Study Sheet!

Woah, long time no post. IB and extra curriculars are starting to pick up, so I haven't been able to get around to a "trivial" thing such as a blog in a while.

We've got an oral project for French due tomorrow, and of course, being procrastinators (fourth indication of an IB student: proffesional procrastination), my partner and I left most of it for tonight. It's actually come together quite nicely for an oral report on cow poop. Yes, that's right, cow poop. The project is that each pair gets a different innovative form or alternative energy as their topic, and we have to research, write up, and present a report including an explanation of the method and each person's opinion on it. It's actually come together rather nicely and I think I'll publish the written when we're done.

I also have a Math test on Polynomial Functions tomorrow, and considering I only got 11.5/12, out of a possible 15 (bonus marks), I'm a little stressed. I know exactly what I did wrong, but I still fear making stupid mistakes. I think I'll start posting up review sheets of sorts here for my tests, a few days ahead of time. It'll help me review (and not procrastinate as much) and it might be useful f or anyone in my class. I tried to start up a homework-posting website last year, but it failed as I got overwhelemed with other stuff and generally neglected my homework anyway. This is a lot less demanding and I think I can keep up.

So, here goes the study sheet for tomorrow, for those of you who like the late-night study sessions:

Polynomial Functions

Definition: A series of terms raised to a natural number with one variable per term.

General form: y=a_nxⁿ + a_n-1x^n-1 + ... a₂x² + a₁x + a₀

• End behaviour:
• odd degree functions behave like the graph of y=x, extend from the third quadrant to the first
• even degree functions behave like the graph of y=x², extend from the second quadrant to the first
• "a" value : The lead coefficient (provided decreasing powers), if negative, means a reflection in the x-axis

• Turning points: There is a maximum of (n-1) turning points, where n is the degree of the function. Turning points are crests and troughs, not points of inflection (s-shape in the graph of y=x³).

• Zeros: There is a maximum of n "real" zeros.

• Order of factors:
  • A cubed factor is a "factor of order three" and appears grapically as an s-shape
  • A squared factor is a "factor of order two" and appears grapically as "touching" the x-axis and "bouncing" off
  • A factor raised to the power of one is a "factor of order one" and appears grapically as passing through the axis

• How to graph a function by it's factored form equation:
  1. Determine the "end behaviour" by "a" value and the degree of the function
  2. Mark the x-intercepts or zeros
  3. Use "order of factors" to determine intercept behaviour
  4. Find the y-intercept (sub x=0)

• Miscelaneous:
  • To find "standard" or "expanded" form from factored form, simply expand the function.
  • The degree of a function is equal to which of it's constant difference tables is constant.
  • The number of factors is at most one more than the number of turning points

How to factor:

• The constant in the standard form equation is the product of the roots. Sub facotrs of it into f(x) and solve. If it is equal to zero, then x minus the number is a factor. (eg. (x-q) when k is a factor of the product of the roots.)
• Factor Theorem: (x-q) is a factor of y=f(x) <=> f(x)=0
• You can then use this factor to conduct long division on the equation to find the remaining roots.

Remainder Theorem

• "Find the remainder when (insert function here) is divided by (x-q)."
• Sub f(q) and solve, that is your remainder.
• Dividend = (Quotient)(Divisor) + Remainder
  f(x) = (Quotient)(x-q) + Remainder
  When you sub q in for x, the divisor becomes zero, and all you are left with is the remainder.
• If (x-q) id divided into some y=f(x) then the remainder will be f(q)

Even and Odd Functions

• NOT TO BE CONFUSED WITH EVEN AND ODD DEGREE
• If f(x) = f(-x) then the function is symetric about the y-axis
• If f(x) = -f(x) then the function is symetric about the origin
• If neither of the above are true, the function is neither even nor odd.

That seems like it. Hope it was accurate. Any questions feel free to comment, I'll get back to you. But not in time for this particular test, I'm off to bed. I promise earlier study sheets next time!