Thursday, December 26, 2013

Antiderivatives and when all starts to fit together

As I near the end marker of this journey of learning calculus I meet anti-derivatives. And I am glad I did. Somehow this concept helped me put into perspective all the concepts that have come before it and makes me feel I am back on track again. Just in case you were wondering where I lost my way, it was somewhere into L'hopital's Rule.

First things first: What is an Anti-derivative?

Well an anti-derivative answers the question of: what is this formula a derivative of? Or from what original formula could we have gotten this derivative. For example, if we have x2 (the squaring function), it's derivative is 2x. Therefore, x2 is an anti-derivative of 2x. (For formal definition go here)

Notice that I wrote that x2 is an anti-derivative of 2x. This is important because there can be many anti-derivatives for a given function. If we consider this formula for an anti-derivative: xn+1/n+1+C, (which looks a lot prettier in pictures, see bellow) there is a constant C that is introduced. The way I understand it is that there is only so much information a anti-derivative can give you. In order to recover a specific formula from it's derivative you need to know where that functions "started".

In the anti derivative formula, if x=0 the all we get is C. For example, imagine that x is time, then x=0 is time 0 or your starting point. At that starting point then f(0)=C.

To complete the argument above, let now imagine this scenario. I know that -2x+5 is a derivative of a function I am interested in knowing. Using the ati-derivative formula I get that the function I am interested in is:

But what is C? I have no clue with the information I was given. Let's say I know C is a whole number between 1 and 4. A graph can show me what to expect the graph to be.

But until I know what that constant actually is, I will not know the original formula.

In the picture opposite I have 4 possible graphs, each passing through the y (vertical axis) at 1,2,3 and 4. All 4 graphs are exact copies of the formula I am looking for, but they "start" at different points C.

For a great explanation about how C is relates to anti-differntiation in terms of position and velocity. Check out this video from my Mooculus course. I liked this video not only because Dr. Fowler seemed to have had had too much coffee, but because his explanation incorporates the steps to solve an anti-differentiation equation that has "physical" applications.

Anti-differentiation is an important bridge in my road to understand calculus. At least that is the promise that was made by my professor when he introduced the topic. Whether that is the case or not, I find it fascinating that having information about a function, I can derive other functions that are related, and give me additional information about the original one.

Let me know what you think in the comments.


Sunday, December 1, 2013

My understanding of the chain rule and 5 videos in case I blew it

A few weeks ago I was introduced to the chain rule in my Coursera/Mooculus course. And I found it to be one of those concepts that is straight forward to understand but hard to our into practice.

The chain rule deals with composition of functions. In other words, it deals with the derivative of functions within functions.

The best example I can think of to explain this is,the following: Imagine you are a salesperson whose job is to call customers, set up an appointment for a consult, give them a sales presentation and close a sale.

That sale depends on how many presentations are given, which are dependent of how many appointments were made, which are in turn dependent on how many calls were made. If you had a formula that described this process and you wanted to know how do a change in calls affect sales you might need to use the chain rule.

Let's imagine that such a formula exists. We will use the following variables for it: Sale (S), Presentation (P), Appointments (A), and Calls will be our (x). The formula is S(x)= P(A(x)), in order to know how changes in x affect S(x) we will need to differentiate the function with using the chain rule:

S'(x) = P'(A(x))(A'(x)) which is the same as saying we are taking the derivative of the outside function evaluated at the inside option and we multiply that with the derivative if the inside option.

Let's imagine than in the example above these formulas can be substituted for S(x)= √(3/4x). How will a change in x affect S(x)?

S'(x)= [1/2(3/4x)^-1/2](3/4) and simplifying we get:

S'(x)= 3


Let's imagine the sales person makes 100 calls a day according to the original formula he will get approximately 8.66 sales. What if the sales person makes 50 extra calls by what amount would sales change?

S'(150)= 3


According to the derivative sales would change .03535 times multiplied by the 50 extra calls which would yield approximately 1.76 extra sales. Now that's the way I understood it. But, since I might be wrong, here are five videos explaining the chain rule.

Professor Jim Fowler produced this amazing video explaining the concept. It is 10 minutes long but for someone strugling to understand what the chain rule is and how it works, its worth the time.


Here is the chain rule introduction by Khan Academy.

Chain rule introduction:


I liked this example from That Tutor Guy


Another straightforward example from


This one is from the IntegralCalc channel in YouTube


Reference for derivatives and limits

The hardest things  for me to find while studying are good reference sheets. Some people might object to their use and see them as crutches one uses instead of trying to understand the subjects studied. I see them as facilitators. I love to look at reference sheets for the big picture. There I can look for pattern and similarities between the concept studies. And instead of just relying on them, they make me want to explore more and go further.

 I include a Slideshare presentation with a reference sheet on derivatives and limits in the hopes that other feel as I do and can find it helpful. Let me know what you think in the comments. If you cannot see the slide, follow the link at the bottom of the page.

Words of Encouragment


On derivatives and rote learning

The derivative of a function gives us important information about the function being examined. That is a very cool idea. It is basically metadata about how the original function will behave. The derivative of a function, using a very basic explanation which is what I can manage with my knowledge, describes whether a function is increasing, decreasing, and/or changing direction, in other words, it gives you the rate of change.

The derivative also gives us the slope of the tangent line for any point along a function that is differentiable. In other words, if we know the derivative and a point along the function, we can calculate the tangent line for that point.

It bet there a lot of other things the derivative tells us, but these two are the ones that fascinated me the most.

The more I think about it the more it sinks in that calculus is about change. Or in the words of professor Jim Fowler from Mooculus, about how "wiggling the inputs affect the outputs".

Let's do an example: Imagine we have a toy rocket which we shoot up into the sky and watch it fall to the ground. A formula that describes the trajectory of the rocket is: y=-x^2+4x where x=time in seconds and y is feet. In one second it's 3 feet high, in 2 seconds it's 4 feet high. It's average climb from zero to 1second is 3 feet per second and from 1 to 2 seconds it's 1 feet per second. But what how fast was the rocket climbing at exactly 1 second?

