Graphs of Polynomials with Repeated Roots and How I Helped My Friend

Graphing polynomials is a tricky subject. I have a friend who asked me what the graph of looks like. He explained to me that he knows what happens on the graph when there is a double root, such as . But what does the graph of a septa root look like?

Just for some context - he teaches a class in preparing for the CSET, which is the subject matter test to get a teacher credential in that subject. He often asks me some questions on the subtler details of mathematics.

The basic polynomial graphs that are emphasized over and over in school are .

The graphs of these look like:


Where is a parabola and becomes recognizable as the "cubic curve."

The interesting thing about powers, that is, functions of the form of , where the exponent n is an integer, is that there is a nice property regarding whether the exponent is an even or odd integer. Recall that an even power makes gives a positive output for any positive input, and that an odd power will have a positive output for positive input and negative output for negative input. Let's look at some power functions on the same graph:


Notice that all the functions with even powers are making a nice U shape, and all the functions with odd powers are making a shape similar to the cubic curve. But as the power increases, while the shape is similar, the graphs are getting steeper. This is because we are multiplying the same number together more times, so we are getting a bigger number at the same input value, or x-value.

So, to answer my friend's question, we will shift the graph over 7:


You can think of these as shifted power functions, or polynomials with one root repeated. Notice that these graphs have the same shape as before, but they are shifted to the left 7 because of the +7.

To summarize, a polynomial with only one root can be thought of a shifted power function. Because of that, it has the shape of either a parabola or a cubic curve, but steeper as the exponent increases.

That answers the question about , and about polynomial functions with the same root repeated many times (in this case the the root is -7, repeated 4 times).

You might be wondering about polynomial functions with multiple roots. For a discussion on graphing polynomials in general, I recommend a very long but helpful discussion on Purple Math's website:
http://www.purplemath.com/modules/polyends.htm


Graphs in this post are from:
www.wolframalpha.com

Making an Herbal Tea Blend and How I Used Math to Do It

I'm trying to make as many Christmas presents as I can and as I have a great love of herbal medicine and teas, I thought I'd make some tea for Xmas. I'd share the actual recipe but, some people who might read this might be receiving some. So as not to give away the surprise, I'll just give the proportions.

Did I just use a math term? PROPORTIONS. I love math. And I used it while shopping for the herbs. I'll explain below.

I found a recipe that I was really excited about. It called for:
1/2 cup of herb #1
3/4 cup of herb #2
1/4 cup of herb #3
1 cup of herb #4
1/4 cup of herb #5

When I got to the health food store, I realized they didn't have cups to measure the portion of bulk herbs one wanted to buy. They did have scales. So, volume and weight are exactly the same thing. But, in the case of these herbs, they were all leafy, none of them were heavy roots or bulky like cloves. I figured they were close enough to being the same weight per volume. So, I decided to use the scale to get my proportions.

But did I need to know how much weight would get me 1/2 a cup? No, a recipe is just a proportional relationship between the ingredients. We have all heard of doubling recipes. You can half a recipe, make a third of one, or make 7 fifths of a recipe. As long as you keep the correct proportions it should turn out good.

For herb #1, I scooped what I thought was about 1/2 cup into a bag. On the scale, it weighed 0.05 pounds. How was I going to figure out how much of the other herbs to get?

Well, the recipe called for 1/2=0.5 cups of herb #1, for which I had measured out 0.05 pounds. 0.05 is just one tenth of 0.5. So the ratio I was looking for was 1:10.
So I also got:
0.075 pounds (3/4=0.75 cups) of herb #2
0.025 pounds (1/4=0.25 cups) of herb #3
0.1 pounds (1 cup) of herb #4
0.25 (1/4=0.25 cups) of herb #5

The weight in pounds I got for each herb was 1/10 the volume in cups. Yay for math! I can't wait for everyone to try it. I'll post the actual herbs after Christmas. ;)

Baking, Math and Calories

A late night working, I found myself craving some brownies (gluten-free of course). I pulled out a box of Arrowhead Mills Gluten Free Brownie Mix (http://www.arrowheadmills.com/product/gluten-free-brownie-mix), which I was skeptical about, not having tried it. I mixed the ingredients, popped it in the oven, and looked over the box one more time...

