Algebraic Function
Tricks of the trade
What to expect when graphing algebraic functions. Algebraic
functions that are neither polynomials nor rational functions may
have one or more distinguishing characteristics. Unlike polynomial
or rational functions, the domains of algebraic functions may consist
of one or more finite or halfinfinite intervals and/or—surprisingly—isolated
points. In addition, algebraic functions may have up to two horizontal
asymptotes (rational functions can only have one), vertical asymptotes
from one halfplane side only (rational functions approach their
vertical asymptotes from both halfplane sides), vertical tangents
(onesided and twosided), vertical cusps, corners, and 2hole
jumps (jump discontinuities that are not defined at the jump).
How might you approach graphing algebraic
functions? Graphing algebraic functions that are neither polynomial
nor rational functions requires a decidedly different approach.
Students who rely solely on plotting points often miss important
features and get incorrect graphs. The approach we recommend is
to learn various special cases and then use these as models for
other similar algebraic function you encounter. The following collection
of examples has been selected to "model" each of the
distinguishing characteristics that algebraic functions exhibit.
Afterwards, we give a typical application that
uses this approach.
Examples of algebraic functions
that exhibit various characteristics .
The square root function. The graph of
the basic square root function, sqrt(x)=x^{1/2}, can
be used as a model for graphing any even
root function. For graphing purposes, a function like y=x^{7/4}=(x^{7})^{1/4} can
be treated as an even root function. In the graph below, the solid
filled small circle at x=0 indicates that the
function is defined at that endpoint. This is an example of a function
defined on only a halfinfinite interval, [0,∞).
This function has a (onesided) vertical tangent
at x=0 in the sense that the slopes of the tangent lines
moving along the curve toward the origin (0,0) (from the right
side) are tending toward +∞. To verify the steepness of the
curve at the origin, you need to zoom in. Here, we show the result
of using the Smart Zooming feature over the interval
(−0.02,0.15).
The cube root function. The graph of
the basic cube root function, y=x^{1/3},
can be used as a model for graphing any odd
root function. For graphing purposes, a function like y=x^{4/3}=(x^{4})^{1/3} can
be treated as an odd root function. It has a vertical tangent at x=0 in
the sense that the slopes of the tangent lines along the curve toward
the origin (0,0) from both sides jointly approaches +∞ (or,
as in the case of −x^{1/3},
they jointly approach −∞). No polynomial or rational
function can have a vertical tangent. (Rational functions do have
vertical asymptotes, but unlike the case of a vertical tangent, rational
functions are never defined at their vertical asymptotes.)
The absolute value function: an example of
a corner. The graph of the basic absolute value function, abs(x)=(x^{2})^{1/2},
can be used as a model for graphing many functions that use absolute
value. It has a corner at x=0. Here, the
slopes of the tangent lines along the curve toward the origin (0,0)
from the right are all equal to +1 and from the left are all equal
to –1.
An example of a function with a jump discontinuity. The
function y=x/x has a 2hole jump discontinuity
at x=0. Here, the size of the jump is 1–(–1)=2. (The
two dashed lines shown in the graph indicate that the graph also
has two horizontal asymptotes.) It is notable that the range of this
algebraic function consists of just two points: {–1,1}. Algebraic
functions that have jump discontinuities are always of the 2hole
type while nonalgebraic (transcendental) functions can have jump
discontinuities with only one hole (see for example the graph of y=cot^{1}(x)).
Example of a vertical cusp. The function y=x^{1/3} has
a vertical cusp at (0,0). This means that it has a vertical tangent
at x=0 and the slopes of the tangent lines along the curve
approaching (0,0) from one side tend to +∞ and from the other
side tend to −∞.
Example of a function with an isolated point. The
function y=(x^{2}(x^{2}–1))^{1/2} is
defined at the number x=0 as well as on the
two halfinfinite intervals (−∞,–1] and [1,+∞). We
call the point (0,0) isolated because it is at a distance
from any other point in the domain.
Example of a continuous function with two
horizontal asymptotes. The function y=x/(x^{2}+1)^{1/2} has
two horizontal asymptotes: y=±1. It
is also notable that the range is a finite open interval, (−1,1).
Example of a function with a vertical asymptote
from one halfplane side only. The function y=(x/(x–5))^{1/2} has
a vertical asymptote from one halfplane side only (the right side)
at x=5, a (onesided) vertical tangent at x=0 (the
left side), a horizontal asymptote, y=1, and two halfinfinite
intervals that make up its domain: (–∞,0] and (5,∞).
Example of a function with a finite interval
domain. The function y=sqrt(1–x^{2}) has
a finite interval for its domain, [–1,1], and
(onesided) vertical tangents at each of the domain endpoints, x=±1.
The solid filled circles at the endpoints indicate that the function
is defined at the endpoints ((±1)^{2}1)^{1/2}=0^{1/2}=0).
Graphing
an Algebraic Function:
Example of a square root function:
y=sqrt(x^{2}–1)
= ( x^{2}–1)^{(1/2)}.
First determine the domain. Here, what is required
is that: x^{2}–1≥0. Adding
1 to both sides, we solve the resulting inequality x^{2}≥1 and
get the solution: x≥1 or x≤–1. Thus,
the domain consists of the two intervals (–∞,–1] and [1,∞). To
determine the xintercepts, set y=0 and solve the equation x^{2}–1=0. Adding
1 to both sides, we solve the resulting equation x^{2}=1,
taking the square root on both sides and get the solution: x=±1. Observe
that if you replace x by –x in the function,
you get an identity: sqrt((–x)^{2}–1)=sqrt(x^{2}–1).
Thus, the graph is symmetric with respect to the yaxis. The
plot of sqrt(x) can be used as a model for sketching
the graph of sqrt(x^{2}–1) on
the interval [1,∞) beginning at x=+1.
Since the graph is symmetric with respect to
the yaxis, we can combine this graph with the reflection
of this graph across the yaxis and sketch both curves that
make up the graph.
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United States Patent Numbers 7,432,926, 7,595,801, & 7,889,199.
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