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)

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

Combining or Collecting Like Terms

The phrases “combine like terms” or “collect like terms” are used a lot in algebra, and for good reason. The process they describe is used a lot in solving algebra problems. Two approaches, one intuitive and the other algebraic, can help in understanding why some terms are “like” terms, and others are not.

Quantities With Units

Suppose you are sitting in front of a table that holds three piles of fruit:
– five apples
– three oranges
– four apples
If someone asks you “What do you see on the table?”, how would you answer the question?

Chances are you answered “nine apples and three oranges”. Why did you combine the two piles of apples with one another, but not with the oranges? How did you know that you could do that?

The quantities of apples may be combined because addition or subtraction only work with  Continue reading Combining or Collecting Like Terms

Simplifying Fractions

Three concepts help explain the process of simplifying fractions:

  1. Multiplying a quantity by 1 has no effect
  2. A fraction whose numerator is exactly the same as its denominator is equal to 1 (unless the denominator equals zero)
    \dfrac{17a^2b}{17a^2b}~~=~~1~~~~a\ne 0,~~b\ne 0
  3. A product of two fractions can be rewritten as a fraction of two products (and vice versa)
    \dfrac{a}{b} \cdot \dfrac{c}{d}~~=~~\dfrac{ac}{bd}\\*~\\*\dfrac{ac}{bd}~~=~~\dfrac{a}{b} \cdot \dfrac{c}{d}

To simplify a fraction:

  • Rewrite both numerator and denominator as products of factors (if they are not already factored)
  • Examine both numerator and denominator to see if they share any factors
  • If they do share factors, use concept (3) above to move the shared factors into a separate fraction
  • That separate fraction should now have a numerator that is exactly the same as its denominator, which by concept (2) above means that it must equal 1, therefore by concept (1) above we can drop it from the expression

Consider the following fraction… can it be simplified? Continue reading Simplifying Fractions

Negative Differences

Algebra is a set of rules that allow us to change the appearance of an expression without changing the quantitative relationship that it represents. Sometimes the changes in appearance are greater than expected, causing us to doubt whether two expressions really do represent the same quantitative relationship.  The ways in which negative differences can be rewritten seem to surprise people until they become accustomed to them.

Consider a difference that is being subtracted:

b-(a-3)

If we wish to eventually drop the parentheses, we’ll have to distribute the negative sign that is in front of them first.  Leaving the parentheses in place while Continue reading Negative Differences