Angle Bisector
This is how to create an Angle Bisector, by using circles and intersections. You can prove that this construction works by ITT (with triangles ADE and AEF) and SAS ( with triangles ADF and AEF), these triangle theorems show Triangle ADF is congruent to Triangle AEF, and so by CPCTC, we know that the angles (alpha and beta) are the same. Showing that AF is indeed the bisector of angle A. |
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Parabola - M241 Worksheet
We create a parabola, using the definition: A parabola is a curve formed by the set of points in a plane that are all equally distant from both a given line (called the directrix) and a given point (called the focus) that is not on the line. Line AB is the directrix and Point C is the focus. By construction, point E is equidistant from point C and point D. Therefore the construction fits the definition of a parabola, which is why a curve is formed. |
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M241 - Relation between f(x), f'(x), and tangent.
The two intersection points show when f'(x) = f(x). The relationship between f(x) and f'(x), is that the derivative, f'(x), finds the slope of the function, which is used also for the tangent line. This can also show if the function is increasing or decreasing. The tangent line uses the slope the derivative function provides and the number provided by the slider created. Which then creates a line that touches the parabola at that one specific point along the curve, which changes as x0 changes. |
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Cosine and Sine Graphs
The construction for (t,sin(t)) and (t,cos(t)) works, because these functions come from the unit circle, we must first form a circle. The cos(x) lies on the y-axis, because cos(0) = 1, it gives us the point (1,0). Sin(x) however lies on the x-axis, because sin(pi/2)=1, which gives us the point (0,1). The construction remains the same because you still rely on the intersection with x-t which relies on the curve from the point of cosine or sine. Which allows us to find the points (t,sin(t)) and (t,cos(t)) because we have the proper placement of the points when x=0 and we have x=t (curve). |
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Carbon Dating Graph
The function f(x)=1000*(1/2)^(x/6000) is the formula to find the half life of a sample of carbon on planet frisbee. The initial sample has 1000 atoms with 10% of decaying carbon 14 molecules. On planet Frisbee molecules of carbon 14 has a half life of 6000 years. In this equation, the total amount of carbon in the atmosphere is labeled on the y-axis and time is labeled on the x-axis. In order to discover how much time has past since the initial amount, set the equation equal to the new sample size. The amount of samples you have and the carbon dating left, is found by the intersection of the c(x) function and line a, which gives you point A (time, sample). |
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