The derivative can help us with that. Using the derivative rules:

The derivative of -x^2+4x is -2x+4. And once we know the derivative, and understand that the derivative is the slope of the tangent line at a given point, and remember that the derivative at a point gives us the instantaneous rate of change of a function, we can do some algebra to know how fast the rocket was climbing at exactly 1 second:

-2(1)+4= instantaneous speed

-2+4= 2 feet per second

So at 1 second the rocket is climbing 2 feet per second.

What happens at 2 seconds?


At 2 seconds the rocket is moving at 0 feet per second, therefore it's standing still.

And at 3 seconds?


Interesting, at 3 seconds the rocket is not climbing but declining at 2 feet per second.

We know the rocket hits the ground at 4 seconds. How fast was it going?


The rocket hit the ground declining at 8 feet per second.

When we look at the graph we can see these numbers make sense. The parabola starts steeply (the rocket is climbing fast but slowing down), then levels off to a point where there is no more climb (the rocket stands still), and then the parabola declines steeply, (the rocket falls accelerating until it reaches the ground). That was really cool. Now if we look at the graph formula of the derivative by itself, it would have told some key elements of the original function.

-2x+4 is a line. And this line is positive from time 0 to 2 seconds, at 2 seconds it crosses the y axis and is negative until 4 seconds. Looking at that graph and using the first derivative test, we can conclude that from time 0 to 2 seconds the original graph -x^2+4x is increasing in the interval, that at 2 seconds there is a local extreme value where the graph changes direction, and we can also see that from 2 to 4 second the graph is decreasing in the interval. What this means is the if I had the function of the derivative but not the original function I could make some intelligent guesses as to how the original function would behave. Now that to me is amazing.

Like I mentioned before, I have been taking a calculus course for about two months now. Therefore the basics of the derivative is something that has long passed. If fact, it took me a couple of hours to find an explanation of it that I could use in order to post it. This problem got me thinking about the depth of my learning. I know I already covered some of these insights in earlier posts about learning for testing and grades, versus learning to understand.

You see, I was feeling quite confident going week to week in my calculus course, at least at the beginning. That was because I was doing the minimum study required to pass the quizzes. And then, when the concepts started to stack up, I found myself lost. I fear this is what is wrong with the way we educate in Puerto Rico and the United States. If we are teaching to tests, then the students will study to pass those tests and might not explore what the knowledge in itself means. That might mean that the best product we could be creating is excellent test takers instead of excellent thinkers.


Thursday, November 28, 2013

The last 6 months

I have always found it amazing how time vanishes. Six months ago I was talking about continuity, and here I am today looking back in disbelief that so much time had passed. All I did back then was take a summer break. I thought to myself after five full months of math I would take pause for a month or two. And so I did.
To anyone looking at this blog I have abandoned my goal after less than half a year. But like an unknown function, just checking the endpoints will not tell you what is happening along the way. Therefore, I am a happy to record that I have been doing some Calculus behind the scenes.
On July 24, I receive a comment on this blog saying this:
You might be interested in
I followed the link and was blown away by what I saw. It was just the course I needed and I could do it through Coursera, the same platform I took my pre-calculus course.
The course is from Ohio State University and it features,I kid you not, the coolest professors I have ever met: Dr. Bart Snapp and Dr. Jim Fowler. Their lecture videos are short, clear and, more times than not, hilarious. I find it increadibly comforting to watch a video on a frustrating difficult topic that is explained with such weird, electric enthusiasm. I particularly enjoy Dr. Fowler's lectures, which are the majority, he is so intense and smart and lives math!
That is what I have been doing since the end of August. I got through 8 weeks of following the course and doing all my quizzes. I even took my fist mid-term. In all this time I have learned about, derivatives, differentiation, the chain rule, the power and product rules and L'hopital's rule. That's when I started to run into trouble. I didn't have a lot of time to practice more and the concepts started to gang up on me. I'm sad to say I could not keep up with the course since then. But a have continued watching the videos.
I have returned to this blog through a series of fortunate events. In the intervening time, I started taking a certification course in online teaching. I am on my sixth and next to last week. The assigment for this week: to create a blog or continue one that is already created. So here I am, I had planned to return in the middle of December to wrap it up but this feels better. It feels like destiny.
By the way, I just realized that the person who had left me the comment of July 24, was Dr. Jim Fowler.
Woah. Mind. Blown. Thanks, Professor.
Let me know what you think in the comments.

Monday, June 10, 2013

Continuity- both for limits and life

If you can draw a graph on a piece of paper without lifting your pencil once, that graph is continuous. In other words, there are no holes, jumps or asymptotes that block its process. My project of learning calculus is, by this definition, not continuous.

Continuity seems to be important enough in Calculus that every book I've seen has a section dedicated to it. However, it's heartening that, as I have learned, continuity is not crucial for determining limits. As long as the graph is approaching the same number from both sides there will be a limit even if there is a hole at that number. Therefore, I estipulate that as long as I am approaching my goal of learning calculus this year, I am allowed a couple of holes here and there.

When a graph is continuous, the limit of a function as x approaches a is f(a). In other words, if I know function x+1 is always continuous, then the limit of x+1 as x approaches 2 is f(2) which equals 3. The information that the graph is continuous let's me know that I can substitute the number I want to take a limit of into an equation without running the risk getting an undefined expression like dividing by zero.

Most of the calculus I have been doing so far deals with discontinuities. And I imagine that in the end it is discontinuities that will be a major part of my studies.

For a good brief intro to basic calculus and discontinuities you can try this video. I found its series in YouTube and really like it since it reminds me of the format of my coursera Pre-Calculus videos.

I plan to minimize future discontinuities in my studies, if possible, and get back to a more productive rhythm. This calculus stuff is getting ever more fascinating and I want to get and see it all.

Let me know your thoughts.


Thursday, May 30, 2013

Limit existence and infinity

One if the awesome things about studying on my own, is the flexibility of choosing my own schedule and resources. The drawback, as I have mentioned before, is that different resources have different points, methods and even explanations. If your resources also vary in timeframe, those issues become more apparent. Let's take for example the limits of 1/x as x approaches 0 from the negative and from the positive side.

