Session ETD 545
Visual Analysis: Identities and Forms of Calculus
Andrew Grossfield, Ph.D., P.E.
Vaughn College of Technology
Introduction
The most important concept of mathematical form is too often neglected in the elementary math
syllabus. Mathematical objects can have many forms but need to be described in a form in order
to be grasped and transferred. Professional mathematicians who swim in an ocean of forms take
the concept as axiomatic, but a student who is learning to factor quadratic functions is only
taught a procedure. Testing to discover which students have mastered a procedure is easy. It is
not as easy to test if a student has grasped a concept.
Identities are fantastic equations! Identities state that two different and seemingly unrelated sets
of operations actually produce the same function. For every value of the argument, both sets of
operations yield the same value. Therefore the functions on each side of the identity have the
same table and describe the same curve in a coordinate system. All of the properties of the curve;
slopes, areas, arc lengths, etc. must be the same regardless of which side of the identity was used
for their computation. In order to solve algebraic and differential equations, and to compute
rates of change, arc lengths, areas and volumes, these identities are needed to change to the form
most suitable for computation.
This paper is an application of the visual techniques described in the paper “Visual Analysis and
the Composition of Functions” to study and verify some common identities needed in calculus
courses. 8 The most important of these techniques rely on the mathematical principle that smooth
operations on smooth curves yield resultant smooth curves. In particular the curves resulting
from sums, products differences and quotients of smooth curves will be smooth except at points
where a denominator is zero. The identities examined will be grouped according to the kinds of
function involved: polynomials, rational functions, and algebraic, trig, exponential and
logarithmic functions.
The exercises serve to develop a student’s abilities in visual analysis. The student is encouraged
to visualize the construction mentally and read the accompanying text only if he/she seeks
additional explanation. The written description is cumbersome to compose and must be tedious
to read. This paper is not intended to be read. The reader should focus on the relationship
between the algebraic description and the graphical constructs.
In the arena of changing forms, a student reader should acquire an appreciation of the clarity,
brevity, flexibility, fluidity and power of algebraic notation as it has evolved over the centuries.
And while algebraic notation is by far superior to ordinary English prose, the ability to visualize
mathematical concepts provides far more insight than can be acquired from algebraic notation.
However because visualization is volatile, it is necessary to pin down our mental constructs. And
since prose is too cumbersome, we must rely on algebraic notation.
Proceedings of the 2017 Conference for Industry and Education Collaboration
Copyright ©2017 American Society for Engineering Education”
ETD 555
The following score or so of examples of identities are intended to compare the graphical, the
algebraic and the English prose descriptions of the concept of a mathematical function, named
and notated by Leonhard Euler in 18th century. 4
Identities of Trigonometric Functions
Polynomial Identities
1
2x – 6 = 2(x – 3)
2.
x2 – 2x = x (x – 2)
3.
x – 8x + 15 = (x – 3) (x – 5)
3a.
x – 8x + 15 = (x – 4) – 1
4.
5.
2
2
2
x3 – 4x = x(x2 – 4) = (x + 2) x (x – 2)
14.
sin (α) + cos (α) = √2 sin (α + π/4)
15.
sin 2 (α) and cos2 (α)
15a.
sin 2 (α) + cos2 (α) = 1
16.
double angle identities
cos2 ( α ) = ½{1 + cos( 2α }
(x – 2) = x – 6x + 12x – 8
3
3
2
sin2 ( α ) = ½{1 – cos( 2α }
cos (2α) = cos2 (α) – sin2 (α)
Identities of Rational Functions
17.
6.
7.
8.
9.
8
x2 – 16
1
x2
=
–2x + 2
x−1
x2 –2x + 2
x3 −2
x−2
1
x−4
=
=
−
x+4
1
.
18a.
x−1
(x−1)2 +1
1
x−2
Identities of Algebraic Functions
10.
11.
√16 − x 2 = √4 − x √4 + x
1
1−√x
=
1+√x
1− x
1
12.
{x 2 } 2 =√x 2 = x
13
{ x 2}
1
2
Identities of the Exponential Functions
(x−1)2 +1
= x2 + x + 1 +
sin(2α) = 2 sin(α) cos(α)
1
(2x+2) = (2x) (22) = 4 (2x)
(2–x) =
18b.
18c.
18d.
1
2x
e 2x = {ex}2 = ex ex
sin (x)
2x
= (2–x) sin(x)
Identities of Logarithmic Functions
19a.
log( x2 ) = 2log(x)
19b.
log( x2 ) = 2log(x)
20a.
20b.
log (x2 – x)= log (x) + log (x – 1)
log (x2 – x)= log (x) + log (x – 1)
2
= {√x} = x
an identity in x; x = ln(e x) and
an identity in y;
y = e ln(y)
Session ETD 545
My hope is that the visual description of these concepts which have been condemned by
American mathematicians in this past century will help future technology and engineering
students gain insight into their study of analytic functions.
The Straight Line: A Simple Beginning
1
To begin the first identity is 2x – 6 = 2(x – 3).
A mathematician might say that this
is an example of the distributive law and move on but the equation viewed as an identity states
that a diagonal line first displaced 3 units to the right and then doubled yields the same result as
first doubling the vertical values and then subtracting 6. The generalization to other straight lines
is mx – mxo = m(x – xo).. We find that the resulting line will, in both cases, have the same slope
and that the vertical intercept is the negative of the slope times the horizontal intercept. ( I wish
the English language permitted a simpler description. ) Graphs of the left and right hand sides of
the identity are shown in Figure 1. I will use the abbreviations, “LHS” to mean the” left hand
side” of an equation and “RHS” for the right. All of the following graphs were constructed using
the wonderful free equation plotting software called “WINPLOT.” 14
y
y


