Transcendental (Non-Algebraic) Function
Tricks of the trade

What to expect when graphing transcendental (non-algebraic) functions. For graphing purposes, transcendental functions can be broken down into two categories: those that involve one or more of the six trigonometric functions and those that do not. In effect, the ones that do not involve trigonometric functions can be graphed completely with all their mathematical features included: all x- and y-intercepts, all maxima and minima, all points of inflections, all holes, all jump discontinuities, all vertical asymptotes, all horizontal asymptotes, all vertical tangents, all vertical cusps, all corners, and all isolated points. On the other hand, if the function includes any trigonometric function, the graph may have infinitely many such mathematical features. For example, sin(x) and sin(1/x) have infinitely many extrema and points of inflection; both sin(x) and sin(1/x) have them over the entire real line while sin(1/x) additionally has them over any finite interval that includes the origin. Neither of these functions can be graphed completely using standard graphics.

What is the best approach to graph transcendental functions? Graphing transcendental functions requires that you first learn to recognize their basic characteristics and graphs. We have found it useful to learn about these graphs in conjunction with their inverses. This focuses on creating a mental image of the complete graph as well as its inverse. We rely on the graphical interpretation of inverse as the reflection of the graph across the line y=x. While plotting certain points is indispensable, developing a mental image of the complete graph is vital to successful graphing. To introduce you to transcendental functions, the following collection of examples has been selected to help "model" the transcendental functions met most commonly. It is not intended to be exhaustive.

Examples of transcendental functions that exhibit various characteristics.

Exponential functions. Any number b>0, b≠1, gives rise to an exponential function y = bx, where b is called the base. This function is defined for all values of x, has a horizontal asymptote given by the equation y=0 (the x-axis), goes through the point (0,1), and has a range of (0,∞). There are two similar graphs—but with different orientations—corresponding to the cases: (1) b>1 and (2) 0<b<1. We illustrate these cases for bases b=2 and b=1/2. Each graph is the reflection of the other graph across the y-axis. To see this analytically, if you start with y=2x and replace x by x, you get y=2x=(2–1)x=(1/2)x, which is the "other" graph. Do this one more time (start with y=2x and replace x by –x) and you get back to where you started.

Natural base. There is a special number, denoted by the letter e, which occurs frequently in mathematics. It is used as a base for the exponential function y=ex and is referred to as the natural base. Its value can be computed, for instance, as the limiting value of the expression (1+1/n)n as n→∞, which is approximately 2.718. Like the special number π, the decimal expansion of e never repeats.

The exponential function y=ex has the unusual property that for each number x, the slope of the tangent line to its curve at (x,ex) equals its y-value, ex.

Logarithmic functions. The inverse of the exponential function y=bx (not the reciprocal, which is the inverse with respect to multiplication, but the inverse with respect to composition) is called the logarithmic function, y = log(b,x). For the special case where b=e, the logarithm is referred to as the natural logarithm, and there are special notations that we use here: ln(x)=log(x)=log(e,x). Below, we illustrate the exponential on the same graph with its inverse for each of the cases b=e and b=1/e. In each case, the graphs of the logarithm and exponential functions can be seen as reflections of each other across the line y=x. Note that in both cases, the logarithm has a vertical asymptote at x=0, corresponding to the reflection across the line y=x of the horizontal asymptote given by the equation y=0 (the x-axis) for its inverse, the exponential. Also, in each case the domain of the logarithm is the half-infinite interval (0,∞), corresponding to the range of exponential, while the range of the logarithm is (–∞,∞), corresponding to the domain of the exponential.

