This week I completed 3 units! I mentioned before that, for me to get into the KA calculus courses by the start of the New Year, I’ll need to start finishing between 2 to 2.5 units per week so I’m glad to have got through 3 in Week 30. The units I got through were Polynomial Graphs, Rational Exponents, and Exponential Models. Being that I’m out of work right now and have much more free time on my hands, I should be able to get ahead in units. I’ve noticed and am happy that I’m beginning to be able to concentrate better with everything that’s going on, though I’m continuously getting more worried about the Covid-19 pandemic.

The virus hit the U.S. hard this week – New York in particular. From what I can tell, the rate of their testing per capita pales in comparison to other countries around the world which is a bad sign considering their confirmed cases are now the highest in the world, exceeding 100, 000. I heard that Florida will soon get extremely high numbers of confirmed cases since their beaches and businesses all stayed open over Spring Break. I certainly hope it’s not a bad as I think it might be for them. I think Toronto has done a fairly good job shutting everything down for the most part. Like everywhere else in the world, our confirmed cases are still growing exponentially. I hope our healthcare system is able to hold up as the wave of positive cases surges in the coming weeks. I think this upcoming week will be noticeably different than last week in terms of people needing to go to the hospital. I also think the general population is going to realize how bad this crisis is and start to become as worried and scared about it as I am.

KA has been a good distraction for me. It’s nice to have a productive and positive distraction. The first unit I got through this week was Polynomial Graphs. This unit 1) reintroduced me to what are known as the ‘zeros’ of a function, 2) taught me about ‘multiplicity’, and 3) taught me what the ‘end behavior’ of a function means.

• The ‘zeros’ of a polynomial equation are the value(s) of x that make the function equal 0. (The way I think of this is, the ‘zeros’ are the places on a graph where the function crosses the x-axis and therefore where y = 0.)
  • P(x) = (x – 1)(x + 2)(x – 3)(x + 4)
    • The ‘zeros’ of this equation would be {1, -2, 3, -4}.
    • The above are ‘zeros’ of the equation because using any one of those numbers for the value of x would make the function equal to 0. Each of those numbers would make one of the binomials (a.k.a. linear factors) equal 0 and, since any number multiplied by 0 equals 0, the value of the function itself would equal 0.
    • This means that if you were to graph this function, at the points x = {-4, -2, 1, 3}, y = 0.
• Multiplicity of a function refers to the number of times a linear factor (a.k.a. binomial) occurs within the function. If a certain linear function shows up twice, it has a multiplicity of 2; if it shows up thrice, it has a multiplicity of 3, etc.
  • P(x) = (x – 1)(x + 2)(x + 2)(x – 3)(x – 3)(x – 3)
    • (x – 1) has a multiplicity of 1.
    • (x + 2) has a multiplicity of 2.
    • (x – 3) has a multiplicity of 3.
  • If the multiplicity of a linear factor is even (i.e. the linear factor shows up in the function 2, 4, 6, etc. times) then the line of the function touches the x-axis at that ‘zero’ x-coordinate but doesn’t cross over it (i.e. it looks like a ‘U’).
  • If the multiplicity of a linear factor is odd then the line of the function crosses over the x-axis at that ‘zero’ x-coordinate.
    • In the example of P(x) = (x – 1)(x + 2)(x + 2)(x – 3)(x – 3)(x – 3),
      • At x = 1, the function crosses over the x axis since the multiplicity is 1, an odd number.
      • At x = -2, the function touches the x axis but does not cross over since the multiplicity is 2, an odd number.
      • At x = 3, the function crosses over the x axis since the multiplicity is 3, an odd number.
• ‘End behavior’ of a polynomial is a phrase that refers to which y-axis direction (either up or down) the ‘ends’ of the polynomial go the further you go in the x-axis direction (left or right).
  • End behavior is determined by taking a x-coordinate of a function that’s past the final ‘zero’ coordinate in both the positive and negative directions, inputting it into the polynomial and determining if the function is positive (i.e. the end it going ‘up’) or negative (i.e. the end is going ‘down’).

The next unit I went through, Rational Exponents and Radicals, was the most challenging but also the most enjoyable once I got the hang of it. This unit taught me how to add, subtract, and multiply/divide exponents on polynomials. It started off by going through the difference between regular exponents, fractional exponents, and negative exponents:

• Regular exponents
  • x^2
    • = x * x
  • x^3
    • = x * x * x
  • x^4
    • = x * x * x * x
• Fractional exponents
  • Exponents that are raised to a fractional power are the same thing as that number under a radical to the denominators root value.
    • x^1/2
      • = 2{x}
      • = the square root of x
    • x^1/3
      • = 3{x}
      • = the cubed root of x
    • x^4
      • = 4{x}
      • = the fourth root of x
    • Ex. 125^1/3 = 3{125} = 3{5 * 5 * 5} = 5
• Negative Exponents
  • Exponents that are raised to a negative power are the same thing as “one over” that number to the positive power of the exponent.
    • x^-2
      • = 1/x^2
    • x^-3
      • = 1/x^3
    • x^-4
      • =1/x^4

I found that working through the questions that were given, most of the time I had to figure out a way to make the ‘base’ of two numbers the same in order to add, subtract, and/or multiply the exponents together. The formulas used to add, subtract and multiply exponents together (which you can only do when the numbers have same base!) are:

• x^b * x^c = x^(b + c)
  • 2^2 * 2*3 = 2^(2 + 3) = 2^5
• (x^b)^c = x^(b * c) = x^bc
  • (3^2)^4 = 3^8
• x^b/x^c = x^(b – c)
  • 4^5/4^3 = 4^(5 – 3) = 4^2

Another rule I learned that is important to remember is that if two numbers have the same exponent but different bases, you’re able to multiply and divide the bases as you normally would and keep the product raised to the same power.

• x^a * y^a = (x * y)^a
  • 2^3 * 3^3 = (2 * 3)^3 = 6^3
• x^a/y^a = (x/y)^a
  • (14^3/4)/(7^3/4) = (14/7)^3/4 = 2^3/4

The final unit I got through this week, Exponential Models, was fairly short and not too difficult. The unit worked on equations of the form:

• F(n) = b * (r)^n
  • F(n) = the functions value given n,
  • b = the value of the function when n = 0,
  • r = the common ratio, and
  • n = the term being calculated.

The unit was mostly made up of word questions which I had to read and decipher to figure out how to construct the above equation with the values given in the question. The questions became more difficult when they asked me to alter the term number used for the exponent in some way based on how the question was phrased. As an example, some questions would give me the common ratio per decade and then ask me to find out the common ratio for 3 years. I would then have to find the 10^th root of r and take that number to the power 3 to get the answer.

Overall this past week was the right amount of challenging. I likely spent close to 10 hours working through KA which I’m fairly proud about. This coming week I once again have a goal to get through 2 units, Logarithms (0/900 M.P.) and Transformations of Functions (0/1000 M.P.). I’ve heard the word logarithm before but have no clue what it means. I have a feeling it will be a tough subject. Like other things I currently have bad feelings about, I hope I’m wrong.