y= x
y= x


x
x








y= x- 3



Figure 1
y = 2x - 6







Identity 1 LHS
y = 2(x - 3)
Identity 1 RHS
Identities of Polynomial Functions
In the study of curves in Cartesian coordinates, it is usual to use the variable x to locate
horizontal displacements from the origin and the variable y to locate vertical displacements. A
“smooth” curve has a tangent line at every point. I use the word smooth instead of the word
“differentiable” which is used in most conventional calculus texts.
It should be noted that the graphs of all polynomials are single-valued, smooth curves that start
on the left at infinity in either the 2nd or 3rd quadrants, perhaps oscillate as they move to the right
Proceedings of the 2017 Conference for Industry and Education Collaboration
Copyright ©2017 American Society for Engineering Education”
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and then head off to infinity in either the 1st or 4th quadrants. This behavior will be observed in
the following four examples.
2.
Our next example is the 2nd degree polynomial whose graph is a vertical parabola
extending toward infinity in the remote regions of the 1st and 2nd quadrants. The identity
x2 – 2x = x (x – 2)
asserts that the expanded form on the LHS represents the same curve described by the factored
form on the RHS. The graphs of both sides of the identity are displayed in Figure 2.
y
y


y = x
2


y = x
x

x


2
y = x - 2x

y = x (x - 2)

y = x- 2
y = - 2x
y = 2(x - 3)
Figure 2

Identity 2 LHS
y = 2(x - 3)
Identity 2 RHS
The LHS of the equation states subtract a straight line from the parabola y = x2, whose vertex is
at the origin. The RHS side of the equation represents the product of the vertical values of two
straight lines. The subtraction of the straight line has the effect of pulling the RHS of the
parabola down and raising the LHS of the parabola. The result is still a 2nd degree polynomial, a
parabola but one whose vertex has been lowered and shifted toward the right.
The initial parabola has a double zero at the origin but the resulting parabola has two separate
zeros, one at the origin and the other at x = 2. We will see in examples 3, 4 and 5 that the
factored form discloses the zeros of the polynomial while the zeros are usually hidden in the
expanded form. It is this presentation of the roots that makes the factored form desirable. For
this reason students need to develop their polynomial factorization skills.
Constructing the parabola as the product of the vertical values of two rising straight lines, we
should see that wherever either of the lines is zero the product will be zero, whenever the vertical
values of both lines have the same sign then the product will be positive and whenever the signs
of the vertical values of both lines differ then the product will be negative. The result in this case
will be a parabola which is negative between the two zeros and positive otherwise. It should be
observed that the axis of symmetry and the vertex are midway between the zeros.
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3.
Let us next examine the polynomial identity x2 – 8x + 15 = (x – 3) (x – 5) which also
equates the expanded form of a 2nd degree polynomial with its factored form. What is to be
learned from the expanded form on the LHS is that the leading term is positive so the vertical
values grow positively for values of x far from the origin. We also see in the expanded form that
when x is zero the y-value, called the y-intercept will be 15. We learn from the factored form
that the zeros occur at the zeros of the factors which are the values of either 3 or 5. The axis of
symmetry will be midway between 3 and 5, which is x = 4. The y coordinate of the vertex can
be obtained by setting x = 4 in either of the two forms. As we see either side will yield the value
y = –1. The coordinates of the vertex are found to be (4, –1).
y=x
2
y=x-3
2
y = x - 8x + 15
y = (x - 3)(x - 5)
2
y=x -5
y = x - 8x
y = - 8x
2
Figure 3
Expanded form of the parabola
Factored form of the parabola
3a.
Continuing with the same parabola, we will examine the associated identity whose graph
is in Figure 3a:
x2 – 8x + 15 = (x – 4) 2 – 1
y=x
2
2
y = (x - 4)
2
y = (x - 4) -1
Figure 3a
The translated – vertex form of the parabola
2
This identity equates the expanded form of the parabola and a form (called the vertex form)
which exhibits the vertex (4, –1). The vertex form can be seen as the result of shifting the
parabola y = x2 by 4 units to the right and 1 unit down. The relationship between the
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displacement of a curve in a Cartesian coordinate system and the effect on its equation is
reviewed in the above paper on Visual Analysis and many algebra texts. 8 The procedure for
computing the vertex form of the parabola, called “completing the square”, can be found in
conventional algebra textbooks.
4.
Our next polynomial example is the cubic identity
x3 – 4x = x(x2 – 4) = (x + 2) x (x – 2) .
This identity equates the expanded form of a polynomial with its factored form. The positive
leading term x3 of the expanded form on the left indicates that the curve starts at – infinity in
the third quadrant and ends at + infinity in the first quadrant. The curve has odd symmetry and is
seen as a cubic which is added to the descending straight line y = – 4x as shown in the left side of
figure 4. For small positive values of x, adding the descending line initially pulls the cubic down
but ultimately since the cubic rises faster than the straight line the resulting curve must attain a
minimum and then begin to rise.
The factored form on the right indicates that the curve has zeroes at the values x = –2, x = 0 and
x = +2. It should be noted that wherever an odd number of the factors is negative the resulting
product is negative as is seen when the values of x are less than –2 or in the interval between 0
and +2. Both cases support the contention that the results of adding and multiplying smooth
curves are smooth curves.



