Math: Week 1
DISCLAIMER: All solutions that you will be seeing below might not be necessarily right as I didn’t particularly double-check all of them, yet. With that said, “Yes, I am aware of my answers.” :>
[Not edited nor proofread]
- Derive an equation for the parabola in case 2.
a. What are the coordinates of F?
b. What is the equation for the directrix?
c. Let A(x,y) be a point on the parabola. What point on the directrix is needed to determine the distance of A from the directrix? Label this point B and identify its coordinates.
d. Set-up an equation using the fact that A is equidistant to the focus and the directrix.
e. Simplify the equation in the previous step to complete the derivation.
2. Write down the similarities and differences and come up with a way of identifying which equation goes with which parabola.
One similarity between the four parabolas is the fact that the focus inside the bound area of the parabola and the directrix is outside (2p units away from the focus). A difference: the direction of the openings for each parabola, which depend on the signum of p.
My way of identifying which equation goes to which parabola is first remembering the standard equation of a parabola I learned in Grade 8, which is…
y = x²
…and that this is opening up. So, when x² is seen in the equation, I’d assume that its axis of symmetry would then be vertical; therefore, x² = 4py is opening up, unless p is negative (which would then open down). This principle applies the same for the parabolas y² = 4px and y² = -4px, but the directions are either right or left.
3. Identify the opening (up, down, left or right) of the given parabola: (italicized text are my answers)
a. y² = -12x →left
b. x² = -8y → down
c. x² -1/4y = 0 → up
d. -x² = -12y → up
4. Give an equation for each of the parabola described below. Assume the vertex is at the origin.
a. directrix is y = -4 → x² = 16y
b. focus is (-3,0) → y² = -12x
c. opens to the right with the focus is 8 units away from the directrix → y² = 12x
d. opens downward with the directrix 6 units away from the vertex → x² = -20y
- Find an expression in terms of p for the length of the latus rectum of any parabola.
a. If the distance of the focus from the directrix is |2p|, and considering that the latus rectum is a chord through the focus that is parallel to the directrix, what is the distance of an endpoint of a latus rectum from the directrix?
b. The endpoints of a latus rectum are on the parabola. Thus, the distance of an endpoint to the directrix must be equal to the distance of this endpoint to the focus. Using your answer in the previous question, what is the distance of an endpoint of a latus rectum to the focus?
c. Since the focus is the midpoint of the latus rectum by symmetry, what is the length of the latus rectum?
2. Give an equation in standard form for the following parabolas. If more than one parabola satisfies the description, give an equation for all parabolas. (okay, Jvie note: I wasn’t sure what to do with the “more than one parabolas” thing, so I only included one equation by the time I made this. I might add the others later, if there are some.)
a. Directrix is x =2 and focus is (10, -5)
b. Vertex is (-3, 5) and directrix is x-axis
c. Axis of symmetry is y = -1 and (1,3) is an endpoint of the latus rectum.
- Identify the equation (in standard form) of the parabola with focus at (-1,1) and directrix y = -5. Sketch the graph and give the vertex, axis of symmetry, and the length and endpoints of the latus rectum of this parabola.
2. Identify the orientation, directrix, vertex, focus, the axis of symmetry, and length and endpoints of the latus rectum of the parabola whose equation is 4x + 20 + y² = 8y.
- Sketch the graph of the parabola (y-3)² = x-2 and then use this to sketch the graph of y = 3 + sqrt(x-2) and y = 3-sqrt(x-2) on separate Cartesian coordinate systems.
2. Give the vertex of the parabola whose focus is (-5,8) and whose directrix is x = 3.
3. Give the focus of the parabola x²+2x-4y+21=0.
4. The latus rectum of a parabola with focus at the (-2,3) is 20 units. If the parabola opens upward, what is an equation (in standard form) for the parabola?
5. Give an equation (in standard form) for all parabolas with the axis of symmetry of x = -2, vertex on the line y = 2, and a latus rectum of length 28 units.
So, for this week, I read and did the exercises of Modules 1–3: Parabolas.
Learning about parabolas are important because these can be applied in a variety of real-life projects such as satellite dishes, or maybe bridges. They’re also very helpful in focusing light and radio waves. Fountain designing, too, is a good application of parabolas.
I recall this shape very much because I have worked on this back in Grade 8. Ma’am Freda was always particular in plotting the points for the graphs, but I enjoyed it thoroughly back then. Looking back on this now, I realized how much more there are to parabolas. It’s not just about solving for points and plotting them one by one on the plane.
As for the exercises that I did, the only patterns I was able to distinguish are my ways of distinguishing which parabola equation goes with which shape as I mentioned above. And it’s mentioned in the module, but it helped that I knew about how the focus should be inside the area of the parabola, while the directrix is outside the area. (Just like how a really thin stick would like to envelop a small ant in its arms).
Well, I think I understood the lessons pretty well. So far, so good. But seeing how I struggled a bit with the questions for the quiz, I’d be better off practicing more, especially with confusing problems or problems with a bit of trick in them. So, that’s what I’ll be doing next. Practicing and practicing more (whenever I would have the time to do so). Personally, Math is always about practicing and mastery while understanding the logic behind it. So, yeah.
On to the next modules!!
