Monday, December 19, 2011

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 $f(x)=(x+7)^7$ looks like. He explained to me that he knows what happens on the graph when there is a double root, such as $f(x)=(x+7)^2$. 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 $x^2, x^3$.

The graphs of these look like:

Where $x^2$ is a parabola and $x^3$ becomes recognizable as the "cubic curve."

The interesting thing about powers, that is, functions of the form of $x^n$, 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 $f(x)=(x+7)^7$, 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

Sunday, December 18, 2011

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. ;)

Saturday, December 17, 2011

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.