y=x
3
y = (x + 2) x (x - 2)
3
y = (x - 2)



3




y=x -4x


y=x+2
3
y = (x - 2)





y=x



y=x-2


y = - 4x




y= x
Figure 4 Adding a descending line to a cubic
y= x+ 2
Multiplying three straight lines
5.
Our last polynomial identity is the cube of a linear factor: (x – 2) 3 = x3 – 6x2 + 12x – 8
The LHS discloses that the curve is the cubic y = x3, displaced two units to the right and so rises
monotonically but crosses the horizontal axis with a horizontal tangent line as shown in figure 5.
The right side of figure 4 shows some of the components of the expanded form of the cubic.
Displayed are the curves y = x3, y= – 6x2 and their sum y = x3 – 6x2 . When the straight line
y = 12x is added to x3 – 6x2 the curve that results is seen to be 8 units more than the translated
cubic shown on the left side of figure 5. I would not claim that seeing the relationship of the
curves in the right side of figure 5 is easy, but I think it is worth a student’s effort.
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






3


y = (x - 2)





3

2
y = x - 6 x + 12x

y=x

3
2
y = x - 6x + 12x

3


















Figure 5 A cubic shifted to the right by 2 units
and then raised by 8 units



3
y = x - 6x
y = - 6x

2
2
Curves in the expanded form of the cubic
Identities of Rational Functions
The next four examples are identities of rational functions. Rational functions are quotients of
polynomials, say y = (5x3 – 2x2 +5x – 6) / (x2 – x + 3).
It should be noted that the graphs of all rational functions are single-valued and smooth curves
except at the finite number of points where the horizontal coordinate produces a zero in the
denominator. These special points where the denominators are zero are called singularities.
These points are of two types: In the one type of singularity, called a pole or vertical asymptote,
the vertical values become infinite on both sides of the singularity as in the function y =
1
1
x
or the
function y = 2 . In the second kind of singularity, called a point gap, it appears as if a point was
x
removed from a continuous curve. This behavior disturbed mathematicians in the early 19th
century since the function values were not defined at the point gaps. The vertical coordinates of
the point gaps are called the limiting values of the function at the point in question. An example
of a function with a point gap is y =
x2 – 9
x–3
0
. It can be seen that when x = 3, then y = , whose
0
value at x=3 cannot be computed. The reader can find a discussion of the point gap phenomena
in the paper “Mathematical Definitions: What is this thing”. 9
If the numerator degree is less that the denominator degree then when x is large the values
computed for y will be small and the curve will converge to the horizontal axis asymptotically.
In this case the rational function is described with the word “proper.”
Otherwise the rational function is described as “improper.” If the numerator and denominator
degrees are equal the curve will approach some nonzero value asymptotically. As an example,
for the function y= (4 x2 – 2) / (x2 + 1), when x is large, y will approach the value 4.
Improper rational functions can be decomposed by polynomial division into the sum of two
unique parts. One part is a polynomial and the other is a proper rational function. For large x,
since the vertical values of the proper part converge to zero and the rational function will
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asymptotically approach the polynomial part. This behavior will be seen in example 9 below for
the function:
y=
x3 − 2
x−1
1
= x2 + x + 1 −
x−1
This function has one pole at x = 1which is the zero of the denominator of the proper part of the
decomposition. As x becomes extremely large, the vertical values of the rational function y(x)
above approach the polynomial y = x2 +x+ 1.
Identities of rational functions can always be transformed into polynomial identities by
multiplying both sides of the equation by the denominators. We observe that
8
x2
6.
– 16
1
=
1
−
x−4
can by transformed into 8 = (x + 4) – (x – 4).
x+4
We continue to examine the equation
8
x2
1
=
– 16
−
x−4
1
x+4
. On the LHS the
number 8 is divided by quadratic with two roots. On the RHS side, a reciprocal of a linear term is
subtracted from a second such reciprocal. Since the equation is an identity both sides describe
the same proper rational function. All of the terms are proper rational functions causing the
vertical values of both sides to approach zero for ± large horizontal values. When x = ± 4, both
sides become infinite and have poles. When x has values different from either + 4 or −4 both
sides are single-valued and smooth. The vertical values of both sides are negative for |x| < 4 and
positive for |x| > 4.


