Initially, my goal was to get through every course on KA starting from Arithmetic and ending at Multivariable Calculus by February, 2020. I wanted to be able to apply to an undergrad program in one field or another that required a strong foundational understanding of math and, in order to apply for the following year, I would have needed to submit my application by February. Now I’m not as certain I want to apply to an undergrad program. Regardless of if I’d like to apply in the future or not, looking at the rate at which I’ve been getting through each course, I’ve revised my timeline and now want to get through KA in 1 year. Being that I just finished Week 13, I’m happy to say I’m 25% of the way there!
This week I finished two units; Sequences and Absolute Value & Piecewise Functions. The first unit, Sequences, taught me that a sequence is a set of values that relate to each other. I learned the following terminology that are important words for describing sequences:
- Finite sequence
- A sequence which has a limited set of terms.
- Ex. {1, 4, 7, 10}
- Infinite Sequence
- A sequence which goes on infinitely in one direction or another.
- Ex. {…, -2, 0, 2} or {3, 5, 7, …}
- Arithmetic Sequence
- A sequence in which each successive term is a fixed amount, a.k.a. the common difference, higher or lower than the previous term.
- Ex. {3, 6, 9, 12, 15}
- Geometric Sequence
- A sequence where the terms are multiplied by a fixed number, a.k.a. the common ratio.
- Ex. {2, 6, 18, 54, …}
- Series
- The sum of a sequence.
- Ex. {2, 6, 18. 54, …} = 80 (I’m not sure if this is the correct way to write it, however).
I learned two different formulas for writing out either a arithmetic or geometric sequence; the explicit formula and the recursive formula:
- f = Function of
- n = Term number
- A = First term in the sequence
- B = Common difference
- R = Common Ratio
- Explicit Formula
- Arithmetic
- f(n) = A + B(n – 1)
- Geometric
- f(n) = A x R^n-1
- Arithmetic
- Reclusive Formula
- Arithmetic
- f(1) = A
- f(n) = A(n – 1) + B
- Geometric
- f(1) = A
- f(n) = f(n – 1) x R
- Arithmetic
The following unit, Absolute Value & Piecewise Functions, was relatively brief. I worked on questions where I was given an equation with an absolute value in it and asked to graph it. I was also shown a graph and asked to come up with the formula to describe it which would include aa absolute value in it. A few things worth remembering when working on these questions are:
- In the equation f(x) = a|x – h| + K,
- a indicates if the graph “stretches”/”compresses” (if a > 1 it “compresses” and if a < 1 it “widens”) and if the function is above the X-axis (a is positive) or below the X-axis (a is negative).
- h indicates if the vertex (no idea if that’s the right word to use) shifts to the right (– h) or to the left (+ h)on the x-axis.
- K indicates if the vertex (again, no clue if that’s the right word) is shifted up (+ K) on the Y-axis or down (– K) on the Y-axis.
I finished the week working on Piecewise Functions which I learned are functions that, when drawn on a graph, their line doesn’t breaks apart. These are sometimes known as step functions because when the line is drawn it can resemble a set of stairs. An example formula of a Piecewise function might be:
- f(x) =
- {-0.125x + 4.75, -10 < x < -2
- {x + 7, -2 < x < -1
- {-12/11x + 54/11, -1 < x < 10
- This equation states that:
- When x is between –10 and -2 (-10 < x < -2), the formula to solve for f(x) is -0.125x + 4.75,
- When x is between –2 and –1 (-2 < x < -1), the formula to solve for f(x) is x + 7,
- When x is between –1 and 10 (-10 < x < -2), the formula to solve for f(x) is -12/11x + 54/11.
This coming week my goal is to get through the units Exponents & Radicals (730/900 M.P.) and Exponential Growth & Decay (0/1300 M.P.). In recent units I’ve been given questions with exponents and struggle to remember how to solve for the exponent. I’m glad to get a refresher on them this coming week and hope the “rules” on how to solve radicals stick with me a bit more afterwards. If they do, that would be rad(…ical).