If we look at a graph of 1/x we can see that as x approaches 0 from the negative side, the graph starts to approach the y axis but not touch it. Since x cannot be 0 (1/0 is undefined), then the y axis, where x=0, is an asymptote, a line that will never be touched by the graph. That means that our graph will get closer and closer to the y axis for all infinity without touching it. Therefore the limit of 1/x as x approaches 0 from the negative side, in the strictest definition of a limit, does not exist and that is how my text book defines it.

My online class however, also defines this limit as negative infinity and as positive infinity when you approach 0 from the positive side. The Profesor explained that it was a more informative way to describe the limit of 1/x as x approaches 0.

My problem came along when I did my textbook homework after my online lesson. While I was checking my answers I noticed I had gotten two incorrect ones. They were the limits described above. In the textbook the right answer was that the limits did not exist, I had answered that they were positive and negative Infiniti. Being a student on my own, I had to retrace my steps, watched my class again and do a lot of research to find out if I had gotten the right answer. When I reviewed my online lesson I heard the explanation the professor made about these limits not existing and how using infinity gave us more info about what was happening.

If I had not been able to re-watch my class, I would probably still be looking for the answer. In a traditional classroom my doubt would have been put to rest in a second by the teacher. On my own I had to figure it out myself. I see advantages and disadvantages to both situations.

How about you?


Monday, May 13, 2013

On limits and life

These last two weeks I've been studying limits in various ways, from various sources, and different media. I've used videos, sites, ebooks, paper books, ad apps. It is great to see how each reference approaches the topic differently, and how the central concepts emerge from the gathering of information.

I am starting to see why many people regard Calculus as beautiful. Being able to find the equation of a line that is tangent to a curve using limits is breathtaking. I am not kidding. There was something awe inspiring when out of some algebra and some elementary Calculus I arrived at a formula, that when graphed, touched a curve at exactly one point before continuing in its path. Can you see the significance of that? These two functions for an instant, touched, and then moved on.

To delve deeper into this existential stream of consciousness about Calculus, consider that a limit tells us about where a function is heading. It doesn't care about what happens when the function gets there. It cares more about what happens as it gets closer and closer to a given point. Therefore, in Calculus as in life, the most important thing is the journey rather than the destination.That gets me to think about the people we meet briefly once in our lives. Those chance encounters might not change our paths, but there is a cosmic record that they happened. At that moment when two people meet briefly, you can describe each of them as being together. In other words we could say that at 9:00pm on Sunday, May 19th, 2013, John met Mary and Mary met John. Therefore John and Mary became part, in that instant, of the greater formula of life as variables sharing the same time and space coordinates. And that point in time, much like a limit, gives us information about where each of them is and how they are behaving.

I am liking this calculus stuff more and more each day.

Let me know your thoughts.


Sunday, May 5, 2013

Slight Detour: The story of a cube

As I was gathering resources and reference for Calculus. I ran by a page (that I cannot locate now), of a math tutor that made a small comment about how he also had a Rubik's cube solving page.


Flash back at least 11 years ago.


For a holiday (I can't remember which) my girlfriend of at least 8 years years (who is now my wife of 10 years) gave me a picture cube as a gift.

After briefly looking at all the pictures of our life together, in fast, deliberate motions, I scrambled the cube. The look of horror on my future's wife face is still etched in my mind.


I looked down at the cube and shared her concern. I had never solved a Rubik's Cube. We knew no one who had, therefore this cube of our pictures would never be rearranged again.


I gave it all I had, for hours and days and weeks I tried to solve it. But I could not. The cube sat in my car for at least two years. One day, after we got married, my wife found it and gave it a sad forlorn look. She took it and stored it with the rest of our pictures and heirlooms. I felt like the worst person on earth for being foolish enough to scramble that picture cube.


Flash forward to two weeks ago.


When I saw that page on how to solve a Rubik's Cube using algorithms. I knew what I had to do.


As fate had it, one day we were shopping at Party City and they were selling mini Rubik's cube for 89 cents. I grabbed one as casually as I could and bought it.


Like I said, I never found the first page I used, but afterwards I got these others and these were the ones that helped me the most. Beginner's Solution to the Rubik's Cube. Beginner's Rubik's Cube Solutions. In the beginning, as I read these I felt overwhelmed, almost as overwhelmed as I felt doing Calculus without precalculus back in January. They were talking about Faces, primes, clockwise and counter-clockwise moves. They also mentiones middle pieces, corner pieces, edge pieces. Then they would give me string of algorithms that looked like this: R2 U F B' R2 F' B U R2.


There was no way this could work, could it? My first attempt with my mini cube, after at least 3 hours, ended in utter failure. My wife, who thought I had gotten my self another hobby besides calculus was not impressed. I had kept my real intentions secret from her. And on top of everything, she thought I was just cheating using formulas to solve a cube. At the time, so did I. I felt like I was painting by numbers. All I had to do was follow those instructions to the letter and I would solve a cube.


By the time I had solved my mini cube, I felt otherwise. This was not painting by numbers. Solving a Rubik's cube with the beginner method was about recognizing patterns and executing moves to make the pieces go where you wanted them to...without messing with the other pieces you had already done.


After solving my mini cube, I did not feel ready enough to tackle our picture cube. So I looked for more practice. I got the Rubik's cube app for my iPhone and Ipad and bought a "magic cube" from China on EBay. My goal was to practice with these as much as I could.


As I waited for my Chinese cube to arrive, I practiced with the app. I found it hard to use at first because the controls were weird. But after a while, I was "fluent" in it.

Still remembering the combination of moves that I needed to make and when I was supposed to make them still took me a long time.


My first time solving a cube in the app took me 1 hour 49 minutes. And I had to refer back and forward to my notes. More than once my wife found scraps of paper with algorithms on them...I got some weird looks from her.


The great thing about having the app is that I could fire it up at a moments notice if I was waiting in line, in an elevator, or in my lunch break. I could even pause the game and pick it up later at night right before bed. I was getting good practice out of this. After 5 tries I could keep my time under an hour. After 10 tries I could solve it a shade under 15 minutes. In the last 5 tries I could solve a cube in about 6 minutes.


