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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 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