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S4w6 FC Intro to conic sections

Last updated 8 months ago

11 questions

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Question 1

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Here is the video we watched part of in class, let me know if you have any questions, you can skip to minute 9 something, which is where we left off.

Question 2

2.

go to https://www.geogebra.org/m/hGvXW3Dy. play with it.

Question 3

3.

Remember at the beginning of the year I told you that I thought that precalculus should have the name that my precalc class was called - Introduction of Analysis. And we talked about finding the connections between different ways to approach math - algebraically, graphically, geometrically, and numerically. You can see that the proof above takes three of those approaches and proves they are the same. It also introduces us to a new type of math - the conic section.

explain the relationship between \beta and \alpha to determine whether you are getting a circle, elllipse, parabola, hyperbola, or simply two crossed lines (also known as a dengenerate hyperbola.

Question 4

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Question 5

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now mess with the position slider at the bottom between messing with the beta slider. When you get to the point where the parabola turns into an ellipse? when the ellipse turns into a circle? when the cutting plane gives you a point? when the cutting plane gives you a line?

Question 6

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Question 7

7.

Draggable item		Corresponding Item

Question 8

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Question 9

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one of the three main ways discussed to define an ellipse is a stretched circle. The equation for a unit circle is
for an ellipse,

Play with this desmos graph https://www.desmos.com/calculator/5icfipfjie that gives you the equations to find the foci and the center of the ellipse. What are the semi-major axis and semi minor axis of an ellipse?

Question 10

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Why do I, on the desmos graph above, have four inputs with a label 'focus", but only ever have two foci on my graph?

Question 11

11.

do some research on hyperbolas. Create a desmos graph as shown above that shows me the parts of a hyperbola - the center, the vertices, and the foci. You need only make a hyperbola going side to side. This website may help. share a link to your desmos in the answer. https://courses.lumenlearning.com/waymakercollegealgebra/chapter/equations-of-hyperbolas/

look carefully at the hyperbola and the parabola. Does half a hyperbola look exactly like a parabola? do you think they are the same?

yes

no

we think of a cone like an ice cream cone, or a funnel. In mathematics, a cone is typically referring to the figure in this geogebra. Sometimes called a double cone, but is simply the shape you get when you take a line through the origin and rotate it around an axis. They are made up of the top nappe, and the bottom nappe. when cut by a plane, the plane sometimes only hit one nappe, sometimes you stop hitting the nappe, sometimes you hit both. how would you categorize these?

point

line

crossed lines

circle

ellipse

parabola

hyperbola

hits one nappe infinitely 

hits one nappe finitely

hits both napes infinitely

hits both napes finitely

Match the equation of the conic section to the relationship between the point on the graph and the focus/foci

The sum of the distances from any point to each of the foci is always the same

the distance from any point to the focus is always the same

The distance from any point to the focus is the same as the shortest distance from that point to a line called a directrix

the difference of the distances from any point to the two foci is always the same

These are the equations for conic sections in standard form. Which of the following are expressed by the ordered pair (h,k)

center

focus

foci

directrix

vertex