I now felt ready to solve our picture cube.


All last week I have been asking my wife if she remembered were we had put that picture cube. I still did not tell her what I wanted to do, but I knew she knew. What I bet she did not know was that I could solve it this time. After a few attempts at finding it, and making a mess of our closet, I got my hands on it. The cube felt weirdly solid in my hand. As if it had gathered the mass of 10+ years just waiting for this moment.


I looked at it and remembered how hard I had tried to solve it the last time. Yet, all that was solved was a single face. It showed a picture of a kiss I gave my future wife after I had caught the garter from my brother-in-laws wedding. I would propose to María soon after that. It seemed fitting that I would attempt solve this cube on the year of our 10th wedding anniversary.


However, after I look a the rest of the pictures, I panicked. I could not tell which pieces belonged together. Many of the pictures had parts that were almost the same color. This would not be as easy as I had hoped. María was walking around the house doing some stuff and would check in me from time to time. When she did I asked her questions that must not have been reassuring to her like: where do you think this piece goes? She would give me her best guess and walk on.


After about 20 minutes it all clicked and I was back on track. A minute later I had solved all the pieces. However, I was not done. This is a picture cube which has a key difference than a regular cube.

In a picture cube the center pieces need to be rotated to fit the rest of the pattern. If you don't, you will have a solved cube looking like my mini cube did: I bet Jessie would love to have her face back in its proper orientation. Before you say anything...yes, my mini cube was of Toy Story...moving on!

Having centers not rotated properly can be quite distressing to many people. Because you can actually see it happening while you are solving the cube. If you type, Rotate center pieces, in google, you will get dozens of hits on this problem and how to solve it. The one I found to help me most was this video. The process involved a very simple algorithm repeated 5 times to turn a center piece 90 degrees clockwise and another center piece counterclockwise.

To finish my cube I would have to do the algorithm twice. Since I had to do 4 moves to complete the instructions once. That meant I had to do 40 moves to rotate all center pieces to their proper place. If I make a single mistake, I could scramble the cube in such a way that all my effort would be wasted. And guess what, at move 38, I lost track of my next move. I had no idea what I had to do next and the cube looked horribly scrambled.

I was heartbroken. I knew I was really close, but if I made the wrong move It was over. I just sat there, looking at all the faces of the cube and see if I could get back on track. A week ago I would have been lost, but after all that practice, I could see the relationships between pieces clearer. I felt a rush of adrenaline as my brain registered that I had already solved the cube. It must be the same rush a chess player gets when he or she knows the game has already been won a few moves ahead. I took a deep breath and mave the last two turns.

And there it was...solved. After all those years.

I showed it to María and the look on her face was amazing. I sat her down and told her how this cube had been on my mind for many years. how I always wanted to solve it. To rearrange all those pictures of our lives together. It was symbolic for me. And then she told me something that melted my heart.

A Rubik's cube has 54 squares. 9 squares per face per 6 faces. María told me that When our picture cube arrived the first time her hands. She was disappointed. The sticker that had the pictures looked really fragile to her. She knew that unless she did something, it would deteriorate and fade really soon. So she cut 54 squares of clear adhesive paper and covered all of them, one by one. It must have taken her hours of painstaking and risky work to get that cube to me. No wonder she was so chocked when I just scambled it the first time, and so sad when she found it still unsolved.

Now that we had it back we put it in a place of honor. Our bedroom table. To be looked upon by us and our little girl for many years to come.


Thursday, May 2, 2013

Pedagogical Quandary

It's been a while since I last wrote. I have been busy studying but not writing about it. There's no good reason for me not writing, even when I have made little progress. I guess if I wrote about what I've been doing it would be repetitive.
I am starting Calculus on my own with no formal teacher or course. I am relying on three textbooks and video lessons. Still after starting Calculus a couple of weeks ago, I have not gotten that far because all three books start with pre-calculus. The only resource I have that starts with Calculus are the video lessons but they don't give me any homework exercises to practice.
Therefore here is my quandary: should I skip the precalculus and go right to calculus since I have 10 weeks worth precalc? Or should I glance over what the books have to say about precalculus just in case they shed light into how they will cover calculus?
I have been inclined to do the former and check out the preparation chapters. That means that I have studied the precursors of limits (slopes of secant lines) at least in three different ways. At least, each time I get the introduction to limits I understand them better.
My other quandary is with practice exercises, I want to get as much homework as I can but all my books only have the answers for the odd numbered items. That took me back to school. I remember being assigned odd numbered items at home for practice and even numbered items for hand in assignments. It always made me anxious when I could not check wether or not I had done the exercises well.
In high school, Mr. Quintero, our notorious but brilliant math teacher changed all that. He had no problem assigning odd numbered exercises to hand in. He realized the back of the book only gave us the answers, so he would put all the weight in the process. That way you could either get the exercise right and check the answer, or get it wrong and work through it to find out what happened. I found this approach far more effective and instructive.
Without a teacher's help, all I have is the answers provided by the book to know if I am right. I guess that is a trade off I'll have to work with.
What do you think?

