4.5 - Implicitly Defined Functions

Overview

Most functions so far were written explicitly, like

y = x^2 + 1

, with

y

alone on one side. Many important curves, like circles, come from equations where

x

and

y

are mixed together, like

x^2 + y^2 = 25

. These equations implicitly describe one or more functions. This topic covers how to graph them, how to pull functions out of them, and how

x

and

y

change together along the curve.

Explicit vs. implicit

Explicit: one variable is written as a function of the other, like $y = \sqrt{x}$ or $x = y^2 - 4$ .

Implicit: an equation involving two variables where neither one is isolated, like $x^2 + y^2 = 25$ or $x^2 + xy + y^2 = 3$ .

An equation involving two variables can implicitly describe one or more functions:

$xy = 4$ describes one function, $y = \frac{4}{x}$ .

$x^2 + y^2 = 25$ describes two functions of $x$ : $y = \sqrt{25 - x^2}$ (the top half of a circle) and $y = -\sqrt{25 - x^2}$ (the bottom half).

The whole graph of

x^2 + y^2 = 25

fails the vertical line test, so it isn't a function of

x

. But it's made of pieces that are. Functions that come out of an equation this way are called implicitly defined functions.

Graphing by finding solutions

The graph of an equation in two variables is the set of all ordered pairs

(x, y)

that make the equation true. So you can graph it by finding solutions:

Pick a value for one variable (often $x$ ).

Substitute it and solve for the other variable. There may be zero, one, or several solutions.

Plot every solution you find, then repeat with more values.

Connect the points smoothly, using any symmetry or extreme points you notice.

For example, in

x^2 + y^2 = 25

, choosing

x = 3

gives

y^2 = 16

, so

y = 4

y = -4

: two points,

(3, 4)

and

(3, -4)

. Choosing

x = 6

gives

y^2 = -11

, which has no real solution, so the graph has no points with

x = 6

Solving for one variable

Solving the equation for one of the variables defines a function whose graph is part or all of the graph of the equation.

Solving $x^2 + y^2 = 25$ for $y$ gives $y = \pm\sqrt{25 - x^2}$ . Each sign is its own function, and each graph is half of the circle.

Solving the same equation for $x$ gives $x = \pm\sqrt{25 - y^2}$ : two functions of $y$ , the right half and the left half.

Solving $xy = 4$ for $y$ gives $y = \frac{4}{x}$ , whose graph is all of the equation's graph.

This is also how you graph an implicit equation on a calculator that only accepts

y =

functions: solve for

y

and enter each piece, like Y1

= \sqrt{25 - x^2}

and Y2

= -\sqrt{25 - x^2}

. Graphing apps like Desmos can graph the original equation directly.

How $x$ and $y$ change together

Along the graph of an implicitly defined function,

x

and

y

vary together. Take two points on the graph that are close together and compare the change in

y

to the change in

x

If the ratio $\frac{\Delta y}{\Delta x}$ is...	then...
positive	$x$ and $y$ increase together, or decrease together
negative	as one variable increases, the other decreases

This is the sign of a slope, read locally, one small piece of the curve at a time. Different parts of the same curve can behave in opposite ways.

Where the graph is horizontal or vertical

The rate of change can also be zero:

If the rate of change of $y$ with respect to $x$ is $0$ , then $y$ isn't changing while $x$ changes. The graph is horizontal there, like the top or bottom of a circle.

If the rate of change of $x$ with respect to $y$ is $0$ , then $x$ isn't changing while $y$ changes. The graph is vertical there, like the leftmost or rightmost point of a circle.

Example 1: The circle

Use the equation

x^2 + y^2 = 25

. (a) Make a table of solutions and identify two functions the equation implicitly defines. (b) Near

(3, 4)

and near

(3, -4)

, describe how

y

changes as

x

increases. (c) Where is the graph horizontal, and where is it vertical?

