Practice: Modeling Events Using Periodic Functions

The following problems rely on the data contained in this spreadsheet. It has four tabs, and each problem will refer you to the appropriate tab. Note that this is real world data, not data generated for an “easy” math problem. Your model is NOT likely to hit all the data points perfectly.

Sunrise

1 – Copy the full year of Sunrise data from columns B and C on the “Sun + Moon” tab of the spreadsheet linked to above into your preferred graphing software.

2 – Analyze the data and develop a simple model for sunrise times in Portland, Maine as a function of the Day #, using one Sine or Cosine function and only one instance of your variable.

3 – Graph the sunrise times and your model’s results on the same graph

4 – What does your model predict well? Why?

5 – What does your model not predict well? What do you know (or learn from research) that would explain why sunrise times in some seasons in Maine differ from your simple model?

Birthday Tides

1a – Open the “All Tides” tab in the spreadsheet that is linked to above, find your birthday in the left column, and make a note of the “Elapsed Days” and “Water Height” of the first four tides on your birthday on your working document. The first tide could be high or low, depending on the time of year.

1b – Develop a simple model for the four tide Heights you noted as a function of Elapsed Days, using one Sine or Cosine function and only one instance of your variable.

1c – Use a calculator or software to overlay a graph of your model with a graph of the four points

1d – Which points does your model predict well? Why?

1e – Which points does your model not predict well? Why?

1f – Can you think of a way to improve your model if the restrictions were dropped, i.e. using more than one Sine or Cosine function and/or more than one instance of your variable?

2a – Starting with your model from question 1, but without restricting yourself to one Sine or Cosine function and only one instance of your variable, improve your model’s predictions using the ideas you came up with in the last step of question 1. In particular, can you improve your model by using functions of your variable instead of constants for the amplitude and/or vertical translation?

2b – Use a calculator or graphing software to overlay a graph of both models with a graph of your four data points

2c – Which model fits your four data points data better?

2d – Which model do you think will predict earlier or later data points better? Why?

3a – Expand your data set by including either the previous or the next 8 tide Elapsed Days and Heights (before or after those used above) from the “All Tides” spreadsheet.

3b – Add the 8 new points to your graph of both models

3c – Which model fits all 12 data points best? Why?

3d – Develop a model that fits all 12 data points better than either of the above models by making the amplitude a periodic function of your variable.

3f – Use your calculator or graphing software to compare the accuracy of your models’ predictions. Which model looks as though it fits the data better?

Function Transformations: Dilation

This post assumes you already familiar with analyzing function translations. Even if you are, reading Function Transformations: Translation may be a useful introduction, as it uses this same approach to understanding transformations. Note that
– Translations move a graph, but do not change its shape
– Dilations change the shape of a graph, often causing “movement” in the process

The red curve in the image above is a “transformation” of the green one. It has been “dilated” (or stretched) horizontally by a factor of 3. A dilation is a stretching or shrinking about an axis caused by multiplication or division. You can think of a dilation as the result of drawing a graph on rubberized paper, stapling an axis in place, then either stretching the graph away from the axis in both directions, or squeezing it towards the axis from both sides.

Transformations are often easiest to analyze by focusing on how the location of specific  points on the curve have changed. In the image above, the point A_1 on the green curve “corresponds” to point A_2 on the red curve. By this we mean that the transformation has moved point A_1 on the green graph to be at A_2 on the red graph.

Horizontal Dilations

In looking at the coordinates of the two corresponding points identified in the graph above, you can see that Continue reading Function Transformations: Dilation

Function Transformations: Translation

The red curve above is a “transformation” of the green one. It has been “translated” (or shifted) four units to the right. A translation is a change in position resulting from addition or subtraction, one that does not rotate or change the size or shape in any way.

Transformations are often easiest to analyze by focusing on how the location of specific  points on the curve have changed. In the image above, the point A_1 on the green curve “corresponds” to point A_2 on the red curve. By this we mean that the transformation has moved point A_1 to A_2.

Horizontal Translations

In looking at the coordinates of the two corresponding points identified in the graph above, you can see that Continue reading Function Transformations: Translation

Practice Problems: Ugly Linear Equations

The title of this post reflects how I categorize problems. The solution to each of the following problems is 7. Focus on finding the most helpful series of algebraic steps to take someone reading your work from the problem as stated to the solution. As the problems begin to include more and more terms, be cautious about doing too much in any one step – as that is how errors often arise.

  1.   2(15-a)~=~4(a-3)
  2.   b-9+3b~=~10+5b+50-8b-20
  3.   3(m-1)-(m+3)~=~2(5-m)+(m+5)
  4. Continue reading Practice Problems: Ugly Linear Equations

Using Corresponding Points to Determine Dilation Factors and Translation Amounts

Two earlier posts provide background information for this one: Function Translations and Function Dilations. If you are not already familiar with these topics, you may benefit from reading those first.

Given two points on a curve and their corresponding points after transformation, how does one determine the underlying transformations? Since two dilations and two translations may be taking place, it can be complex to try to separate the effects of dilation from those of translation.

As an example, consider the two curves above. The green curve is the graph of

y_1(x_1)~=~(x_1 -1)^2+1

and the red curve is a transformation of the green one. Two points are labeled on the green curve:

A_1:(1,1)\\*~\\* B_1:(2,2)

and their corresponding transformed points are labeled on the red curve: Continue reading Using Corresponding Points to Determine Dilation Factors and Translation Amounts

Pi Notation (Product Notation)

The Pi symbol, \prod, is a capital letter in the Greek alphabet call “Pi”, and corresponds to “P” in our alphabet. It is used in mathematics to represent the product (think of the starting sound of the word “product”: Pppi = Ppproduct) of a bunch of factors.