Wednesday, April 24, 2013

The two questions of Calculus

At last on day 113, I dive in into Calculus. What I write bellow is my interpretation of what I have read me researched. If you are using this post as reference, I suggest you double check my statements. I am by no means a reference source on calculus, just another student trying to learn it.
Calculus, from what I have learned far, seems to be the study of change. It deals mostly with two major subjects differentiation and integration. If fact I finds some sources that specifically reference differential calculus versus integral calculus.
If I use Professor Edward Burger's approach to explain calculus from one his Thinkwell videos, then calculus is the mathematical discipline created to answer two mayor questions: What is the instantaneous velocity of an object? And What is the area or volume of an exotically-shaped object? The first question is in the realm of diffential calculus and the second belongs to integral calculus.
All courses I am reviewing right now, start with differential calculus.
In differential calculus I read that the original question that started the discipline was finding the slope of tangent line of a curve. I found this surprising because I remember drawing or working with tangent lines and alopes back in intermediate school geometry. However, when I read deeper I was blown away with the reason the slope of a tangent was such a problem.
A tangent is a line that intersects an object, like a circle or a curve, at one (and only one) point. That's the rub. Way back in my geometry class, I learned that you needed at least two points to make a line. The difference between this two point will give us the rise and run of the line, which is its slope. So how can Impossibly find the slope of a tangent line to a curve if a tangent is only one point? The answer, learn calculus. The short answer, and the heart of all calculus it seems, is to find another point in the curve that is sooooooooooooooo close to the first point that the distance between them is infinitesimal and therefore negligible.
In The chart opposite, as point point B in the curve get closer and closer to point A that is also in the curve, the line that passes through them looks more and more like the tangent of the curve.
Now imagine the curve actually represents a car's velocity traveling on a straight line. Then point A is we're the car is at time x. In algebra we can find the average velocity between A and B1 by subtracting the miles traveled by the time taken to travel those miles. However, how can I find the exact velocity (Instantaneous velocity) of the car at point A? If I make the time interval between A and B small enough as to make close to an instant, then I can Algebraically compute an approximation of instantaneous velocity.
In integral calculus, the questions searched are a little different.
What is the area of an exotic shape? And from what I have studied, exotic shapes just mean shapes other than the ones we have formulas for. We have formulas for squares, circles, triangles, cubes, spheres, pyramids and if my memory doesn't fail me, cones. I bet we also have formulas for many other shapes, but how about the area of the shape in the chart below.
How do we measure this shape's area?
Well, we could inscribe it in a grid of squares of a given size and count the squares that the shape fills.That would give us an approximation of the area we are looking for.
If we start to make the squares smaller and smaller, more of the shape is inscribed within it.
We can repeat the process of making the squares smaller and smaller, to get better approximations. If we make those squares infinitely small, the value of the area of the shape will be so close to its real area that the difference would be negligible.
Therefore, the previous act of getting the tangent of the curve and now the act of getting the area of an exotic shape, relied on the same procedure to get answered. We used infinitesimally small numbers. In fact, I have run across various references to calculus as infinitesimal calculus. If we want to make the jump from algebra, which can only give us approximations of the answers to these questions, to calculus , where we can get the exact answers we are looking for, we must go through Limits.
And so must I, next time.
What do to think?

Saturday, April 20, 2013

Beginning Calculus in fits and starts

After 10 weeks of taking a class, it has been hard for me to figure out a way to start Calculus. The past few days I've been collecting resources to help me study. I have videos, ITunes courses, websites and more. What I now need is structure.
I guess I got too used to being given material instead of going to look for it. Maybe I am just holding myself back in fear of what's to come. Two posts ago I, finished with a line that sums up my feelings. I illustrate it bellow.
According to some accounts, ancient map makers used to put inscriptions at the edges of their maps that read: "beyond here there be (insert your mythological beast here)". I guess it was their way of saying, "don't know what's beyond so it must be sea monsters, dragons and some other weird stuff." It could have also meant they feared what was there.
Do I fear calculus?
A little. But I fear not knowing what's beyond even more.
It's about time Imjust dive in. Training's over. A part of me knows I am ready for this.
So here it goes...

Thursday, April 18, 2013

Statement of accomplishment

I got my statement of accomplishment from the UCIRVINE pre-calculus class I took at Coursera. It looks clean and simple. It has my name and the signature of both professors in the class. The only grievance I have is the note at the bottom.
It reads:
This is my first Coursera course so I do not know if all other universities use the same language. If they do, shame on them. I understand all the caveats they must state, specially the part about this course being in no way eligible for college credit and the fact that in the free version of the course they could not verify your identity. I guess what bothers me is that they focused on all the negatives, none of the positives.
I will make a suggestion in how to phrase the bottom of those certificates.
Please Note: This online offering while covering essential topics of the subject studied is different from the curriculum offered to students enrolled at UC Irvine. The statement is given to the student named above under the assumption that said student abided by the honor code explained in the syllabus and handed in their own work. Unfortunately we cannot verify this students's identity. This course is not eligible for college credit, grade or degree at UC Irvine. This statement does not affirm that this student was enrolled as a student of UC Irvine.
But maybe I'm just overly sensitive. Good news is I passed the course...and at least I can verify my identity. At least I hope I can. Let's check:
Fernando Santiago = the guy who spend 10 weeks doing an UC Irvine precalculus class at Coursera.
Identity verified.
This post should give me closure.
Your thoughts?

Saturday, April 13, 2013

Autopsy of a test result: How panic, exhaustion and arithmetic don't mix

Since my school days I've had this rule on test results: If you can understand why your answers are incorrect, you are OK. The reasoning behind it is that if you are able to figure out what went wrong and why, that means the learning had taken place but there was a mistake along the way to execution. On the other hand if you have no clue as to why your answer is incorrect, that should send you back to review the concepts being tested.
As I related in my previous post I scored a 24.5 out of 34 in my Pre-Calculus final. The one that was 2.5 hours long and I had to take twice back-to-back because I thought I had set the time running on the final attempt by mistake. I took this week to understand what wheat wrong in those 10.5 items I got wrong. The answers surprised me. But they really shouldn't have.
In one of my Encouragement pictures posted here there is a quote that reads: 10 out of 9 times it's arithmetic that gets you. No one seems to get the joke... But this test is testament to that. 9 out of the 11 items missed were due to arithmetic, but not the way that you think. Here is the list of reasons my answers were incorrect.
  1. Wrote the wrong sign when copying problem.
  2. Used full angle when I needed the co-terminal angle. Which I knew was only in Q4 or Q1.
  3. Wrote the wrong sign when copying problem.
  4. Wrote a 28 that looked like a 78, I couldn't read my own writing!
  5. This time I used a sign incorrectly (summed -3+10 and wrote -7).
  6. Used a parenthesis instead of a bracket in an interval, even when I knew the number was included in the interval.
  7. Error copying line in a problem, I wrote a 5 instead of a 1 in the line below.
  8. I got the answer write but the program would only accept the variables py as p*y. A clearer head would have realized this.
  9. Boldly stated that the square of 9 was 49...which is the square of 7.
  10. Forgot to find the squarer root of the hypotenuse in a Pythagoras theorem solve. I am actually good at solving these problems, it was a huge oversight.
  11. I was too exhausted to simplify the solution of a half-angle cos identity.
With the possible exception of 2 and 6, the rest of my mistakes were due to my exhaustion and my panic to solve the test. Most of them were due to sloppy, hurried writing. The others were due to foggy thinking. I could find none that showed a flawed understanding of the material. In fact when I got back to doing these again for review, the moment I found my mistake was invariably followed by a "well, duh!", or "that was a stupid mistake". None of them were followed by a "why is this incorrect".
Before you think I am saying that I deserved a higher score, rest assured I feel I got the score I deserved. The mistakes I made were mistakes nonetheless, and they prove I need to pay more attention to what I am doing. And prove that I shouldn't take 5 hour long tests at 3am on a Saturday!
What do you think?

