I didn’t make much progress getting further into Precalculus this week. I only just got through the unit Conic Sections on Sunday morning. I did get a much better handle on each type of conic section and their respective formulas, however. This is partly because it took me FOUR attempts to get through the unit test! 😡 As often happens, even though I didn’t make as much progress through KA as I’d hoped, I’m happy that I have a much stronger understanding of conic sections now. Being that I only managed to get through a few videos and exercises before getting into the unit test, this post is a bit shorter than usual.

The first and only thing I learned this week was about foci on hyperbolas. I watched two videos on how they work. The first video explained that the way foci relate to the hyperbola is that the absolute value of the difference between the distance from any point on the hyperbola to each foci equals 2a. Sal explained why this is the case in the video but at this point I don’t really understand why it’s true.

Sal also mentioned in the first video that the formula to calculate the distance the foci are from the origin is f = √(a^2 + b^2). The second video proved this formula with a lot of algebra which I was able to understand as he went through it but not well enough to reiterate here. 

One final thing I learned about hyperbolas (although I don’t really understand it either) is that the slope of the asymptotes is equal to (+/-) b^2/a^2:

I first attempted the unit test on Friday morning which was 14 questions long. I found it hard to remember the equations for circles, ellipses, and parabolas which was disappointing since I had just worked through them the week before. Although it took me longer than I would have liked to remember which formulas to use at what times, I was eventually able to recall them without having to look them up in my notes which I was happy about. Each of the three times I failed the test I actually understood the most difficult parts of the questions and got those parts correct but made simple, careless mistakes along the way. The first time I failed was because I inputted the (x, y) coordinates backwards as (2, 0) instead of (0, 2), and both the second and third time I failed was because I used the wrong (+/-) sign. As frustrating as it was to have to redo the unit test four times because of simple mistakes, it really helped me to memorize and understand the formulas for each conic section:

Circle

• Formula
  • (x – (h))^2 + (y – (k))^2 = r^2
    • (x, y) are any points on perimeter of circle.
    • (h, k) are center coordinates of circle.
    • r is the radius of the circle.
  • (Note: this is the same thing as saying a^2 + b^2 = c^2.)

Ellipse

• Formula
  • (x – (d_ox))^2/a^2 + (y – (d_oy))^2/b^2 = 1
    • (x, y) are any points on perimeter of ellipse. 
    • d_ox and d_oy are the distance on the x-axis and y-axis that the center of the ellipse is from the origin.
    • a and b are the radii along the x-axis and y-axis respectively.
• Foci on Ellipse
  • f = √(a^2 – b^2)
    • f is the distance to the focus.
    • a is the major radius (could be in either the x- or y-direction).
    • b is the minor radius.

Parabola

• Formula
  • √(y – (k))^2 = √((x – a)^2 + (y – b)^2)
    • (x, y) are any points on the parabola
    • k is the directrix.
    • (a, b) are the coordinates for the focus.
  • (Note: the equation can also be √(x – (k))^2) which would indicate that the parabola opens up in the positive or negative x-direction whereas √(y – (k))^2) indicates that the parabola opens up in the positive or negative y-direction.)

Hyperbola

• Formula
  • (x – (d_ox))^2/a^2 – (y – (d_oy))^2/b^2 = 1
    • (x, y) are any points on perimeter of hyperbola. 
    • d_ox and d_oy are the distance on the x-axis and y-axis that the center point of the hyperbola is from the origin.
    • a is the vertex of both arcs of the hyperbola from the center point of the hyperbola.
  • (Note: if the equation is (x^2… – y^2…) than the hyperbola opens up in the x-direction whereas of the equation is (y^2… – x^2…) than the hyperbola opens up in the y-direction.)
• Foci on Hyperbola
  • f = √(a^2 + b^2)
    • f is the distance to the focus.
    • a is the vertex of both arcs of the hyperbola from the center point of the hyperbola. Not sure what b is.

Although I still want to review all of the final unit in Precalculus, Probability and Combinatorics (800/800 M.P.), I still think it’s possible for me to get through this course by the end of this coming week. I’m looking forward to reviewing this final unit and am hoping that I’ll have a stronger grasp on permutations, combinations, and probability, in general, than I did when I went through it initially. As I’ve mentioned before, this is often the case so my fingers are crossed! 🤞🏼