The base e≈2.718>1

The base 1/e≈0.368<1

The sine function, y=sin(x), and its inverse, y=sin-1(x)=arcsin(x). Despite the notation that uses the power "–1", sin-1(x) does not mean 1/sin(x); instead, it is always used to refer to the inverse with respect to composition. To refer to the reciprocal (inverse with respect to multiplication), syntax requires that you use parentheses: (sin(x))–1. The graph of the basic sine function is a "wave-like" pattern, called a sine wave, that repeats on every interval of length 2π. It has an amplitude of (1/2)(1–(–1))=1 (where 1 is the maximum y-value and –1 is the minimum y-value) and is defined everywhere.

Observe that the reflection across the line y=x produces a sine wave along the y-axis, and as such it is not a function (it does not pass the vertical line test). To obtain an inverse, we need to restrict the sine curve to where it is one to one (a restricted domain where it takes on all its y-values exactly once). While there are many ways this could be done, it is standard to use the smallest interval that contains the origin, [–π/2,π/2]. Here is the graph of the sine curve and its inverse, arcsin(x).

The cosine function, y=cos(x), and its inverse, y=cos-1(x)=arccos(x). The graph of the basic cosine function is similar to that of the sine function. It is a "wave-like" pattern that repeats on every interval of length 2π. It is actually a translation of the sine curve (shift the sine curve left by π/2 units and you get the cosine curve). Like the sine, the cosine has an amplitude of 1 and is defined everywhere. In this case, to obtain the inverse, the standard restricted domain where cos(x) is one to one is taken to be the interval [0,π]. The range, [–1,1], then becomes the domain of the inverse, arccos(x). Once again, observe that these graphs are reflections of each other across the line y=x. This can be quite tricky to grasp, since the inverse cosine is always positive, but recall that this range, [0,π], equals the "restricted" domain of cosine.

The tangent function, y=tan(x), and its inverse, y=tan-1(x)=arctan(x). Your understanding of the graph of the basic tangent function can be anchored by interpreting the identity: tan(x)=sin(x)/cos(x). In particular, everywhere the cosine is equal to 0, the tangent is undefined since division by zero is not permitted. In each such instance, this gives rise to a vertical asymptote. There is no amplitude since the tangent has no maximum or minimum. The tangent curve also repeats but unlike the sine and cosine curves, it repeats over intervals of length π. Thus, at the endpoints of the interval (–π/2,π/2) the tangent has vertical asymptotes since cos(±π/2)=0. Also, since the tangent function is one to one on this interval, it is used to define the inverse. For the inverse, arctan(x), the vertical asymptotes, when reflected across the line y=x, become horizontal asymptotes, as can be seen in the graph that follows.


Graphing a Transcendental Function Over the Interval [0,2π]:
Example of a Sine Function: y=2sin(3x+
π/2).

Since the sine function is periodic, it repeats every time its argument, in this case 3x+π/2, takes on all values through an interval of length 2π:

0≤3x+π/2≤2π.

We can solve this inequality for x in two steps. First, we subtract π/2 (equivalently, add –π/2) to each term,

–π/2≤(3x+π/2)–π/2≤2π–π/2, which simplifies to –π/2≤3x≤3π/2.

Second, we divide each term by 3 (equivalently, multiply each term by 1/3),

–π/6≤x≤π/2.

Here, the period is the length of the resulting interval, π/2–(–π/6)=2π/3. At this point, we can sketch one complete period of the sine wave over this interval. Note that this is a sketch, so there is no need to plot more than the beginning and ending points.

The factor of 2 in the expression y=2sin(3x+π/2) is the amplitude and it represents the maximum y-value, and as well, –2 represents the minimum y-value. Without plotting again, we can mark off and label tick marks at –2 and 2 along the y-axis.

Noting that the next complete wave starts at π/2 and ends at π/2+2π/3=7π/6, and further, that the complete wave after that starts at 7π/6 and ends at 7π/6+2π/3=11π/6, we can extend the curve through a bit more than 2 periods to 2π. We alternate different colors in the graph below to help distinguish how each complete period is sketched.

 

 

       

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United States Patent Numbers 7,432,926, 7,595,801, & 7,889,199.
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