Solving Exponential Equations + Looking at exponential graphs.

I would like to start this off by saying sorry for the late blog post. Oh and that ..
ROXANNE WILL BE NEXT SCRIBE.
K thanks.

We started off our lesson with Mr. K's great line, "You are always looking for a pattern .."
SLIDE 3/10
Then the two questions. The one that people may have had problems with is the question with the solution in green.
Basically the same steps to solving exponential equations, but maybe a little more thinking.

49 becomes 7 ^ 2 .. All to the power of (x-2)
-You would then multiply 2 to (x-2)
.. 2(x-2)
.. 2x - 4
So the equation is now:
7 ^ 2x-4 = 7(7^(1/2))

I'm sure the square root caught most off guard. Both numbers have the same base so you add the exponents .. Which results in:
7 ^ 2x-4 = 7^(3/2)
..2x-4=3/2

Fractions are a nuisance at times, so we can multiply every term by two to get rid of it.
..4x-8=3

Thus solving it you'll find that x = 11/4

** YOU MAY ONLY MULTIPLY BASE NUMBERS IF EXPONENTS ARE SAME.
** YOU MAY ONLY ADD EXPONENTS OF BASE NUMBERS ARE THE SAME.

SLIDE 4/10

I was only caught off guard by the last question on this slide. It was a tricky one!
Again, you CANNOT multiply the base numbers unless the exponents are the same!
You had to take an extra step, and because all of us are smarter than the average bear, someone did.
The equation was divided by 2 to get rid of the useless 2.
..250/2 = (2(5))^2x-1/2
Now you have:
125 = 5^2x-1

Now it's the equation has become simply easy to solve.
-Work towards having the same base number, then solving for X.

5^3 = 5^2x-1

3 = 2x - 1
4 = 2x
2 = x

SLIDE 5/10

"Always looking for a pattern .."

May be tricky, new to the eyes, but definitely not new to the brain.

The equation is in QUADRATIC FORM. Meaning you must factor, but how? Where? When? .... Why?
There's another way to look at this equation. 3^2x is also ..
(3^x)^2
Making the equation become:
(3^x)^2 - 6(3^x) + 9 = 0

Ahhh... Now you see!

It would be easier, and you can do this. Let (3^x) = a .. or b .. or z .. or George
a^2 - 3a -3a + 9 = 0
(a - 3)(a - 3) = 0
Therefore, a = 3
Because we let (3^x) = a
.. (3^x) = 3^1
..X = 1

The second question is the same! Of course everyone got confused and did not accept REJECTION. It's not possible for this:
4^x = -1
So you simply ..REJECT!

The third question is more of an "INSPECTION" and there are other ways to prove it.
So this question is more of a check than a proof.
..4^x-9^x = 0

We got the same exponents ..

..(2^2)^x-(3^2)^x = 0
2x - 2x = 0
x = 0
If we apply that..
..4^0 - 9^0 = 0
1-1 = 0
0=0
//

SLIDE 8/10

Then we started to look over the graphing world of exponential graphs!
I personally found it interesting!
We were first asked to graph what you see on this slide.

SEE THE PATTERN?! .. AWESOME... AWESOME POSSUM =)

SLIDE 9/10

These are how the graphs look like!
y = 0 is the asymptote.
It will never touch the X axis. EVERY GRAPH HAS THE POINT (1,0)
We tried things such as a different coefficient and vertical and horizontal shifts.
We predicted how the graph would move according to the coefficient.
MASTER JABBAMATHEE predicted that the graph will slowly get bigger, after the point (1,0) it will quickly get bigger. As seen with the graph:
f(x) = 5^x
compared to f(x) = 2^x

SLIDE 10/10

Now we fiddle with our calculators!
We looked at Y = (-2)^x
REMEMBER, BRACKETS!

Then pressed ZOOM - and pressed "4"

Which gave us the graph you now see on the slide.

Sometimes there isn't a Y value there. Why? Because it doesn't exist. (IMAGINARY NUMBER)
.. The equation (-2)^x ..
As you see from the 2 examples. .6 and .5 Let's test them:

(-2) ^ (6/10)
= (-2)^(3/5)
Which allows the answer to be negative and existing. Meaning there is a Y value.

(-2)^(5/10)
= (-2)^(1/2)
.. IMPOSSIBLE! UNREAL ANSWER.

That's all we did for today!

HOMEWORK WAS EXERCISE 19 AND 20.
INCLUDING THE QUESTIONS THAT WE DID NOT DO BEFORE!

THIS IS NUMBER ELEVEN, signing out.

Intro to Exponential Functions

Hi everybody this is benofschool and here is today's scribe.

Well we're back from spring break and everybody is ready to do some math! Just as a reminder everybody's Flickr pictures are due on the midnight of Friday. Just post a direct link to your Flickr page onto the blog. Remember to tag it with your name, flickr, and trigonometry. For the picture do not contrive it or draw it and say that it is math. It is math but it isn't naturally occuring. The picture must exist as it is naturally. One example of a contrived shot would be a drawing of a parabola on a piece of paper. It is drawn but a natural one would be on that I had taken last year:
My picture wasn't contrived as in I didn't draw a parabola on the floor but I saw a parabola on this chair. I hope that explained everything. Okay back to our new unit.

