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
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)
. 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". Before finding the function for the tangent line, we will need to first find its "y" value, simply by substituting the value "-2.25" into the original function, f(x).The slope of the tangent line is also required to find out its linear function, and we can do this by substituting the value "-2.25" into the first derivative function, f'(x). Calculations for both of these are shown as below.)
(Now that we know the values for x, y, and m, we can simply substitute them into the linear function, and solve for "c", then we would have the function of the tangent line.)
Plug the point, (-2.25, 4.21875) into the function
(After finding out the function of the tangent line, draw this linear function with the original cubic function together into the graph calculator, then we can use this tool in the calculator to find out their intersecting point.)
Intersection = (1.5,0)
(Up to here so far, I have observed that the tangent ...
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(Now that we know the values for x, y, and m, we can simply substitute them into the linear function, and solve for "c", then we would have the function of the tangent line.)
Plug the point, (-2.25, 4.21875) into the function
(After finding out the function of the tangent line, draw this linear function with the original cubic function together into the graph calculator, then we can use this tool in the calculator to find out their intersecting point.)
Intersection = (1.5,0)
(Up to here so far, I have observed that the tangent line at the average of the two roots, -3 & -1.5 happen to intersect the cubic function at its third root, but I am not sure if this is always the case, so I will be following up the same steps but with other two combination of roots.)
2. Roots -1.5 & 1.5
Average of the two roots =
Plug the point, (0, -13.5) into the function
Intersection = (3,0)
3. Roots -3 & 1.5
Average of the two roots =
Plug the point, (-0.75, -7.59375) into the function
Intersection = (-1.5,0)
(From this example, I have found out that the tangent line always intersects the original function again at its third root, no matter which two roots I averaged to get the tangent point, so based on this observation, I assume that this is going to work with all other cubic functions. Therefore, I am going to test it out with a few more cubic functions to see if my hypothesis is true.)
Example 2-
Roots = 7, 3, -4
. 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
. 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 )
Functions with One Root
Example 1-
Root = 0
. Roots 0 & 0
Average of the two roots =
Plug the point, (0, 0) into the function
Intersection = (0,0)
Example 2-
Root = -2
. Roots -2 & -2
Average of the two roots =
Plug the point, (0, 0) into the function
Intersection = (-2,0)
Example 3-
Root = 6
. Roots 6 & 6
Average of the two roots =
Plug the point, (6, 0) into the function
Intersection = (6,0)
(With the three very different examples of one root functions, where the first one being the simplest cubic function, the second one being a horizontal translation of the first function, and the third being a negative function and translated both horizontally and vertically from the first one, I have found out that their tangent line always goes through the root of the original function, and since the functions have only one root after all, but with a multiplicity of three, my hypothesis have also proven this correct.)
Functions with Two Roots
Example 1-
Roots = -2, 2
. Roots -2 & 2
Average of the two roots =
Plug the point, (0, -8) into the function
Intersection = (-2,0)
2. Roots -2 & -2
Average of the two roots =
Plug the point, (-2, 0) into the function
Intersection = (2,0)
Example 2-
Roots = 3, -5
. Roots 3 & -5
Average of the two roots =
Plug the point, (-1, -64) into the function
Intersection = (3,0)
2. Roots 3 & 3
Average of the two roots =
Plug the point, (3, 0) into the function
Intersection = (-5,0)
(Two roots functions also always work similarly as the previous functions, which match my hypothesis quite well. I have noticed that when the tangent is at the average of the two same roots, since they are the same 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.