20 servings per box? And they told me to put it in an 8x8 pan? I could cut one side into 5 rows and the other side in 4 columns, but...that's hard to eyeball. I was mathematically inclined to cut both sides into 4. That would make 2in x 2in pieces...how many of them? Yes, 16 pieces.

I'm always curious exactly how good or bad food is for me. I started looking over the nutritional information. At the very top, it said 150 calories per serving. But the suggested number of servings is more than how I wanted to cut it. So how many calories are in my 2in x 2in servings?

So, first, I calculated how many calories are in the whole batch.
20 servings x 150 calories per serving=3000 calories (Don't eat the whole box in one day!)

Next, divide that by my number of servings: 16.
3000 calories/16 servings=187.5 calories per serving

Now, let's recap. First I multiplied by 20, then I divided by 16. We can actually combine this into one step: multiplying by 20/16. That's called a scaling factor; scaling the size of the serving scaled the number of calories. (It's actually kind of a relative scaling, since the serving size is based on the whole size, but you can understand the idea that the bigger the piece of brownie you eat, the more calories you are consuming.) Why stop there? We can figure out all the nutritional information for my preferred serving size. But before we make all those conversions, let's simplify our scaling factor.
20/16=(4x5)/(4x4)=5/4

So instead of multiplying all the nutritional information by 20/16, we can multiply by 5/4. We'd get the same answer anyway, because they are equivalent fractions.

Original Serving Size:
Fat 1.5g
Sodium 110mg
Carbohydrate 21g
Dietary Fiber 1g
Sugars 16g
Protein 1g

My 16 2inx2in servings:
Fat 1.5gx5/4=1.875g
Sodium 110mgx5/4=137.5mg
Carbohydrate 21gx5/4=26.25g
Dietary Fiber 1gx5/4=1.25g
Sugars 16gx5/4=20g
Protein 1gx5/4=1.25g

Well, this was enough to make me only eat one piece. Okay, I ate two - but at least I didn't eat the whole box! That's a lot of sodium and sugar! They were very tasty though.

101 Uses of the Quadradic Equation

http://plus.maths.org/content/os/issue30/features/quadratic/index

From astronomy, to car crashes to singing in the shower, the authors show how one of the most famous equations helps us out. For many young students, who can't let go of their cell phones, I like to remind them that they wouldn't have them without mathematics. This article shows how the quadratic equation lead to mathematical developments that allowed for cell phones, via imaginary numbers.

So, like I also tell my students, if you can't remember the quadratic equation get it tattooed on your wrist*! It deserves that special honor!

*I say that as a joke - I allow them to have a sheet of notes on tests to rewrite formulas and the likes.

Samovar, August 26th 2011

I have been here a handful of times pre-food-restrictions. I've always enjoyed their menu, particularly their tea services, which run in the 20's but come with a nice assortment of odds and ends plus tea. Their cheese and fruit plate, with honey comb, was always one of my favorite things to order there. For your average diner, this place is a wonderful treat, although a bit expensive. But today I will review it from the perspective of someone dining with food allergies or restrictions.



Today, I already had lunch and mostly came in for the delicious soy Matcha shake. At $8 it's a bit expensive, and you don't get that much. But I was in the mood to spoil myself a little, and cheat from the Paleo diet (soy is a bean, not a caveman food item).







The Gluten-Free Nitty Gritty:

As an Asian cuisine spot, this has a lot of potential to be mostly dairy free and gluten free, but potential gluten traps like soy sauce.



I decided to get a snack too and looked over the food items. To my delight, they had a key on the bottom to indicate which items are vegan and gluten free! Scanning through for item marked with that beautiful g next it, I was sad to see there were only a few. Of those few, some had soy sauce. Great! This must mean they have gluten free soy sauce. As a Celiac, you have to be extra cautious, to the extent where I've read advice on calling and talking to the chef in advance, asking tons of questions, and leaving if you aren't satisfied. Being new to this, I'm taking some baby steps. I pretty much settled on the marinated portabello salad, but in case it was marinated in gluten free soy sauce, I wanted to make sure they had gluten free soy sauce. The waitress, however, did not know. She was nice enough to go ask the kitchen, and I overheard the person she asked, ask another person, who did confirm that they have gluten free soy sauce. I got the portabello salad, thinking that I should have asked specially about that. I did ask the waitress if the salad was already gluten free and she made some show of checking the key. Perhaps this is all very new, and she is learning. Ideally, she would know the answers to most of my gluten and allergy related questions already.