Well we continued on what we did on the last day of class before spring break. We were asked to find other ways to write down several numbers with exponents and only exponents. Mr.K showed us many examples such as using negative exponents. Like in grade 9 math, negative exponents means find the power of the reciprocal of the number. For example:

The number 4 could be written as 16 to the power of 1/2 which is the same as the square root of 16. The second example 1/16 is a fraction so to get from that into a number with a denominator of 1 instead just use a negative exponent to get the reciprocal of 1/16. Then rest of the exponent says 1/2 meaning to get the square root of 16 which is 4. As a reminder of exponents and powers lets look at this picture of the anatomy of a power:

So a is the base which was 16 in the example above, b is the exponent which was 1/2 above and c is the power which is the result, 4.

Okay after find some of those we went back to solving for x but x was in a different place. x was an exponent. If you look at the slides 4 and 5 x is in the exponent spot. So to solve these just try to get the same base on both sides by using the techniques used to find other forms of numbers in exponents like earlier explained. When that is complete since the bases are equal then so must the exponents. So seperately just solve for x like in basic algebra if needed. Sometimes you end up with an algebraic expression like 2x-1 as an exponent. So if the base that has that exponent needs to be changed into another base make sure to multiply the new exponent into 2x-1. Then again solve for x.



Afternoon class. We had a pre-test on identities. Like the normal procedure several minutes to complete the pre-test and then we got into groups and work as a team to solve the questions not completed earlier. After handing in the test we went over the questions. The slides for the pretest is on the second slide of April 7, 2008. The first question nobody had trouble with but we went over the second. The questions asked for the sin(Π+Θ). So first to find sine we first found the sine of Θ which was the y-coordinate, n, over the square root of m squared plus n squared. But the question asked for the sin(Π+Θ) so add that value found earlier to Π and we get a negative value which was answer a. The 3rd question was simple so we did not go over it but the trick to do is was to draw a triangle and find the lengths of the sides. Then you should notice that A and B are the same angle meaning they have the same sides. Then just use the difference identity for cosine and all shall be found well just the answer really. Okay the next question was one that we went over. First just treat this like a normal algebraic equation for a quadratic and factor it. Then we get two answers but one is rejected because it isn't in the domain specified in the question. Then the other one was accepted, which was the answer. Now onto the long answer.

The long answer question was a pretty difficult. The tricky part was actually seeing the double identity on the left side. Change the sin2x into its identity and for the cosine one there are three different identities. On the sixth slide there is only one that would give us a tan by itself and it was the third one on the slide. Everything reduces nice and it leaves with tanx. The right side is a simple identity and we get tan also on the right side. Yay we proved it so remember QED.

We went back to practicing more identity problems. the one we had to do is on the next slide. We had to solve for x. All we have to do is just notice the identity on the left side and change the sin^2 (x) into 1-cos^2(x) and then it becomes a quadratic. Just solve for x like normal. The next slide shows how to find a variable on the calculator. So all we have to do is plug the left side of the equation onto Y1 and the right side onto Y2 and find the intercepts. Make sure to change the window's max and min values into -2Π to 2Π. The intercepts are the answers.

The next few lines are just identity practice for tomorrow's test. So good luck on the test everyone and homework is exercise #20 omit questions #10,11,12. The next scribe will be Eleven. Good night everybody and don't forget the flickr pics.

New Unit: Exponential Functions

Today was a new day, and today was a new unit. The unit was called exponential functions. Although we didn't do much on the new unit, we were introduced by trying to find 4 different ways to write the following numbers: 2, 3, 4/9, 1/4 with exponents. For an example Mr. Kuropatwa gave us two ways to write the number 2 in exponential form: 2^1 and (1/2)^-1.

While working on these questions we got all are previous tests and quizzes back. Everyone was pretty much down on their marks. Naturally, nice Mr. Kuropatwa discussed marks and how the number on your report card isn't very accurate because that was a number reflecting your knowledge at that point in time, and a given number of days or weeks later it would obviously be different, because in those number of days or weeks you most likely obtain a fuller understanding and a more complete idea on that one concept that you didn't do so well on at first.

Mr. Kuropatwa then talked about "How to be an Expert" http://expertvoices.blogspot.com/search/label/Assignment

It consisted of people who would try a certain thing or try to grasp a concept and realize that they can't do it so they give up. These people were considered to be in the "suck threshold". Then there came the amateur who would figure out how to do something, but just do it the same way over and over, not really ever moving anywhere just staying at that one point. Experts are different then these people because after they figure out this certain concept, they continuously try to improve on it and get better and better. These people are the true experts.

Leading to the end of class, we were given a project that can either be completed solo, in a pair, or with 3 members in a group. It was an assignment on the creation of 4, 5, or 6 math questions depending on the number of people in each group. With a soloist creating 4 questions, a pair creating 5 questions and so on. That required you to explain how you created each problem, and how you solve it step by step. These questions should expand over 2 different units and these should be units that you have had trouble on, in order to expand your knowledge even further. This whole assignment is worth 20% and a due date of your choice, with a final due date of around mid June. Sample problems which reflect the questions to be made for the final projects have to be chosen by April 7, 2008. These can be COPIED from sources such as your excerise book. The picture with trigonometric concepts, such as reflection, waves, etc. is also due by this date. That is all we did in pre-calculus class on the last day of school before spring break. Everyone enjoy their holiday!

Go YELLOOWWW! I'm Out.
-Francis