Before sitting down to write this post I assumed I’d be starting it off saying something along the lines of “Week 16 was one of my least productive weeks so far.” My hope was to get at least ~75% of the way through Quadratics Functions & Equations but at this point I am closer to 50%. Looking back over my notebook, however, I realized I’d gone through 15 pages of notes and, although I didn’t get as many of the practice modules completed as I’d hoped, I came a long way in understanding quadratics and the parabolas they create when graphed.

The unit started off with an introduction to parabolas which are curved, symmetrical lines on a graph. The curve either opens upwards or downwards and is created via a quadratic equation (i.e. a second degree equation, i.e. one of the terms is raised to the second power and no other term is raised to a higher power, ex. y = 4(x + 6)(x + 4) = 4(x^2 + 10x +24)). Working through practice questions where I had to graph the parabola was interesting and really helped me to understand the concept.

Next I learned about the Zero Product Property which seems like it will be very important going forward. I still need to learn more about the ZPP to have more confidence explaining it but, as far as I can tell, the ZPP works by setting a quadratic equation to equal 0 and, by doing so, you’re then able to figure out the two X-intercepts of a parabola (if the parabola crosses the x-axis anyways). An example of the ZPP would be:

• x^2 – 3x – 54 = 0
• x^2 + 6x – 9x -54 = 0
• x(x + 6) -9(x + 6) = 0
• (x – 9)(x + 6) = 0, therefore, because one of the two expressions must equal 0 for the equation to be true,
• x = 9 OR x = -6

I was reintroduced to square roots through quadratic equations and had to solve equations by taking the square root of certain terms. From this I learned that the square root of a number must always be written as plus or minus (+/-) as both the resulting positive or negative number squared will equal the initial number (ex. 8^2 and -8^2 both equal 64, therefore the square root of 64 must be written as +/-8 to show that it could be either). Working through quadratic equation questions, I learned that the +/- results I got were the essential component to finding the two x-intercepts of a parabola. As a side note, understanding why a square root has a +/- answer helped me understand that a negative number does not have a real square root since the product of any number multiplied by itself will always be positive.

As another side note, this seems like a good time to mention two terms that I’ve seen but haven’t yet written down:

1. The “product” of two numbers is equal to those two numbers multiplied by each other, and
2. The “sum” of two number is equal to those two numbers added to each other.

I learned about different forms of quadratics (i.e. the way they’re written down). These include:

1. Standard Form
   1. y = 3x^2 + 12x -15
2. Factor Form
   1. y = 3(x – 1)(x + 5)
3. Vertex Form
   1. y = x(x + 2)^2 -27

I still don’t have a great understanding of the pros/cons of each form. My gut tells me knowing the names of the different types of forms and understanding how to get from one to the other is going to be important to build on going forward.

The last thing I did this week was look at one video on the Quadratic Formula. When introducing the quadratic formula, Sal said something along the lines of “the quadratic formula is one of the top 5 formulas in all of mathematics”. It seems clear that this formula is a big deal, though at this point I don’t exactly know why. The quadratic formula is as follows:

• ax^2 + bx + c = 0
• (the brackets { and } equal square root)
  • = Q.F. = x = (-b +/- {b^2 -4ac})/2a

The above equation for the Q.F. doesn’t look exactly the way it does on the videos I watch because I don’t know how to add the radical symbol, plus it’s tough to properly show that the top half of the equation is divided by 2a. Based on Sal emphasizing how important this equation is, I’m interested to learn more about it and see why it’s such a big deal.

I have nine days until the end of December. My goal was to get through Algebra 1 before the New Year and I still think it’s doable. Last week, not only was I fairly busy but I found getting through material in this unit took longer than in past units. I still have ~50% of this unit to finish, the unit challenge, the next unit (Irrational Numbers) and the final course challenge, all of which needs to be done in 9 days to hit my goal. Merry Christmas to me!