The Dairy-Free Deal



While mostly Asian based, there are some salads, sandwiches etc that come with dairy. They also have a v* marking, that shows that items can be made vegan. This wouldn't mostly be by making without adding cheese. There are plenty of exciting asian style items without dairy on the menu though!





The Paleo Low-Down



There are nice salad options, although most menu items are either sandwiches or grain based meals. Many items come with crackers, and probably wouldn't be too fun wit hot the crackers. Also, soy is of course abundant on the menu. The portabello salad was probably the only purely paleo item on the food menu. Of course, with the teas, you could get almost anything except the Genmaicha, which is a brown rice green tea drink. For a Paleo dieter, I would say mostly come here if you want to have tea. In that regards, they have a wide variety of very high quality teas. I recommend their Earl Grey. It is exquisitely Bergamot flavored.



Ratings



Normal Dining Rating: 4/5 stars.

Minus a star for being a little pricey and pretentious without have whatever it takes (flavor, pizazz) to make it seem worth the price.



Food Allergy Dining Rating: 3/5 stars.

Minus a star for the waitress not have thorough knowledge of the gluten free options.





NOTE:

3/5 is high for the food allergy dining these days! There is must room for the restaurant industry to learn and grow with respect to the needs to food allergies. Restaurants are picking up on this; the change is happening now.

K&L Bistro, August 20th 2011

At 7pm, this place was busy, but we were offered a seat a lone table by the front windows. A small cozy joint, with friendly wait staff and a titilating menu with lots of meyer lemon, I was excited to try the place for the first time.





The Gluten-Free Nitty Gritty

As someone with many food restrictions, any conversation with the wait staff begins by spelling out my food allergies and asking about particular items I am interested in. This waitress "got it." I wanted the tuna tartar, which came with crackers. she pointed out that it has Ponzu sauce (has soy sauce in it) and would ask to have it made without, and also suggested susbtituting the crackers with lettuce. This is the kind of service that I would like for every dining experience! My partner and I had picked out 3 entrees and asked her which was gluten free. Out of the 3, 2 were already gluten free, so we ordered those. Even though the waitress was very knowledgeable, when delivering our food, she informed that she has let the kitchen know about my food allergies.





The Paleo Low Down

We had a steak with cumin fries and sword fish with caponata, a Sicilian cooked sweet and sour salad. The later was the most delicious thing we ordered, their version being made of eggplant, peppers and zuccini among other, with a nice tang to it. As being new to eating steaks, I requested for it to be cooked medium, which was probably a mistake since it was overcooked. The fries and sword fish were good, but not amazing. We also had a nice house red wine, a blend of Merlot and Cabernet. These items all fir into the Paleo diet...although, does wine really? Did cavemen have wine? Sure they had grapes, but goodness knows we can't go without some form of alcohol while fine dining. It goes hand in hand. Beer is clearly out of the question for gluten-free and paleo dining since the base ingredients were not part of a caveman's diet.



The Dairy-Free Deal



There were plenty of dairy free options. Many of the salads had dairy on them. This is usually disappointing to have without, since they often only have a few other items on them. My partner is fine ordering salads without the dairy, but it seems like a bummer to me. Our entrees were both originally dairy free and very satisfying.





Rating



Normal Rating: 4/5 stars

Minus 1 for good but not amazing food. The ambiance and service were good.



Allergy Dining Rating: 4/5 stars

Although it's nice to have the menu clearly marked which items are gluten free, or even to have a gluten free menu, the waitress was so knowledgeable and reassuring that it was ok. Preferably, I would like to be able to talk to kitchen staff in advance, tour kitchens etc. As a novice food critic, I am getting there. As a society we are just becoming aware of the needs of Celiac disease family members, friends, and customers. As this is a learning curve for all of us, I am positive they have room to grow to 5/5 stars.