y = -1/ (x+4)

y = -1/ (x+4)

y= x- 4

2
y = (x - 16) / 8






























y= -(x+4)

y = 1 / (x-4)

2

y = -1/ (x+4)
y = 8/ (x - 16)


y =-6x
2
y =3x 2- 6x
y =-6x
2
Figure 6 The reciprocal of a 2 root parabola
y =3x 2- 6x
The sum of two vertical hyperbolas
These graphs illustrate the general principles that the reciprocals of large numbers are small and
vice versa. And in addition the reciprocals of positive values are positive and the reciprocals of
negative numbers are negative.
7.
Now we examine the equation
1
x2 –2x + 2
=
1
.
(x−1)2 +1
This function is again the
reciprocal of a parabola but the LHS vertex form of the denominator shows there can be no
zeroes. Therefore, this function has no poles and is smooth everywhere. On the LHS, we see that
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1
the curve crosses the vertical axis at y = . On the RHS we see that the vertex of the
2
denominator parabola is above the horizontal axis and so the denominator has no zeros and
therefore the curve has no poles. The parabola starts infinitely high in the second quadrant
descends smoothly to a minimum at x =1 and then rises toward infinity in the first quadrant.
Since the reciprocal of large numbers is small and vice versa the reciprocal of the parabola rises
smoothly in the second quadrant from zero, reaches a maximum at x = 1 and then descends back
toward the horizontal axis. The line x = 1, which is the axis of symmetry of the parabola, also
serves as the axis of symmetry of the reciprocal.


2
2
y = x - 2x +2
y = (x - 1) + 1




2
2
y = 1 /{( x - 1) + 1}
y = 1 / ( x - 2x +2)

Figure 7
8.
This example


y =-6x
x−1
x2 –2x + 2
2
=
y = x - 6x3 2y = 8/ (x - 16)
x−1
(x−1)2 +1

y =-6x
2
y = x - 6x3 2y = 8/ (x - 16)
is similar to the last except the function is multiplied
by a linear term. As do all proper rational functions, this function approaches zero as |x| becomes
large. On the LHS we observe that the y intercept is − 1/2. On the RHS it can be seen that since
there are no zeros in the denominator, there are no poles and the function is single-valued and
z
smooth everywhere. If you let the variable z = x−1, then the function becomes f(z) = 2 . The
z +1
function f(z) exhibits odd symmetry about the value z = 0. Our function f(x) then exhibits odd
symmetric about the value x = 1, being positive for x > 1 and negative for x < 1. Starting at
x = 1 our function rises from zero, reaches a maximum and then decays toward zero with
increasing values of x.

y = x-1
x-1
y = -------------2
( x - 1) +1
2
y = 1 / ( x - 2x +2)





y =-6x 2
Figure 8
y = x -3 6x
2
2y = 8/ (x - 16)
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9.
Our last example of a rational function identity is the improper rational function
x3 − 2
1
= x2 + x + 1 +
x−1
x−1
which on the LHS is a cubic divided by a linear polynomial. The RHS represents the
decomposition into a quadratic plus the reciprocal of a linear term. The parabolic term reveals
that the vertical values of the curve go to infinity in the 1st and 2nd quadrants. Both sides reveal
the pole at x = 1. The LHS of Figure 9 shows the components of the improper rational function
and the RHS and shows the function hugging the parabola far from the origin.