Tuesday, April 9, 2013

The tale of a test twice taken

Well, the time had actually arrived. I took my pre-calculus test last Saturday. Before I let you know how I did, I must tell you the story of how I came to take a 2.5 hour test twice almost back to back.
In my previous post I was going on about assessment and self-esteem, of how I would feel if I did worse in the exam that I thought I deserved. My conclusion was that it would be hard to do badly but the most important thing is that I know I have learned. I had also made a point about tests being a tool to mostly measure teacher performance rather than student performance. In other words, the tests scores will give more information to the professor about what worked in class than I would to a single student about their performance. Fast-forward a week and you would find me sitting with my laptop, iPad, textbook and notebook ready to start my 3:30am.
Why in earth would I be up at three in the morning to take my final? Simple, I had misread my watch and thought it was 4:30am. I was going to start the test at five, but when I had to check on my baby daughter, after she got out of bed, I decided to start the test early...that as you will see was a lucky mistake.
The final test was worth 80% of the class grade, so all those exercises I had been doing throughout the 10 week course (about 140 items) would count for 20% only. So, if O failed the test I would fail the class. The only solace I had was that the test could be taken twice at any time from Friday to Monday. My plan was to take it early on Saturday, then study up on the segments I had done badly. If I liked my score the first time around, I would take it once. My goal was no only to pass with the required 65% but to get a certificate of completion with distinction. I needed an 85% in the course to qualify for that.
After I have all my tools set, I start the test. The countdown read 2 hours 30minutes and started to descend a little faster that I wanted. I had 35 exercises to finish. My first wake up call came at question number 1. I had no idea what it was asking me to do. I flip through my notes and look through my reference pages to jog my groggy brain into gear. It occurred to me that an hour of sleep would have been welcome. But I soldiered on, I skipped the first couple of questions until I reached one that made sense. I picked up my pencil and calculated away. It would be a long two hours and a half. I could tell.
Halfway through my time I notice I am not hallway through my test yet. I start to get the feeling O will not finish all the questions, which would be bad because I wanted to use my first attempt as a reference and needed all answers graded. At least I had saved an HTML copy of the test in,y hard drive, so even if I do not finish all the items, I would know what they asked.I wanted to make a static PDF copy of the test, but found out my laptop did not have that capability. That would come and bite me later on.
As the clock winds down to the last 5 minutes I still have 4 or 5 exercises to go. I feel tired from all the calculations, and checking my notes, and finding reference pages to use. But at least I would get the majority of the except cowed marked one way or another. When the time stops I wold my breadth for the result.
I scored 15.67, just under 44%.
Feeling a little down, but not that much considering I had practiced very little for test, I proceeded to open the HTML copy I had saved in order to print the questions for review. And that's when disaster struck. As I looked at the HTML copy of the page I had saved, I saw something that made my heart stop. There was a clock at bottom counting down from 2 hours 30 minutes. It seemed I accidentally triggered my second attempt of the test.
On impulse I close the page. Then I freak out thinking I just lost my chance to take the test again. So I do the only thing a sleep deprived, exhausted human being would think of doing, I opened the page again. The clock started counting down from 2 hours 30 minutes once more. However, I could not know if this new clock was real or if the true countdown would be the one starting when I first opened the HTML file. Fully awake now, I figured I had to options: either hope it was a glitch, study all day for the test and retake it the next morning risking not having a second attempt to do; or suck it up, take the test again and try to finish it before that first countdown wound down.
Desperate, and not wanting my grade of this course which I had pit so much into, I decided to take the test again right then and there. Knowing full well that it might all be for nothing, since nothing could guarantee that upon hitting submit after completing all 35 exercises I would not get a message saying:"We are sorry but it appears you have already attempted this test twice." Regardless of that possibility I barreled on. It was already 6:00am.
The new test was slightly different from the first. It had the same questions but the variables and constants changed in most of them. Still, the second time around I was sufficiently awake to start remembering all I had learned in the class. Still, I knew my nervousness and agitation could make me make mistakes. And there was no fixing mistakes this time. It was now or never.
About an hour and a half into the test, at around 7:30am, my baby daughter awoke. She would be hungry and very curious about what daddy was doing. It was an eventuality I knew I would face. My plan was to leave the test were I was and get her breakfast. While she ate I could do some more exercises. Then my wife woke up. She instinctively noticed my predicament and told me to go on with my test. She would make breakfast. My wife is an angel. She woke up early on the day she could get to sleep late to help me pass a test that was, in the grand scheme of things, insignificant. All because she knew it was important for me. Thanks to her I was able to finish the test with 17 minutes to go. With trepidation I hit the submit button. I hoped against hope that the system would accept this attempt. It did.
My second score was 24.5 out of 35, exactly 70%.
I sighed with relief. A day ago, that score would have been a let down. But that day after 4 and a half hour of testing, and scribbling, and checking, and answering, I was exhausted and happy. My technical difficulties were overcome. I had passed the class. While my wife and daughter ate their pancakes, I raised both arms in triumph and gave a muted cheer. They cheered back.
I was done with my pre-calculus review. I would receive my statement of accomplishment a week later. Now, in the distance, through the wall I had just taken down , I could see a sign over the horizon that read: Beyond, there be Calculus.

