This week my studies focused on the vertex of the graph of a quadratic equations. The vertex of a parabola is the point in which the graph changes direction. Imagine the highest point of a roller coaster which you climb before plummeting downwards. That is the vertex. Now imagine a slope between two mountains, the vertex would be the point where you stop coming down one of the mountains and start coming up the other. We can find the vertex of a quadratic equation in standard form by transforming it to vertex form.
The way the vertex transformation works is the following: If you can get your quadratic equation in the form of y =a(x-h)² + k, remembering that (x-h) is equal to (x+(-h)), then (h,k) is the vertex of that equation. If a is positive, then the vertex is the minimum point the parabola reaches before going up, if a is negative then the vertex is the maximum point the parabola reaches before going down.
The vertex is an important point to know since if someone gives you the vertex and another point in the parabola, you can get the standard equation for it. For example, if I know the vertex of my parabola is (3,-4) and that point (2,-3) is in the parabola we can use algebra to get the equation of it.
y =a(x-h)² + k
y =a(x-3)² + (-4) substituting the given vertex into the equation.
-3 =a(2-3)² + (-4) substituting the (x,y) point given we the solve for a.
1=a or a=1
Therefore the equation of the parabola is (since a=1, we omit it):
y =(x-3)² - 4
Or (x-3)(x-3) -4
which gives us the standard equation x²-6x+5.
Now that is cool.
It seems most of my time while going through pre-calculus in order to get to calculus has been spent on quadratic equations. I did not remember there was so much to learn from them. Whenever I am sure I am done with parabolas something else comes up. Ad that something else is amazing to understand.
Right now when I look at x2-6x+5, I can:
Look at the standard form of that quadratic equation and tell at what point the graph passes the y axis (y intercept) and know whether the graph is concave up (like a cup) or concave down (like and umbrella)l
Solve quadratic equations using the quadratic formula and thus getting the points were the graph crosses the x axis (x intercepts).
Use the discriminant in the quadratic formula to know wether the equation has rational, irrational or imaginary solutions as well as the number of x intercepts.
Use factoring (when possible) to get the same results as with the quadratic equation.
Transform that standard form to the vertex form (y = a(x-h)² + k) to find the vertex or the point where the parabola changing direction (its maximum or minimum). I will talk about this form later in the post.
And I can graph it since I know these points:
Y intercept is 5 so (0,5)
2 X intercepts 1 and 5 so (1,0) and (5,0)
The vertex is (3,-4)
Let's graph it:
Notice I added a new point at the top right corner, (6,5). I knew that point should be there since a parabola is simetrical across its vertex. In other words, one side is a mirror image of the other. And since point (0,5) was on the left side, then point (6,5) must exist in the right. If you want to be sure, just plug 6 in the formula substituting it for x. I'll wait until you do so.
Going back to the vertex transformation, that transformation got me to think that someone had to discover this relationship between quadratic equations and their graphs through serious and hard study. I must remind myself that all these formulas, equivalencies, and transformations I am now given, where not always available. It is easy to forget that someone put a lot of effort coming up with these "shortcuts" we now use is class. I thank immensely all those mathematical scholars who made this learning possible.
If you liked this quadratic stuff as much as I did let me know.
Partly due to the movie in theaters at the moment of this writing, and partly because I have always wanted to, I ask for a copy of Les Miserables in audiobook form this past Christmas. It was an amazing unabridged version of 50 hours narrated by Frederick Davidson (Pseudonym). Sadly, after finishing it I went looking for the narrator to see if there was anyway I could contact him and let him know how he made this novel come alive with his voice, I found out he had passed away in 2005. If you have never read Les Miserables, I highly recommended. If the width of the book, at 1,000 plus pages, intimidates you, you are not alone. The book form of the novel is often referred to as "the brick". However, it will be one of the most transcendental books you will read in your entire life. Victor Hugo's tale of grace in the face of misery and love I the face of death will move you. I chose an audiobook form so I could listen to it in my daily commute, the best commute ever in the last couple of weeks.
But wait, this blog is about me learning calculus. Where does les Miserables fit in? Well, as I listened to the book I started to notice that Victor Hugo used many allusions to math, the scientific method, chemical properties to describe characters and situations. I specifically remember his calculated descriptions of different cannons and the parabolas the trajectory of their balls describes in order to let you know which was was better for urban warfare. He would also project his beliefs on knowledge and science as the "light" that dispels the shadows of superstition. Crudely put as per my understanding, the republic, science, mathematics, God as a duty towards fellow humans equaled light while monarchy, superstition, and religion as an excuse to marginalized fellow humans equaled shadow.
Upon further research, I found out Victor Hugo was an excellent student in literature and mathematics. I can nos understand the spirit of his thoughts. He was, in my opinion, a voice influenced by the age of enlightenment. A proponent of education as the savior of humanity. In fact the right for a free, accessible education for all children is repeated throughout the book as a novel and revolutionary idea. It was sobering to imagine a times in which that was not the case. In which ignorance was not a product of a system in need of repair like today, but it came with the wealth of your family. If you had no money, you got no formal education.
Just to think that the reason I am trying to learn calculus is because I felt I could have done better with the education the state provided to me free of charge from primary to high school, and at a very low cost in the public university of Puerto Rico. If I had lived in Victor Hugo's france, calculus could have been unattainable for me in my struggle to go on in life.
I finished my first week of pre-calculus by UCIrvine through Coursera. It was harder than I expected due to the sheer magnitude of the workload. Although it was algebra, exponent rules, radical, rules, polynomial operations and inequalities, the fact that it was 30 videos and quizzes made it labor intensive. However, in the end I grit my teeth and got throughout them.
As I mentioned in a past post, the way mathematical expressions needed to be input into the quizzes was giving me trouble. Yet, thanks to economies of experience, the more exercises I did the better I got, and the faster I was able to do them.
I felt and odd satisfaction when I finished my last quiz on the due date. I knew the first-week homework would not be part of the final grade so I could have very well skipped the exercises and done nothing, but I felt compelled to do it. To be fair, I re-learned (I wonder if that term is trademarked) many concepts that had been dormant since high school.
For example, the arithmetic commutative, associative and distributive properties are basic knowledge almost since elementary school. However, most of the questions I have when solving a problem have to do with these properties. And the fresher I have them in my mind the easier the problem becomes.
Other concepts I re-learned this week:
Roots can be expressed as exponential fractions and be subject to exponent rules.
When you divide or multiply both sides on an inequality by a negative number you need to flip the inequality symbol that joins them.
Factoring polynomials can help you understand the relationship they have with other polynomials.
The "two trains leave the station" type of problems have to do with time, distance and rate...not (necessarily) geography, thank heavens!
I find it thrilling to work with all this math again. As an adult, I have a different perspective on all of this. You see, for many mathematically inclined adults working doing algebra exercises from junior high would be a waste of their time. For them this math is just too simple to be worthwhile and they see no point. The way I see it, before this math became the stuff of junior high, very smart and serious adults were discovering and working with it. What we now find simple, was a breakthrough for scholars in other times. Therefore I am following the path these great minds paved...how's that for self delusion.
Let me know your thoughts on this in the comments.