3
x -2
y = --------x -1

2
y = x +x+1


2
y=x +x+1



















-1
y = --------x -1

Figure 9 Components of improper rational function


Improper rational function and parabola
Identities of Algebraic Functions
While polynomials and rational functions are always single-valued, the curves derived from
algebraic functions, like circles and hyperbolas, may be multi-valued. While polynomials have
no singularities and rational functions have singularities at only a finite number of points,
algebraic curves can disappear over entire intervals where the values inside a square root may be
negative. In the following two examples we will make the function single-valued and consider
only the positive values of square roots.
10.
In the example √16 − x2 = √4 − x √4 + x it is seen that the LHS is the square root of a
vertical parabola that extends down to infinity in the 3rd and 4th quadrants. The graph of the
square root of a linear term is a horizontal parabola. The RHS is the product of two such
parabolas, one of which opens to the left while the other opens to the right. On the LHS it can be
seen that the y-intercept is 4 and that the curve does not exist when |x| > 4. On the RHS it is seen
one of the factors does not exist on the interval where x > 4 and the other disappears whenever
x < – 4 . On the LHS we see that since the square root operation is monotonically increasing, the
function √16 − x 2 increases when 16 – x2 increases and decreases otherwise. For x between –4
and 0 the vertical values increase from 0 to 4. Squaring the both sides of the equation
y = √16 − x 2 reveals that the curve is the upper part of a circle of radius 2.
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
2 .5
y = (16 - x )


y = 16 - x
2




.5
.5
y = (4 + x )
y = (4 - x ) .


2 .5
y = (16 - x )








.5 .




y =-6x
2
y =3 x2 - 6x
y =-6x
2
2
y =3x 2- 6x
2
Figure 10 Two constructions of a circle
11.
The identity
1
1−√x
=
1+√x
1− x
is used in a procedure called “rationalizing the
denominator,” whose object is to describe an equivalent form whose denominator has no
radicals. The curve does not exist when x is not positive. On the LHS the denominator 1 − √x
starts at y = 1 and decreases monotonically to – infinity in the 4th quadrant, passing through the
horizontal axis when x = 1. The reciprocal must then start at y = 1 and increase monotonically to
+ infinity when x = 1. On the other side of the pole, the vertical values start at – infinity and
continue to rise monotonically to zero. On the RHS, for 0 < x< 1 the numerator is positive and
increasing while the denominator is positive and decreasing. The effect of both changes with
increasing x is to produce a curve that rises to infinity. For x larger than 1 the denominator is
now increasing at a faster rate than the numerator but negatively. The result is that the function
values rise monotonically from – infinity to zero.



1
y = --------.5
1- x
.5
1+ x
y = --------1-x

1+ x




.
.


.5





.5
y=1- x



y =1-x


1

y = -------.5
1- x

y =-6x
2
y =3x 2- 6x
Figure 11 Algebraic curve LHS

2

.5
1+ x
y = -------1-x
y =-6x
2
y =3x 2- 6x
RHS
2
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12.
Since we are taught that the square root is the inverse of the square, it should be that the
1
square root of x2 equals x, that is {x 2 } 2 = √x 2 = x. But it is not so. The function x2 is always
positive and the positive square root is the ray y = x when x > 0. However √x 2 is the descending
1
straight line, y = – x when x is negative. This means that {x 2 } 2 = √x 2 = |x|. If the doublevalued square root is allowed then the graph is the union of the two straight lines y = ± x.

y=x
2


y=x
2


y=-x
.

y=+x




y=±x
.


y=±x




y =-6x 2
2y3 = x - 6x
Figure 12 The square root of a square
y =-6x
2
2y3 = x - 6x
The double-valued square root of a square
1
2
2
As another example, if only the positive square root is allowed, then { x 2 } = {√x} = x only
for positive values of x. Since negative numbers do not have square roots the straight line does
not exist in the 3rd quadrant and only the ray in the 1st quadrant remains.
y = + x: x > 0

y=+x

.5
.

Figure 13 The square of a square root

y =-6x
2
y = x - 6x
32
Identities of Trigonometric Functions
14.
The elementary trigonometric identity sin (α) + cos (α) = √2 sin (α + π/4) is an
instance of a basic principle that lies at the heart of AC circuit analysis and the theory of
vibrations. The mathematician C. P. Steinmetz promoted this profound concept, which is
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included in the curricula of all engineering students. The principle is that whenever a finite
number of sinusoids of the same frequency, of any amplitudes and phase shifts, are added, the
result will be a sinusoid of the same frequency unless the sum is always zero. This principle
implies that in any linear circuit whose signal sources have the same frequency, every signal
response will be a sinusoid of the same frequency.
Let us call the vertical value computed on either side y. We see on the LHS the vertical value, y,
is 1 when α is either 0 or π/2. The vertical value on the RHS is also 1 at both points. Can simple
principles predict the shape of the curve between 0 and π/2?
The applicable principle is that the sum of derivatives is the derivative of the sum. For values of
α between 0 and π/4, the sine rises faster than the cosine falls. Our curve starts by rising. At the
point α = π/4, the rate of rise of the sine exactly equals the rate of fall of the cosine and so the
sum reaches a maximum. The value of y(π/4) = √2. For values of α between π/4 and π/2, y
monotonically descends back to the value of 1. A sinusoid whose peak occurs at α = π/4 is either
a sine curve advanced by 45° or a cosine curve delayed by 45°.
y = cos(x) + sin(x)

