Concepts – MathMaine

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 the y-coordinate has not changed and the x-coordinate has been stretched to be three times further away from the y-axis… changing the shape of the curve in the process. If you were to examine any other pair of corresponding points, you would see the exact same scaling factor at work. This can be described algebraically by the equation:

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.

Horizontal Translations

In looking at the coordinates of the two corresponding points identified in the graph above, you can see that the y-coordinate has not changed and the x-coordinate has increased by 4. If you were to examine any other pair of corresponding points, you would see the exact same difference. This can be described algebraically by the equation:

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:

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

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.

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 zero or a positive quantity before the bars may be dropped. While this is fairly straightforward when working with constant values, as shown above, what happens when a pair of absolute value signs contains a variable?

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 quantities that have the same units, such as apples in the example above. If you were asked to add three apples and two oranges, the only answer you could give is “three apples and two oranges”, since neither addition nor subtraction make sense with quantities that have differing units. The concepts of addition and subtraction are based on combining or separating quantities with the same units.

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?

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 distributing the negative produces:

Negative Fractions

Question: Where should I put the negative sign when I am writing a fraction like negative two thirds?

Answer: As long as you write only one negative sign, it does not matter where you put it.

Two ideas are useful to keep in mind during the explanation that follows:
– Subtraction is the same thing as the addition of a negative.
– The negative of a number can be created by multiplying the number by negative one.

These principles apply to fractions as well, so:

$-\dfrac{3}{5}\\*~\\*~\\*=(-1)(\dfrac{3}{5})\\*~\\*~\\*=(\dfrac{-1}{~1})(\dfrac{3}{5})\\*~\\*~\\*=\dfrac{-3}{~5}$

Placing the negative sign before the entire fraction (subtracting the fraction) is equivalent to adding the same fraction, but with a negative numerator. But this is not the only option…

Geometric Sequences and Geometric Series

Geometric Sequences / Progressions

The terms “sequence” and “progression” are interchangeable. A “geometric sequence” is the same thing as a “geometric progression”. This post uses the term “sequence”… but if you live in a place that tends to use the word “progression” instead, it means exactly the same thing. So, let’s investigate how to create a geometric sequence (also known as a geometric progression).

Pick a number, any number, and write it down. For example:

$5$

Now pick a second number, any number (I’ll choose 3), which we will call the common ratio. Now multiply the first number by the common ratio, then write their product down to the right of the first number:

$5,~15$

Now, continue multiplying each product by the common ratio (3 in my example) and writing the result down… over, and over, and over:

$5,~15,~45,~135,~405,~1,215, ...$

By following this process, you have created a “Geometric Sequence”, a sequence of numbers in which the ratio of every two successive terms is the same.

Vocabulary and Notation

In the example above, 5 is the first term (also called the starting term) of the sequence or progression. To refer to the first term of a sequence in a generic way that applies to any sequence, mathematicians use the notation

$a_1$

This notation is read as “A sub one” and means: the 1st value in the sequence or progression represented by “a”. The one is a “subscript” (value written slightly below the line of text), and indicates the position of the term within the sequence. So $a_1$ represents the value of the first term in the sequence (5 in the example above), and $a_5$ represents the value of the fifth term in the sequence (405 in the example above).

Arithmetic Sequences and Arithmetic Series

Arithmetic Sequences / Progressions

The terms “sequence” and “progression” are interchangeable. An “arithmetic sequence” is the same thing as an “arithmetic progression”. This post uses the term “sequence”… but if you live in a place that tends to use the word “progression” instead, it means exactly the same thing. So, let’s investigate how to create an arithmetic sequence (also known as an arithmetic progression).

Pick a number, any number, and write it down. For example:

$5$

Now pick a second number, any number (I’ll choose 3), which we will call the common difference. Now add the common difference to the first number, then write their sum down to the right of the first number:

$5,~8$

Now, continue adding the common difference to the sum and writing the result down… over, and over, and over:

$5,~8,~11,~14,~17,~20,~23,~26,~29, ...$

By following this process, you have created an “Arithmetic Sequence”, a sequence of numbers that are all the same distance apart when graphed on a number line:

Vocabulary and Notation

In the example above 5 is the first term, or starting term, of the sequence. To refer to the starting term of a sequence in a generic way that applies to any sequence, mathematicians use the notation

$a_1$

This notation is read as “A sub one” and means: the 1st value of the sequence represented by “a”. The one is a “subscript” (value written slightly below the line of text), and indicates the position of the term within the sequence. So $a_1$ represents the value of the first term of the sequence (5 in the example above), and $a_7$ represents the value of the seventh term of the sequence (23 in the example above).

Piecewise Functions and Relations

While many relationships in our world can be described using a single mathematical function or relation, there are also many that require either more or less than what one equation describes. The behavior being described might start at a specific time, or its nature changes at one or more points in time. Two examples of such situations could be:

In the graph on the left, note that the blue line starts at the origin. It does not appear to the left of the origin at all. Furthermore, when x = 3 the blue line stops and the green line begins – but with a different slope.

In the graph on the right, note that the blue curve starts at x = 0. It does not appear of the left of the vertical axis at all. And when x = 3 the blue parabola turns into a green line with a very different slope. And the green line stops at x = 5.5, just as it reaches the horizontal axis.

These graphs do not seem to follow all the rules you were taught for graphing lines or parabolas. Instead of being defined over all Real values of x, they start and stop at specific values. The graphs also show (in this case) two very different functions, but in a way that makes them look as though they are meant to represent a single, more complex function. Both of these graphs are examples of “Piecewise Functions”, functions defined in a way that lets you mix and match as many separate components (pieces) as you wish.

Linear Systems: Why Does Linear Combination Work (Graphically)?

A system of linear equations consists of multiple linear equations. You can think of this as multiple lines graphed on one coordinate plane. Three situations can arise when looking at such a graph. Either:
1) No point(s) are shared by all lines shown
2) There is one point that all lines cross through
3) The lines lie on top of one another, so there are many points that the lines have in common.

There are four commonly used tools for solving linear systems:
– graphing
– substitution
– linear combination
– matrices
Each has its own advantages and disadvantages in various situations, however I often used to wonder about why the linear combination approach works. My earlier post explains why it works from an algebraic perspective. This post will try to explain why it works from a graphical perspective.

Consider the linear system:

$\begin{cases}y=-3x+2\\y=x-6\end{cases}$

which, when graphed, looks like:

Domain, Range, and Codomain of a Function

When working with quantitative relationships, three concepts help “set the stage” in your thinking as you seek to understand the relationship’s behavior: domain, range, and codomain.

Domain

The “domain” of a function or relation is:

the set of all values for which it can be evaluated
the set of allowable “input” values
the values along the horizontal axis for which a point can be plotted along the vertical axis

For example, the following functions can be evaluated for any value of “x”:

$f(x)=2x+1\\*~\\*g(x)=x^2+5$

therefore their domains will be “the set of all real numbers”.

The following functions cannot be evaluated for all values of “x”, leading to restrictions on their Domains – as listed to the right of each one:

$h(x)=\dfrac{1}{x}~~~~~~~~~\text{x cannot be zero}\\*~\\*j(x)=\dfrac{1}{(x-2)(x+4)}~~~~~~\text{x cannot be 2 or -4}\\*~\\*k(x)=3x+2,~1<x<10~~~~\text{only values between -1 and 10 may be used for x}$

The values for which a function or relation cannot be Continue reading Domain, Range, and Codomain of a Function