Precalculus Study Guide
Copyright Maplesoft, a division of Waterloo Maple Inc., 2021
Chapter 11: Functions Defined Implicitly
Introduction
Some equations of the form fx,y=0 can be rewritten in the form y=gx, in which case, we say that the original equation defined yx implicitly, and the equation y=gx gives yx explicitly.
Some equations of the form fx,y=0 cannot be rewritten in the form y=gx. For these, there is no combination of algebraic steps that will isolate the letter y in the equation. However, such equations can still define y=gx in principle, even if the values have to be computed numerically. Such functions yx are said to be defined implicitly by the equation fx,y=0.
Some equations of the form fx,y=0 do not define yx implicitly since they represent relations, one-many maps from the domain to the range. However, in such cases, we can define branches along which a function yx is defined implicitly by the equation fx,y=0.
This chapter explores examples of all three types of behaviors, the simple case such as posed by the equation
2x+3y−7=0
where y can easily be isolated; the single-branch case such as posed by the equation
lny+x+y−5=0
where y cannot be isolated; and the multiple-branch case such as posed by
x2+y2=1
the familiar equation of the unit circle.
Chapter Glossary
The following terms in Chapter 11 are linked to the Maple Math Dictionary.
discretization
discriminant
domain
explicit
floating-point
implicit
inequality
interval
nonnegative
parabola
quadratic equation quadratic formula
real number
sequence
square root
Typical Problems
11.1. The equation 5x2+3xy+2y2−4x−7y−9=0 defines one or more functions y=yx implicitly.
(a) Obtain a graph of the implicitly defined functions.
(b) Express explicitly each function in the graph plotted in Part (a).
(c) Use the given equation at x=1 to compute all real values of y1.
11.2. The equation lnxe−y+x+y−5xy+3x+7y+9=0 implicitly defines a function y=yx.
(a) On the interval 1,5, obtain (using some sort of plotting device) a graph of the implicitly defined function.
(b)
In the interval 2,5, substitute each node xk=2+320k,k=0,1,...,20, into the given equation, then solve for the corresponding yxk numerically. Plot the points xk,yxk and compare the graph to the one obtained in Part (a).
Maple Initializations
Before accessing the Maple version of the solutions to the typical problems stated above, initialize Maple by pressing the button provided on the right.
Solutions
Problem 11.1
11.1 - Mathematical Solution
11.1 (a) - Mathematical Solution
Obtaining a graph of the function defined implicitly by the equation
5x2+3xy+2y2−4x−7y−9=0
is a challenging task if done without the aid of an appropriate plotting device. One way to accomplish this by hand is to anticipate Part (b) and obtain explicit expressions for the two branches contained therein. Thus, write the equation in the form
2y2+3x−7y+5x2−4x−9=0
and use the quadratic formula to obtain
y±
=−3x−7±3x−72−425x2−4x−922
=147−3 x±121−10x−31x2
The square roots appearing in each of these branches is real for 121−10x−31x2≥0. Again applying the quadratic formula and sketching the parabola represented by the left-hand side of this inequality, we discover that the domain for each of the two branches of the implicit function is
−5+85931≤x≤−5+85931
or approximately
−2.14≤x≤1.82
Then, from the function values listed in Table 11.1.1, use a sheet of graph paper to sketch a graph akin to the one in Figure 11.1.1.