If you are not familiar or comfortable with Sigma Notation, I suggest you read my post on Sigma Notation first, then come back to this one – because Pi Notation is very similar.

Once you understand the role of the index variable in Sigma Notation, you will see it used exactly the same way with Pi Notation, except that Continue reading Pi Notation (Product Notation)

Sigma Notation (Summation Notation)

The Sigma symbol, \sum, is a capital letter in the Greek alphabet. It corresponds to “S” in our alphabet, and is used in mathematics to describe “summation”, the addition or sum of a bunch of terms (think of the starting sound of the word “sum”: Sssigma = Sssum).

The Sigma symbol can be used all by itself to represent a generic sum… the general idea of a sum, of an unspecified number of unspecified terms:

\displaystyle\sum a_i~\\*\\*=~a_1+a_2+a_3+...

But this is not something that can be evaluated to produce a specific answer, as we have not been told how many terms to include in the sum, nor have we been told how to determine the value of each term.

A more typical use of Sigma notation will include an integer below the Sigma (the “starting term number”), and an integer above the Sigma (the “ending term number”). In the example below, the exact starting and ending numbers don’t matter much since we are being asked to add the same value, two, repeatedly. All that matters in this case is the difference between the starting and ending term numbers… that will determine how many twos we are being asked to add, one two for each term number.

Continue reading Sigma Notation (Summation Notation)

Practice Problems: Three Step Linear Equations

The solution to each of the following problems is 20. Focus on finding the most helpful three or four algebraic steps to take someone reading your work from the problem as stated to the solution.

  1.   3(x+10)~=~90
  2.   2x-20~=~60-2x
  3.   \dfrac{2x}{3}~=~\dfrac{60-x}{3}
  4. Continue reading Practice Problems: Three Step Linear Equations

Practice Problems: One Step Linear Equations

The solution to each of the following twenty problems is 12. Focus on finding the most helpful algebraic step to take a reader from the problem as stated to the solution, and be sure you can explain why that step leads to a solution.

  1.   x+7~=~19
  2.   3+x~=~15
  3.   11~=~-1+x
  4. Continue reading Practice Problems: One Step Linear Equations

Math: Pen vs Pencil

Many math students are given strict instructions by their teachers to do all their work in pencil. I disagree.

The advantage of doing work in pencil is that:

  • it is easier to erase, so students are less likely to be paralyzed by “I am not sure this is correct, so I don’t dare write it down”

The disadvantages of working in pencil are that: Continue reading Math: Pen vs Pencil

11 Ways To Do Better In Math

1) Reflect and Summarize
at the end of each class, at the end of each week, at the end of each month. Review your notes and/or think back over the material that has been covered, then decide which skills or ideas you think are most important. Summarize the material you are learning as concisely as you can, because summarizing helps you learn. Identify any skills or ideas that you are not confident about. Write your reflections and summarizations as part of your notes, with reminders about what needs more work.

2) Use scrap paper
Using scrap paper removes a source of anxiety when Continue reading 11 Ways To Do Better In Math

Interactive Graphs for Linear, Quadratic, Rational, and Trig Functions Moved to GeoGebraTube

Some may have had trouble using my GeoGebra applets in their browsers. I have moved all of them to GeoGebraTube, which will hopefully fix the problem. You may search for them by typing “MathMaine” into the GeoGebraTube search box.

Links to all updated interactive graph applets are below. Comments and suggestions are always welcome!

Linear Functions

GeoGebraBook: Exploring Linear Functions, which contains:

Interactive Linear Function Graph: Slope-Intercept Form

Interactive Linear Function Graph: Point-Slope Form Continue reading Interactive Graphs for Linear, Quadratic, Rational, and Trig Functions Moved to GeoGebraTube

Unit Circle Symmetry: a GeoGebraBook Exploration

Check out this GeoGebraBook of nine applets that will help you explore Unit Circle Symmetries. It contains three applets per type of symmetry on the unit circle, one focusing only on the unit circle, and the other two linking unit circle properties to patterns in the graphs of the sine and cosine functions.

When two angle expressions, such as \theta and (\pi -\theta ), exhibit symmetry on the unit circle, an understanding of unit circle symmetries and reference angles often allow function arguments to be simplified. Mastery of symmetries and reference angles will also be very handy when expanding inverse trigonometric function results to describe all possible answers to a problem.

Suggestions for improvements to these applets, or additional applets, are always welcome via comments on this post.

Absolute Value: Notation, Expressions, Equations

What Does Absolute Value Mean?

The term “Absolute Value” refers to the magnitude of a quantity without regard to sign. In other words, its distance from zero expressed as a positive number.

The notation used to indicate absolute value is a pair of vertical bars surrounding the quantity, sort of like a straight set of parentheses. These bars mean: evaluate what is inside and, if the final result (once the entire expression inside the absolute value signs has been evaluated) is negative, change its sign to make it positive and drop the bars; if the final result inside the bars is zero or positive, you may drop the bars without making any changes:

\lvert ~1-4~ \rvert\\*~\\*=~\lvert ~-3~ \rvert\\*~\\*=~3

Another example is:

\lvert ~4-1~ \rvert\\*~\\*=~\lvert ~3~ \rvert\\*~\\*=~3

Note that absolute value signs do not instruct you to make “all” quantities inside them positive. Only the final result, after evaluating the entire expression inside the absolute value signs, should be made positive.

\lvert ~1-4~ \rvert~\ne~\lvert ~1+4~\rvert~~\text{ Do not make this mistake!}

Absolute Value expressions that contain variables

Just as with parentheses, absolute value symbols serve as grouping symbols: the expression inside the bars must be evaluated and expressed as either Continue reading Absolute Value: Notation, Expressions, Equations