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 half-infinite 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 half-plane side only (rational functions approach their vertical asymptotes from both half-plane sides), vertical tangents (one-sided and two-sided), vertical cusps, corners, and 2-hole 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)=x1/2, can be used as a model for graphing any even root function. For graphing purposes, a function like y=x7/4=(x7)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 half-infinite interval, [0,∞).

This function has a (one-sided) 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=x1/3, can be used as a model for graphing any odd root function. For graphing purposes, a function like y=x4/3=(x4)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 x1/3, 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)=(x2)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 2-hole 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 2-hole type while non-algebraic (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=(x2(x2–1))1/2 is defined at the number x=0 as well as on the two half-infinite 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/(x2+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 half-plane side only. The function y=(x/(x–5))1/2 has a vertical asymptote from one half-plane side only (the right side) at x=5, a (one-sided) vertical tangent at x=0 (the left side), a horizontal asymptote, y=1, and two half-infinite intervals that make up its domain: (–∞,0] and (5,∞).

Example of a function with a finite interval domain. The function y=sqrt(1–x2) has a finite interval for its domain, [–1,1], and (one-sided) 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=01/2=0).

Graphing an Algebraic Function:
Example of a square root function:
y=sqrt(x2–1) = ( x2–1)(1/2).

First determine the domain. Here, what is required is that: x2–10. Adding 1 to both sides, we solve the resulting inequality x21 and get the solution: x1 or x–1. Thus, the domain consists of the two intervals (–∞,–1] and [1,∞). To determine the x-intercepts, set y=0 and solve the equation x2–1=0. Adding 1 to both sides, we solve the resulting equation x2=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(x2–1). Thus, the graph is symmetric with respect to the y-axis. The plot of sqrt(x) can be used as a model for sketching the graph of sqrt(x2–1) on the interval [1,∞) beginning at x=+1.

Since the graph is symmetric with respect to the y-axis, we can combine this graph with the reflection of this graph across the y-axis and sketch both curves that make up the graph.

 

 

       

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