y = sin(x)

















y = cos(x)
Figure 14 The sum of two sinusoids
of the same frequency
y =-6x
2
y = 3x - 6x
2
15.
The magnificent identity sin 2 (α) + cos2 (α) = 1 is usually called the first Pythagorean
identity. The identity has this name because for any right triangle whose legs are parallel to the
axes and whose hypotenuse has length 1.0, the sine is the length of the vertical leg and the cosine
is the length of the horizontal leg. The Pythagorean equation states that for any right triangle the
sum of the squares of the legs equals the square of the hypotenuse, which in this case is 1.
Take any number for α. Add the square of the sine and the square of cosine of the angle. The
result will always be 1. Wow! Imagine this is true for any angle!
A glance at the figure 15a shows that the sin 2(0) = 0 and cos 2(0) = 1. Next notice the curves
dovetail. In the first quadrant, as α increases and the square of the sine grows, the square of the
cosine falls monotonically until the sine squared equals 1 and the cosine squared equals 0.
Similar behavior occurs in the other quadrants. The sum of the squares is always 1.
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2
y = cos (x)
2
y = sin (x)




























y = sin(x)
y = cos(x)
y =-6x2
y =-6x2
y = x3 - 6x2
Figure 15 The sine squared
y = x3 - 6x2
The cosine squared
y=1
2
2
y = sin (x)




y =-6x2
y = cos (x)










y = x3 - 6x2
Figure 15a The sum of the sine squared and the cosine squared always equals 1.
16a. The following three identities relate the function y = cos (2α) to expressions in the
functions cos (α) and sin (α). Examine the first of the three identities;
cos2 ( α ) = ½{1 + cos( 2α }
on the LHS cos2 (α) is continuous, smooth and positive and oscillates between 0 and 1. Since
both (– 1) 2 and (+ 1) 2 equal 1 there will be two maxima in the interval 0 ≤ x < 2π, So the
function cos2 (α) has an of amplitude of ½ and it is raised by a value of ½ and oscillates twice as
fast as cos (α) which is the expression on the RHS.
16b. The second identity sin2 ( α ) = ½{1 – cos( 2α } can be justified by a similar visual
analysis.
16c. If sin2 (α) is subtracted from cos2 (α) the constants cancel and the oscillations add which
produces the third identity.
cos (2α) = cos2 (α) – sin2 (α).
These can be checked by visual analysis and also by choosing random values for x.
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
2

y = -.5cos(2x) + 0.5
y = sin (x)
y = .5cos(2x) +.5
y = cos 2 (x)


y = .5


y = .5






















y = -.5 cos(2x)
y =-6x2
y = x3 - 6x
2





y = x3 - 6x
2
Figure 16a Double angle identies
Figure 16b
2
y = cos (x)
2
y = sin (x)



y = .5 cos(2x)


y =-6x2










y =-6x2
Figure 16c
cos
2(x)
– sin










y = x3 - 6x
2
2(x)
17.
Mathematicians have discovered equations which permit the exchange of trig functions
of multiples of an angle into algebraic expressions of trig functions of the angles alone. A simple
example is the double angle formula sin(2α) = 2 sin(α) cos(α). We note that the product of two
smooth functions is also smooth. The sine function has zeros at multiples of π and the cosine
function has zeros at π/2 added to multiples of π. Since the product has a zero whenever either
one of the factors has a zero, the product will be zero at any integer multiple of α = π/2 . In
every quadrant the factors will be plus or minus the product of an increasing curve and a
decreasing curve whose maximum will be in the middle. When α = π/4, a maximum of 1 is
attained, as we see here; sin(2* π/4) = 2 sin(π/4) cos(π/4) = 2*
1
√2
∗
1
√2
= 1.
The graph shows that sin(2α) ≠ 2 sin(α); that is, it wrong to take the 2 out of the sine function.
However, when α = 0, 2 sin(α) is tangent to sin(2α). When α is small, then 2 sin(α) is just a little
bigger than sin(2α) and is used as an approximation.
ETD 555
y = sin(2x) = 2sin(x)cos(x)


y = sin(x)
y = sin(x)cos(x)
















y = cos(x)
y =-6x2
y = x3 - 6x
2
Figure 17 The double angle sin formula
Identities of the Exponential Functions
18a. For a fixed value of b the exponential function y = bx starts on the far left, just above the
horizontal axis and rises monotonically and smoothly, crossing the vertical axis at y = 1 and
continues monotonically rising toward infinity on the right. The identity
(2x+2) = (2x) (22) = 4 (2x)
states that displacing the exponential curve 2 units to the left produces the same result as
multiplying the vertical values of the curve by 4, as shown in figure 18a. This can be checked by
viewing the above graph ad by choosing random values for x .