Wednesday, April 3, 2013

Testing and self-esteem

This following week is my pre-calculus final exam. It will try to assess all that I have learned in the last 10 weeks. But can it?
Testing has been, in my opinion, one of those necessary "evils" of traditional education. They are necessary in order to keep a record, a benchmark, a log that can be revisited to understand the decisions made. I call it an evil because I know who the test is for: It's for the teacher, and for the school, not necessarily for the student. What I mean is that standardized assessment tools, like tests, work more as a measure of homogeneity in the learning of a group, that of individuals. Taking all the scores together a teacher can assess their own performance. However, gauging individual performances based on them can be tricky and the results can be deceitful. And before anyone thinks I am only writing this post out of apprehension of doing poorly in my test, I can tell you I am not alone in thinking this way.
However, I am worried about doing badly in my test. It would be really disappointing to have spent so much time and effort in class, and have nothing to show for it. Yet, do I really have nothing to show for it? Doesn't this blog chronicle all I have learned in 10 weeks better than any test could assess? Of course I do and of course it does. So why do I still worry? Because of my self esteem.
I recognize that I need that external evaluation to measure my performance. And that might not be a good thing. I shouldn't need a test to tell me how well I learned pre-calculus, but part of me does. And I know that a bad score in the test would take a lot of wind out of my sail. But why? Why should it? I guess because when people ask how you did on a task, the answers we give more often than not are the results, not the journey. We focus on the scores, the grades and the ranks to measure our selves. And what we usually measure our selves against is other people.
I've been doing some research on self-esteem, for other purposes, these last few weeks and I was surprised to find that my self-esteem is not as high as I thought it would be. In fact, my whole project on learning calculus could be seen as a why to correct issues stemming from low self-esteem or more likely low self-efficacy.
Self-efficacy is the confidence you have in the ability of doing something. Self-esteem is your perception of selfworth. And while I believed my self-esteem is always high, it takes me more than a while to get over setbacks, I expect perfection and I am my worst critic. Three factors that denote a less than high self-esteem. However, my self-efficacy had always seemed high because I have never had doubts on my ability to do anything or learn anything. With the exception of calculus. For the longest time I believed I could not do it. I am making that change this year.
So going back to the test, I start to understand where my apprehension lies. My self-efficacy tells me I can do this and that everything will be fine. My self-esteem is worried about the outcome and how to take it. My knowledge of assessment tells me the test is necessary as a tool for me to understand my weaknesses and strengths. My knowledge of test making tells me the test is a tool for quality assurance of course performance. When I take all these things together, I have to conclude that I am stressing over nothing...and that I need to monitor my self-esteem. No matter how I do in the test, I know I have made progress and I am proud of that. Besides, perfection is overrated.
Wish me luck, I'll keep you posted on the results.

Monday, March 25, 2013

The Work

My favorite literary character is Sherlock Holmes. I have read the complete works of this series by Sir Arthur Conan Doyle at least twice and made a point of visiting 221b Baker Street on a vacation to London last year. I have also enjoyed the movies with Robert Downey Jr. and Jude Law. However, I must make special emphasis on my love for BBC's Sherlock series, which transports Holmes and Watson to present day London, but I digress.
What is commonly remembered of Sherlock Holmes is his power of "deduction" (actually it's induction), which is his ability of grabbing the minute details of a subject and arriving at a conclusion. For example: Holmes would take a look at a person's clothes and tell their occupation, errands of the day, mode of transportation, degree of anxiety and town of residence. He would do so by noticing things like tips of mid in the shoes, the pattern their clothes are crumpled, a ticket or stub protruding from a pocket, a button in the shirt not completely fastened, the know of a tie...etc. front hese actions people say That Sherlock Holmes is a brilliant man.
What most people never realize is that Sherlock Holmes spent most of his time on study. He would famously analyze 140 different type of Tobacco ash. Therefore when he got on a scene and saw Tabacco Ash he could say what kind it was. In other words Sherlock Holmes was brilliant because he would do the work (and had an extraordinary brain that could make sense of all he learned and appy it as needed). Lacking the later I have taking to weeks to do the former with my trigonometry.
As promised, I took all this week to study up on trigonometric identities and the values of sin, cos and tan at various degrees. I had to do this because I felt I had reached a wall in my studies that was making it difficult for me to understand the newer concepts I was being given in Pre-calculus. This situation was creating a mental block that was giving me a chance to doubt my abilities to carry on my goal of learning calculus.
I am happy to say that, as I suspected and sincerely hoped, a serious attempt to memorize and analyze trigonometric properties and identities has helped me a lot.
I started with the basic stuff. It was taking me too long to grasp angles quoted in radians. Therefore, I spend a day with degree and radian conversions. I did not want to only memorize that π/6, π/4, π/3, π/2 and π were 30deg, 45deg, 60deg, 90deg and 180deg respectively. I wanted to understand it and see it. As it turns out, a quick drawing of an unit circle is very handy for this, as is knowing your multiples of 90, 180, 270 and 360.
Then I progressed with the value of trig functions at various degrees. The degrees I studied were of course the "nice degrees" as the people at call them. These are angles were sin, cos and tan (along with their reciprocal functions) have exact values that are easy to remember.
I cannot thank the people behind enough since they had in their site the very tool I needed to help me study. It was a chart with all the trig values in the easy angles for all quadrants. However, their chart has all the values hidden from view until you ask it to show them. It does not have a button for hide all or show all, so finding a pattern would be difficult, but it is a great way to test your knowledge. You can find the chart here.
Doing "THE WORK" helped me join together all the concepts I had studied so far in trigonometry. This might sound absurd since I have been doing trig quizzes for my pre-calc class the last 3 weeks, but it's true nevertheless.
All weeks prior, I've been relying on my memory and the examples the professors gave in class. It is very common that quiz questions can be answered using the exact process the Profesor used in class, and that was the case with my homework. Therefore, as long as I followed what the professor did step by step I would get a correct answer without any deep understanding of the process used. In othet words, I had gone through a lot of knowledge in pre-calculus class, but that knowledge didn't stay fresh in my mind for long. Case in point: reference angles.
A few weeks back I studied reference angles, which are acute positive angles that are coterminal to other angles and the x axis. This is a fancy way of saying that if you have an angle that is 150o, it's reference angle is the smallest angle formed from the terminal side of that angle to the x axis. Therefore the reference angle of 150o is 30o. It is a lot easier to understand when you see it in the figure opposite.
The homework for that section was very simple and straight forward. I felt I was getting a free pass. If I had paid attention to the importance of reference angles in Trig values, it would have saved me a lot of headaches.
You see, It was not until I was studying the relationship between trig values and the unit circle that I finally understood the importance of reference angles. Once I had learned sin values from 0° to 90°, I went over to the other quadrant and thought to myself: Since trig functions are cyclical, sin of 120° must equal sin of 30° since 90°+30°=120°. When I went over to a reference chart to check this, I found out I was incorrect.
Sin of 30° was not equal to sin of 120°, sin of 30° is equal to sin of sin of 150°. That threw me for a loop until I put the values in an unit circle. Then it all made sense. Not only did I find a pattern for the values of sin, cos and tangent, but I also understood that the value of all of them is equal to the value of their reference angles before taking their sign value into consideration. Therefore since 30° is the reference angle of 150°, 210° and 330°, sin is 1/2 (or -1/2 depending of the quadrant) in all those angles.
It was an excellent week of study and discovery. Of course these discoveries are small compared to Sherlock Holme's deductions. If fact they are small in terms of what every high school student already knows of trigonometry. But for me that are huge and important because I found them on my own. Nothing can compare to the thrill of figuring things out for yourself.
Let me know how I'm doing or if I got something wrong.

