Mathematician Preemptive Strike 2 Due on Day 1 of Chapter 2

Transcription

1 Mathematician Preemptive Strike Due on Day 1 of Chapter Your first experience with a Wiggle chart. Not to be confused with anything related to the Wiggles Note: It s also called a Sign chart, but since most of you haven t seen the movie Signs, my showing a picture of Mel Gibson with a mean alien can t be as cute. And well, Mel has had some problems On your calculator in y1, graph y1 x Re-write the equation using a wiggle chart so that the absolute value signs are removed. 1. Determine the problem point. Problem points are the x-values that would cause the value in the absolute values to be 0. In this case, the point is x=. They really aren t all that problematic to find.. Draw a number line with only the problem point shown Let s go back in time for a second and remember something that you learned a long time ago You know that 6 6. You should also realize that the following is true: 6 ( 6) 6 Pay attention to that very last equality. To remove an absolute value when the information inside is negative, you are actually making the entire thing negative.. Use the number line to write inequalities using x Either x < or x >. Remove absolute value signs: When x <, the value inside of x will be negative. When x >, the value inside x is positive. Fill in your wiggle chart

2 To remove the absolute value signs, you write When x <, x = - ) = -x + When x >, x = ) = x These two equations should match what you see on your calculator for the graph of y = x Graph y x x If I CHOOSE TO use a wiggle chart, I need to start with a number line that has - and marked. To remove the absolute value signs, I need to consider three different situations. When x is less than -, when x is between - and, and when x is greater than. I need to consider what is inside those absolute value signs. When x < -, the value inside x + is negative, so to remove the absolute value signs, I need to write it as + ). Same thing with x. ). I put this onto my sign chart like this. I perform the same action between and. And the same action when x is greater than.

3 When x, ( x ) ) When x, ( x ) ) When x, ( x ) ) (When we remove the absolute value signs, when x<-, we need to put a negative sign in front of each of them) (The first equation gets a positive and the second one gets a negative) (Both of them get positive signs) Now remove parenthesis - When x, ( x ) ) When x, ( x ) ) When x, ( x ) ) (When we remove the absolute value signs, when x<-, we need to put a negative sign in front of each of them) (The first equation gets a positive and the second one gets a negative) (Both of them get positive signs) Simplify each of the equations and fill in the following piecewise function x f ) x x Then graph the function: Another example Changes occur when x = -6, -, 1, and Points associated with those x values (-6, 0), (-, 5), (1, 15), and (, 7) If I would let you get away with it (which I won t), at this time, you could plot those points and play connect the dots, then figure out the equations of the different lines. x < < x < < x < < x < x > Simplify x < -6 -x-15-6 < x < - -x- - < x < 1 x+1 1 < x < 6x+9 x > x+15

4 Graph the following and use (some kind of) a wiggle chart to remove the absolute value signs. Rewrite in piecewise function form. No work; no credit. 1. y x x 1 x. y x 1 x x 5. y x x x Simplify the following equations. Simplify does not mean expand. It means, Factor it completely and reduce where you can. Borrow someone s TI89 if you don t have one generally, if you type exactly what you see below, the TI89 gives you the reduced equation. THERE IS NO EXCUSE FOR MISSING THESE. BORROW AN 89. COME IN EARLY TO USE ONE OF MINE. NO EXCUSES!!! Note: You are using the 89 to CHECK these. You must show all work just as I have shown for the examples. No work no credit. Example and of course, I am doing an easy one ( x ) ()( x ) ( x) (6)( x ) (5x) ( x ) In the numerator, I see x s on both sides, I see +) s on both sides. I see that I have a on each side. I m going to factor out the parts that I have on both sides. x ( x ) ( x )( x ) ()(5) ( x ) I keep simplifying and NOW I expand what is left inside the parenthesis. x ( x ) ( x )( x 8x 16) 15 ( x ) x x 8x 16x 15 = ( x ) THAT is as simplified as it gets. 1.. ) ) 5 () ) () 7) ) ) ) 10 () () 7) )) (5) )

5 . Relatively difficult 1) 1 1 1) 1 ) 1) 1) 1 1 1) x When will we use Wiggle / Sign charts? You ll soon learn that you are not allowed to use them for your explanations on your AP test So why do you have to learn them? You are allowed to use them for your own use in your work they will help dramatically. Even though the AP graders will never look at the work, we can use Wiggle charts to help us determine when derivatives are positive and negative. When will we have to remove absolute value signs? We need to remove them to determine slopes of lines (derivatives) and we will use them again when we are determining area under curves. When will we have to reduce fractions like the ones that were in this worksheet? You will always need to reduce fractions to the simplest form. The ones that I gave to you are actually derivatives that I found for you and you simplified for me.