A Parabola is a symmetrical open plane curve formed by the intersection of a cone with a plane parallel to its side. The path of a projectile under the influence of gravity ideally follows a curve of this shape. This is u
In this task I will investigate the patterns in the intersections of parabolas and the lines y = x and y = 2x. Forming a conjecture that holds true for the vertex of the parabola being in the first quadrant and then change it so it holds true for the vertex is in any quadrant. Then I will prove my conjectures for other lines like y = 3x and 4x and so on and I will also change the degree of the polynomials and their values to prove the conjuncture to be true for values greater than 3.
Using the dynamic graphing software GeoGebra construct the required graph
Graphical solution for the given equation is given below
(Fig 1.1)
1. ‘Find the four Intersections made my the parabola x2-6x+11 and the lines y=x and y=2x’
The co-ordinates for the intersections of the parabola and the two given lines are
Ans 1. To find the co-ordinates using technology graph the parabola and the two lines required, and note the points of intersection.
Alternatively solve the quadratic equation but substituting the value of y = x and y = 2x
Giving the two equations
To find Equation 1 - (x2 – 7x + 11 = 0)
Change (y = x2 – 6x + 11) to (x = x2 – 6x + 11) by substituting x by y and solve the quadratic equation. The solution for the equation one will give the points (x2, y2) and (x3, y3)
To find Equation 2 - (x2 – 8x + 11 = 0)
Change (y = x2 – 6x + 11) to (2x = x2 – 6x + 11) by substituting 2x by y and solve the quadratic equation. The solution for the equation one will give the points (x1, y1...
... middle of paper ...
...19 0.13 -0.17 0.30 0
The value for D turns out to be zero for all values of the cubic polynomials thereby preventing us to form a conjecture for D = 0
Therefore we can not find a conjecture like the conjecture for a second degree polynomial
6. ‘Consider weather the conjecture might be modified to include higher order polynomials’
Ans 6.
Using the dynamic graphing software GeoGebra construct the required graph.
Below the graph consist of the intersections of the line ‘y = x’ , ‘y = 2x’ with the curve y = x4
The graph has 4 points of intersections with the ‘y=x’ and ‘y=2x’.
Like the graph of x2 the graph of x4 provides four points of intersection.
Therefore we can use the formulae D = | SL - SR |
Bibliography
IGCSE Mathematics – Pimentel & Wall
IBDP Press Mathematics Higher Level [Core] – Nigel Buckle & Iain Dunbar
Upon completion of this task, the students will have photographs of different types of lines, the same lines reproduced on graph paper, the slope of the line, and the equation of the line. They will have at least one page of graphing paper for each line so they can make copies for their entire group and bind them together to use as a resource later in the unit.
on the y. If my prediction is right I should be able to draw a
...does increase, proving the fact that the parabola not only become more definite in shape as the distance increases, but so does the trajectory and height.
3. How small must the combination of F and X be to make this an
words the points all lie on a straight line that goes up from left to
Above is my original data. In the graph, it can be seen that there are
= 5cos Step 3: Substitute all values in the polar coordinate formula. [(dy)/dx] = [((dr)/(d theta) sin theta + r cos theta)/((dr)/(d theta)cos theta - r sin theta)]. =
Dielectric materials are insulators that make it harder for the electric field to penetrate the space within a capacitor; this is due to the theory of polarization. In Lab 4 (Parallel Plate Capacitor), the objective was to measure the dielectric constant (κ) of a textbook (paper) using a makeshift capacitor of aluminum foil. This was done through graphical analysis by the linearization of equation (1). The goal was to construct a linear graph in which the slope and slope error was calculated using the Linest function, the slope than allows for the derivation of the dielectric constant of the paper in a textbook. Error propagation (error formulas) was also used in this lab to account for sources of errors that could have occurred.
1. Describe the two methods of TRIZ that can be used to approach a problem.
gradient of this graph = - (Ea / RT) which can be used to calculate
In this portfolio task I have investigated the patterns in the intersection of parabolas and various lines. I have formed a conjecture to find the value of D of the parabolas, which are intersected by 2 lines, of varying slopes and shown the proof of its validity. I have used the TI-84 graphic display calculator, the software Geoegebra and Microsoft Excel to do my calculations.
The point of collision is the point where the graph of the ticker tape stop...
The four-color conjecture has been one of several unsolved mathematical problems. From 1852 to this day, practically every mathematician has studied the problem long and hard, but to no avail. The conjecture looks as though it has been solved by Wolfgang Haken and Kenneth Appel, both of the University of Illinois. They have used computer technology to prove the conjecture. The calculation itself goes on for about 1200 hours. The staggering length of the computation of the proof is what creates some controversy in the mathematical world. The Appel-Haken Theorem is based on numerous assumptions, “that there is an overwhelmingly great probability that their method of proof must succeed.” [3] It assumes that the theory itself is correct, but the theory itself is also an assumption. You can see why this issue has been wreaking havoc for many years.
The plural of focus is foci. The midpoint of the segment joining the foci is called the center of the ellipse. An ellipse has two axes of symmetry. The longer one is called the major axis, and the shorter one is called the minor axis. The two axes intersect at the center of the ellipseThe center of the ellipse is at (h, k). The radius of ellipses are not a constant distance from the center. To find the distance to the curve from the center you have to find the distance from the center to the curve for the x and y separately, these points are called vertices. The vertices are on the major axis and minor axis. The major axis is the longer axis and the minor axis is the shorter axis through the center of the ellipse. To find the distance from the center in the x direction you take the square root of a2. To find the distance from the center in the y direction you take the square root of b2. You then will have two points on the x direction and two points in the y direction and you use these four points to draw your ellipse. Ellipses are symmetrical across both of there
I started drawing a gun. No, I started drawing a pistol. The exact pistol from the movies I watched called, “Terminator.” I then drew a bullet coming out of the pistol. This was called a projectile. Though I did make the bullet in a form called a parabola. A parabola is a projectile in motion in which it starts and ends in an arch shape, like a cannon and its cannonball being fired out from its barrel, which leaves a “parabolic path.” At the end of where the bullet was going to land, I...