point, (-0.5, 91.875) into the function Intersection = (7,0) (After my second example, I found out that my conjecture works perfectly well, since both examples I've worked with so far are positive functions; thus I think I should try it again with a negative function and see if my hypothesis still works) Example 3- Roots = -2, 1, 4 1. Roots 1 & 4 Average of the two roots = Plug the point, (2.5, 10.125) into the function Intersection = (-2,0) 2. Roots 1 & -2 Average of the two roots = Plug the point, (-0.5, -10.125) into the function Intersection = (4,0) 3. Roots -2 & 4 Average of the two roots = Plug the point, (1, 0) into the function Intersection = (1,0) (My hypothesis works perfectly well with all the cubic functions I have chosen so far, but they are all functions with three roots, thus before I can turn my hypothesis into a theory, I should test it with one root and two root functions ) ...read more.

Conclusion

root, which means that this root has a multiplicity of two, suggesting that the curve does not go across the x-axis, which in other words, this point is either the relative minimum or the relative maximum point of the function, which means that it would have a slope of zero, so the function of the tangent line would always "y=0", which is basically equivalent to the x-axis, and thus the intersecting point of this tangent line and the original cubic function is at its other root.) Proof: Create a cubic function where a, b, c are the three roots of this function. Let "a" and "b" the two roots that we are trying to find the average of. Equation of the tangent line If this tangent line intersects the original cubic function at its third root, then that third root has a "y" value of zero, thus by plugging 0 into the linear equation as follow: If the origin (0, 0) is not a solution of this cubic function Therefore, this tangent line intersects the original cubic function at point (c, 0), which is the third root of the cubic function. Conclusion: Tangent lines at the average of any two roots of any cubic function will always intersect the original cubic function at its third root. ...read more.

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