2 x
y = 2 2 = 4*2
x
y=2
x+ 2



y=2
x





Figure 18a
18b. The identity (e (x)) 2 = ex ex = e ( 2x) states squaring an exponential curve produces the
same curve as would be produced by compressing the horizontal axis 2 times. Both sides
produce a smooth, monotonically rising curve as shown in figure 18b.
ETD 555

y=e

2x
x 2
= {e }



y=e
x
y= 2
x



Figure 18b
1
18c. The identity (2 –x) = x states that the reciprocal of the exponential function produces
2
the same result as reflecting the function about the vertical axis, as shown in figure 18c. This can
be checked by choosing random values for x and by viewing the graph. In figure 18c, it is seen
that as y = 2 x grows smoothly and monotonically, the reciprocal decays monotonically toward
zero.





-x
y=2

1
= ---x
2
x
y=2







Figure 18c
We find an application of this identity in the study of decaying oscillations whose equations have
a form like y = e – α x sin(ω x) and whose graph is shown in figure 18d.

y=e
-x
y=-e
-x

Figure 18d
A decaying oscillation
ETD 555
Identities of Logarithmic Functions
19.
Let us now turn our attention to the logarithmic identity log( x2 ) = 2log(x). The curve
corresponding to the function y = log(x) is defined only in the right half plane, starting at
– infinity for very small positive values of x and rising monotonically and smoothly toward
+ infinity as x increases. For values of x close to zero, the values of y rise fast but as x moves
away from the origin the values of y increase but at a steadily slower rate. Analyzing the LHS of
identity 19 we note the curve of x2 is a parabola, always positive except at the origin. The curve
of y = log( x2 ) has even symmetry and is defined everywhere except at the origin where y
approaches – infinity. The curve begins at the far left at y = + infinity, descends smoothly and
monotonically, crossing the x-axis at x = – 1, and continues smoothly descending to – infinity
when x = 0. Even symmetry dictates that in the right half plane the curve will be the mirror
image of the left half plane. The graph of the function y = log( x2 ) is shown in figure 19 b.
y = 2 ln(x)
y = 2 ln(x)


y=x
y=x
2
2
y = ln(x)




2
2
y = ln(x )
Figure 19a
y = 2 ln(x)
y = ln(x )


Figure 19b
y = ln(x 2)
Now we see a problem. The RHS of identity 19 is not defined for negative values of x. The
graph of the curve y = log( x2 ) is simply the curve of the log function doubled vertically but only
in the right half plane. If it is important that we have agreement between the two sides of the
identity then we could write log( x2 ) = 2log( |x| ). With this modification both sides of the
identity are defined everywhere except at the origin and even symmetry is displayed.
The following few paragraphs should be skipped by students who have not yet studied
differential calculus.
Note should be made here of a point that was briefly mentioned in the abstract. The derivatives
1
1
of differentiable identities also form identities. We know that the derivative of ln(x) = x. But x is
defined everywhere except at the origin while ln(x) is only defined for positive x. The equation
d
dx
1
{ln(x)} = x is only true in the right half plane. For this equation to be identically true
everywhere except the origin the argument of ln(x) must be positive for all x. If the argument is
forced to be positive by using |x| instead of x, then the equation will be identically true
everywhere. We must conclude that for equations to be identically valid everywhere then the
ranges and domains must be compatible for all the functions on either side of the equation.
ETD 555
Let us examine this last property of identities with respect to identity 19: log( x2 ) = 2log( |x| ).
The derivative of the LHS, y = log ( x2 ) is y′ =
The derivative of y = 2 log (|x|) is y ′ =
2
d
x
dx
∗
| |
1
2
x
x
* 2x =
2
.
|x|.
|x|
2
+1; x > 0
dy
2
d
2
|x|
|x|
=
{
=
, we find dx = |x| ∗ dx |x| = |x| ∗ x = , which is the same
dx
−1; x < 0
x
x
result obtained by differentiating the LHS. This serves as an example of the proposition that the
derivative of an identity is also an identity for all values of x.
Since
20.
d
Similar issues will be found in examining the identity log (x2 – x) = log (x) + log (x – 1) .
The argument of the log function in the LHS is a parabola which is negative in the interval
0 < x < 1 . On the RHS we see that both logs are only defined in the infinite interval x > 1.
Again as in identity 19, the issue can be resolved by taking the absolute values of the arguments
of the logs, Identity 20 asserts that log of this parabola can be described as the sum of a log
function and a horizontal translation of the log.

