Until this week I had only ever worked on linear equations which show up as a straight line when graphed. I had seen graphs with curved lines before but had never learned what equations produced those types of graphs. Working through the unit Exponential Growth & Decay this week, I finally got introduced to exponential equations which are what create curved lines when graphed.

I managed to get through two units this week, however, with the first one being Exponents & Radicals. It was quite useful going through this unit which refreshed the “rules” or exponents and radicals for me. Some notable rules include:

• When a negative exponent is attached to a base number, the value of the number equals 1 over the base number to its positive exponent.
  • Ex. 2^-3 = 1/2^3 = 1/8
• When Multiplying two numbers with exponents which have the same base number, you simply add the exponents.
  • Ex. 3^3 x 3^4 = 3^(3 + 4) = 3^7
• When dividing two numbers with exponents which have the same base number, you subtract the exponents.
  • Ex. 4^5/4^3 = 4^(5 – 3) = 4^2
• A radical uses the same symbol as the “square root” symbol but can indicate “cubed root” or the root of any number by simply adding the root-number above the radical symbol on the left.
  • Ex. a radical with a 4 above it and the number 16 inside of it would equal 2.
    • 2 x 2 x 2 x 2 = 16 = 2^4 and 16 under the radical symbol with a 4 equals 2.
  • (I don’t know how or if it’s possible to add the radical symbol on here.)

As soon as I began the unit Exponential Growth & Decay I was immediately taught why curved graphs are curved. When a function has an exponent in its equation, the exponent multiplies the term number every time the term is increased. For example:

• f(g) = A(r)^g
  • f(g) = Function
  • A = Initial Value (i.e. the value of y when x = 0 – not sure why this isn’t referred to as the Y-intercept)
  • R = Common Ratio
  • g = Term Number
• f(g) = 2(3)^g
  • f(0) = 2(3)^0 = 2(1) = 2
  • f(1) = 2(3)^1 = 2(1 x 3) = 6
  • f(2) = 2(3)^2 = 2(1 x 3 x 3) = 18
  • f(3) = 2(3)^3 = 2( 1 x 3 x 3 x 3) = 54
  • f(4) = 2(3)^4 = 2(1 x 3 x 3 x 3 x 3) = 162
• In this example, every time the term number increases by one, the functions value is multiplied by 3. This is why the function appears as a curve when graphed.

When the Common Ratio of an exponential function is >1, the curve/slope of the line increases when moving from left to right (i.e. from negative to positive) on the X-axis. This is known as Exponential Growth. When the Common Ratio is <1, the curve/slope of the line decreases when moving from left to right (again, going from negative to positive) on the X-axis. This is known as Exponential Decay.

Once again, going through the “refresher” unit on exponents and radicals was very helpful for me. Not only did I quickly remember the rules of exponents and radicals, but they seemed easy to understand and felt more concrete in my mind when I finished the unit. Coming up in Week 15 I’ll be taking on Quadratics: Multiplying & Factoring (800/1500 M.P.). I found quadratics tough to understand when I worked on them the first time so I’m hoping the same thing will happen this coming week with quadratics that did last week with exponents and radicals. Fingers qua-ssed!