xy+xy−x−2.04.282.22−1.84.651.55−1.64.851.05−1.44.950.65−1.25.000.30−1.05.000.00−0.84.96−0.26−0.64.89−0.49−0.44.79−0.69−0.24.66−0.86
xy+xy−x0.04.50−1.000.24.31−1.110.44.10−1.200.63.85−1.250.83.56−1.261.03.24−1.241.22.86−1.161.42.40−1.001.61.82−0.721.80.800.00
Table 11.1.1 Function values for branches y+ and y−
Figure 11.1.1 Graph of implicitly-defined yx
11.1 (b) - Mathematical Solution
In Part (a), it was necessary to anticipate Part (b) where the two branches contained implicitly in the equation
are obtained explicitly. Recall that by use of the quadratic formula, applied to
the branches y+ and y− were determined to be
11.1 (c) - Mathematical Solution
If the substitution x=1 is made in the equation
the equation
2y2−4y−8 = 0
results. This equation determines the two values of y that correspond to x=1. Of course, this equation can be solved by the quadratic formula, yielding
=−−4±−42−42−822
=4 ±804
=1 ±5
={3.236067977−1.236067977
11.1 - Maplet Solution
11.1 (a) - Maplet Solution
A graph of the implicit functions defined by the equation
can be obtained with the Implicit Function Tutor .
Clicking this link will launch the tutor with the solution embedded as shown in the thumbnail image in Figure 11.1.2.
For the graph, simply enter the defining equation and click the button labeled Graph. Use the Plot Options button as needed.
To launch the Implicit Function Tutor, click the following link: Implicit Function Tutor
Figure 11.1.2 Thumbnail image of the Implicit Function Tutor
11.1 (b) - Maplet Solution
The explicit rules for the functions defined implicitly by the equation
Clicking this link will launch the tutor with the solution embedded as shown in the thumbnail image Figure 11.1.2.
After entering the defining equation, click the button labeled Explicit Branches. The resulting expressions are then
y±=147−3 x ±121−10x−31x2
11.1 (c) - Maplet Solution
To use the given equation to compute all real values of y1, again use the Implicit Function Tutor .
Clicking this link will launch the tutor with the solution embedded as shown in the thumbnail image in Figure 11.1.3.
In the section Numeric study, enter 1 as the value of x, and for the Interval containing corresponding y, enter y=−2 to 5, as suggested by the graph.
Click the button labeled Equation for y to see the equation that results from substituting x=1 into the original equation.
Click the button labeled Analytic Solution to obtain the two values y=1 ±5 or click the button labeled Numeric Solution to obtain the values of y in floating-point (decimal) form.
Figure 11.1.3 Thumbnail image of the Implicit Function Tutor
11.1 - Interactive Solution
11.1 (a) - Interactive Solution
Type (or Control-drag) the equation.
Context Panel: Plots≻2-D Implicit Plot≻x,y
11.1 (b) - Interactive Solution
Context Panel: Solve≻Solve for Variable≻y
11.1 (c) - Interactive Solution
Type (or Control-drag) the equation. Press the Enter key.
Context Panel: Evaluate at a Point≻x=1
Context Panel: Solve
11.1 - Programmatic Solution
11.1 (a) - Programmatic Solution
The function defined implicitly by the equation
q ≔ 5 x2 + 3 x y + 2 y2 − 4 x − 7 y − 9 = 0
can be plotted with Maple's implicitplot command found in the plots package. Its use leads to Figure 11.1.4.
implicitplotq,x=−3..2,y=−2..5,scaling=constrained
Figure 11.1.4 Graph of the function defined implicitly by 5x2+3xy+2y2−4x−7y−9=0
11.1 (b) - Programmatic Solution
The equation
q
can be solved explicitly for y=yx. Maple's solve command gives
Y ≔ solveq, y
Thus, there are two branches, which we label as
y+= Y1; y−= Y2
These two explicit branches of the implicitly defined function are plotted in Figure 11.1.5, where y+ appears in black, and y− appears in red.
plotY,x=−2.2..2,color=black,red,numpoints=2000,scaling=constrained,legend=typesety+,typesety−
Figure 11.1.5 Graph of branches y+ (black) and y− (red)
The algebra by which these two explicit branches are extracted from the equation
is no more than an application of the quadratic formula. The variable in the quadratic equation is y, and all x's are treated as constants. Thus, the equation is written as
Ay2+By+C=0
where A,B, and C, are respectively
A ≔ coefflhsq, y, 2;B ≔ coefflhsq, y, 1;C ≔ coefflhsq, y