y = ln(x) + ln(x-1)

y = x(x - 1)








y = ln{x(x-1)}
y = ln(x)
y = ln{x(x-1)}








Figure 20a

y = ln(x - 1)

y = ln{x(x-1)}
Figure 20b y = ln(x) + ln(x-1)
Identities of inverse Forms
Now let us examine two identities involving inverse forms of functions.
α)
β)
an identity in x;
an identity in y;
x = ln(e x) and
y = e ln(y)
Here too we come across problems of incompatible domains and ranges. It is easy to check that
the functions y = e x and x = ln(y) comprise an inverse function pair. For x = 0, e x =1 and ln(1)
returns 0. Moreover, try a random value of x and compute y = e x . For every value of x, the
result is that the operation ln(y) returns the value of x. In the identity α, e x is positive and
ETD 555
defined everywhere. Identity α is true everywhere. However, since ln(y) is defined only for
positive values of y, the identity β prevails only for positive values of y.
Summary
The previous studies of identities involving elementary functions are intended as exercises to
assist beginning algebra and calculus students in acquiring insight into the graphical behavior,
quirks and misconceptions regarding these functions.
Visualizations are of significant value to engineering students who will see them used routinely
in their courses. Electrical engineering students will see analogue signals and frequency
response charts as the graphs that are studied in calculus. Civil and mechanical engineering
students will learn to visualize loading, shears, moments and deflections. Engineering students
can acquire in their beginning algebra and calculus courses a natural introduction to the
techniques of visualization of the equations they will confront in their future studies.
Some may consider the previous descriptions of graphical constructions as being tedious and
repetitious. I do not. I see they all support the contention that continuous operations on
continuous curves produce continuous results. I see each of the previous identities as being
special and may in some future routine calculation provide the key needed to produce the form
which solves the problem then being confronted.
For Contemplation
There is no doubt that robotics, drones and other recent advances in modern technology have
caught the attention of our nation’s youth. But the calculus and algebra that these young students
see in their math courses appear disconnected from their interests. The math taught in our high
schools is the result of the successful drive during the 19th century to confront the enormous
difficulties of the real number system and of extremely discontinuous functions. When it became
accepted by mathematicians that visualizations of properties of extremely discontinuous
functions conflicted with the impetuous visualizations of mostly smooth functions the
visualizations became suspect and almost all were to be avoided. A student was not allowed to
just examine a graph and arrive at a conclusion. An argument employing Cauchy’s delta-epsilon
techniques was required for “rigorous” justification. The delta-epsilon approach should be taken
in the study of the real number system and in the study of extreme discontinuities.
However, in the world of technology the use of nominal values and tolerances avoids the
difficulties associated with real numbers. The functions encountered in beginning engineering
and technology courses are smooth almost everywhere. The mathematics curriculum should
begin with the study of these mostly smooth functions. The preceding examples provide the
justification for the limits of trust that can be given to visualizations. The analytics encountered
in technology and engineering courses appear formidable enough without suppressing the
advantages imparted by visualizations.
The knowledge of the discoveries about the real numbers or extremely discontinuous functions is
not required for the professional engineering licensing exam. Why are we allowing a
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pedagogical approach constructed to deal with difficulties of the real number system to pervade
our nation’s beginning mathematics curriculum at the expense of the insight that might be
obtained with clear organization, definitions and visualizations?
References
1.
2.
3.
4.
5.
6.
7.
8.
9
10.
11.
12.
13.
14.
Aleksandrov, et al Mathematics Its Content, Methods and Meaning The MIT Press, Cambridge MA, 1963
Alsina, Claudi & Nelson, Roger B. Math Made Visual
MAA, Washington, 2006
Grossfield, Andrew What are Differential Equations? A Review of Curve Families
ASEE Annual Conference, 1997
Dunham, William, Euler, the Master of Us All MAA, Washington, 1999 p 35
Grossfield, Andrew The Natural Structure of Algebra and Calculus
ASEE Annual Conference, 2010
Grossfield, Andrew Mathematical Forms and Strategies
ASEE Annual Conference, 1999
Grossfield, Andrew Are Functions Real? ASEE Annual Conference, 2005
Grossfield, Andrew Visual Analysis and the Composition of Functions SEE Annual Conference, 2009
Grossfield, Andrew Mathematical Definitions: What is this thing? ASEE Annual Conference, 2000
Grossfield, Andrew Introducing Calculus to the High School Curriculum: Curves, Branches and Functions
ASEE Annual Conference, 2013
Grossfield, Andrew Visual Differential Calculus
ASEE Zone 1 Conference, 2014
Nelson, Roger B. Proofs without Words: Exercises in Visual Thinking
MAA Washington, 1993
Nelson, Roger B. Proofs without Words II: More Exercises in Visual Thinking MAA Washington, 2000
Parris, Rick
WINPLOT, a general-purpose plotting utility,
Peanut Software