Sunday, March 17, 2013

The Wall

For a while, I have been able to see it from a distance. Every day I would check to see if I saw it moving closer, knowing that if it did, I would be in trouble. When I returned this week from vacation I was not surprised to see it right in front of me. I am talking about the wall. That seemingly insurmountable force that wants me to stop my quest to learn calculus by making it look daunting, foolish and, worst yet, insignificant.
I was expecting to have lingering doubts about how consequential my desire to learn calculus is. I knew a day would come when I would ask myself: "What am I doing? This is ridiculous. I am just going to waste my time with this." The posts on words of encouragement were a weekly reminder against this.
What I was not expecting was the attack on my intellectual ability to actually do this. When the wall appeared this week, it had an inscription that says: "You cannot understand any of this anymore. It will just get worse, I promise." It's hard to argue with the inscription when it rings so true. The last two weeks I have been struggling to truly grasp the concepts I have been reviewing. They are mainly pre-calculus properties and transformations. I have a hard time using radians and relating them to trig graphs and the unit circle.
Part of me knows that the reason i am struggling is that I have not done THE WORK. I have not consciously gone throughout the process of memorizing all the trigonometric properties and, more importantly, the values of sin, cos and tan for special angles in the unit circle. Still, a small but potent voice, tells me that I am just rationalizing the real problem: you are incapable of learning this. And for this week I have been to much of coward to find out which voice was right.
Until today.
Today I will begin THE WORK. I will set out to store in my long term memory all the trigonometric properties I can. I see cue cards and pop quizzes in my future. I am confident that the work I put in these days will be invaluable when I start with Calculus.
Wish me luck. I have a wall to turn down. Now, where is my sledgehammer?

Thursday, March 14, 2013

Yes, today is π day 3.14

And to think I almost missed it when I have been battling with inverse trigonometric functions all week!
Check out this site for all things PI.
I promise to post can put a hold on Math blogs!

Saturday, March 2, 2013

On logarithms and funny notations

I remember having trouble with logarithms in high school because I had to do a transformation that hurt my head to understand. Before reviewing this section of pre-calculus, I consulted my memory bank's folio on logs. It read: "Logs will ask you to do stuff to convert them that will feel unnatural." I could not remember what it was exactly, but I could see the numbers bouncing around each other rearranging themselves. As I reviewed the chapter I remember what that is.
Logarithms are just the accepted notation to answer this question: to what power do I need to raise x number to give me answer y? In other words, I might want to know to what power should I raise 2 to get 64. In exponential form I would write 2x=64.
However, in logarithmic form we would grab the x exponent throw it across the equal sign, then take the 2 shrink it, append it to the word "log" and finally take the 64 across the equal sign to join log2. At the end we should have something like this: log264=x.
I hope you can see why it blew my mind as a teenager.
The Log transformation got me to think about math notation, or the way we represent mathematical concepts. I mean, if you think about it, someone at some point had to decide that the letter x would make a good place holder for a number we don't know; or that a cross would signify addition and a a horizontal line subtraction. I bet those first proponents had to explain what the symbols meant:
"Dear colleagues, whenever you see this notation of two parallel horizontal lines "=" it means that the terms on either side of "=" are equal"
I can almost hear the collective "Ahhh"s of fellow mathematicians reveling in not having to write the word "equal" anymore. I am of course being a little facetious. I am bet the introduction of the equal sign did not happen quite that way...but come on, I bet the person who came up with the radical sign (√) for square roots got a long of grief from the long division enthusiasts.
Getting back to logarithms, the logarithmic and exponential, as best as I can explain from what I understood, equations are crucial to describe rapid growth (or decline). Imagine you had a magic ball that would split into two magic balls every hour, then the new balls would also split every hour and so on. How many balls will you have in 24 hours? Pick a number in your mind and make it high. If you number is 16.7 million balls, you are right on the money.
The exponential or logarithmic graph has a segment rises or falls very quickly and another that gets progressively smaller but never touches the point it gets close two. That, if I remember correctly is an asymptote. A basic exponential graph has the x-axis as an asymptote, the logarithmic graph usually has the y-axis.
If you look at the graphs of log2(x)and 2x , the first is logarithmic and the second exponential, they will be mirror images of each other along the y=x line. This happens because in the logarithmic equation, the y values you get are the power you need to raise 2 in order to get the x values, while in the exponential one the y values you get are the result of raising 2 to the x power.
The cousera course videos keep telling me that exponential and logarithmic equations are important to calculus. For the time being, I guess I must take their word for it.
If anyone thinks I am missing something, let me know.
'Till next time.


Tuesday, February 19, 2013

More on quadratic equations and parabolas

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.