(a) Solve for

y

y = \pm\sqrt{25 - x^2}

. Some solutions:

$x$	$-5$	$-4$	$-3$	$0$	$3$	$4$	$5$
$y$	$0$	$\pm 3$	$\pm 4$	$\pm 5$	$\pm 4$	$\pm 3$	$0$

Answer: The equation implicitly defines

y = \sqrt{25 - x^2}

(the top half) and

y = -\sqrt{25 - x^2}

(the bottom half), each on

-5 \le x \le 5

(b) Move

x

from

3

3.1

and find

y

on each half:

\text{top: } y = \sqrt{25 - 3.1^2} = \sqrt{15.39} \approx 3.923, \qquad \frac{\Delta y}{\Delta x} \approx \frac{3.923 - 4}{0.1} = -0.77

\text{bottom: } y = -\sqrt{15.39} \approx -3.923, \qquad \frac{\Delta y}{\Delta x} \approx \frac{-3.923 - (-4)}{0.1} = 0.77

Answer: Near

(3, 4)

the ratio is negative, so

y

decreases as

x

increases. Near

(3, -4)

the ratio is positive, so

y

increases as

x

increases. Same

x

-value, opposite behavior, because the two points are on different implicitly defined functions.

(c) Near the top,

(0, 5)

, move

x

from

0

0.1

y = \sqrt{24.99} \approx 4.999

, so

\frac{\Delta y}{\Delta x} \approx \frac{4.999 - 5}{0.1} = -0.01

. That's already close to

0

, and it gets closer as the step gets smaller. Near the right side,

(5, 0)

, swap roles and move

y

from

0

0.1

x = \sqrt{24.99} \approx 4.999

, so

\frac{\Delta x}{\Delta y} \approx -0.01

, again close to

0

. The bottom and the left side work the same way.

Answer: The graph is horizontal at

(0, 5)

and

(0, -5)

, where the rate of change of

y

with respect to

x

0

. It's vertical at

(5, 0)

and

(-5, 0)

, where the rate of change of

x

with respect to

y

0

Example 2: A tilted curve

Consider the equation

x^2 + xy + y^2 = 3

. (a) Find all solutions with

x = -2, -1, 0, 1, 2

, and sketch the graph. (b) Solve for

y

to find the functions this equation implicitly defines. (c) Where is the graph horizontal, and where is it vertical? (d) On the upper function, how does

y

change as

x

increases from

-2

2

(a) Substitute each

x

-value and solve the quadratic in

y

$x = 0$ : $y^2 = 3$ , so $y = \pm\sqrt{3} \approx \pm 1.73$ .

$x = 1$ : $1 + y + y^2 = 3$ , so $y^2 + y - 2 = 0$ , which factors as $(y + 2)(y - 1) = 0$ . So $y = 1$ or $y = -2$ .

$x = -1$ : $1 - y + y^2 = 3$ , so $y^2 - y - 2 = 0$ , which factors as $(y - 2)(y + 1) = 0$ . So $y = 2$ or $y = -1$ .

$x = 2$ : $4 + 2y + y^2 = 3$ , so $y^2 + 2y + 1 = 0$ , which is $(y + 1)^2 = 0$ . Only one solution: $y = -1$ .

$x = -2$ : $4 - 2y + y^2 = 3$ , so $(y - 1)^2 = 0$ . Only one solution: $y = 1$ .

Plot the eight points and connect them smoothly. The graph is an oval tilted along the line

y = -x

(b) Treat the equation as a quadratic in

y

y^2 + xy + (x^2 - 3) = 0

. The quadratic formula gives

y = \frac{-x \pm \sqrt{x^2 - 4(x^2 - 3)}}{2} = \frac{-x \pm \sqrt{12 - 3x^2}}{2}

Answer: The equation implicitly defines two functions on

-2 \le x \le 2

: the upper function

y = \frac{-x + \sqrt{12 - 3x^2}}{2}

and the lower function

y = \frac{-x - \sqrt{12 - 3x^2}}{2}

. Their graphs meet at

x = -2

and

x = 2

, where the square root is

0

(c) The square root needs

12 - 3x^2 \ge 0

, so

-2 \le x \le 2

. The graph has no points left of

x = -2

or right of

x = 2

, and at those two

x

-values there's only one

y

. So the leftmost and rightmost points,

(-2, 1)

and

(2, -1)

, are where the graph turns vertical. The equation stays the same if you swap

x

and

y

, so the same reasoning gives

-2 \le y \le 2

, with the highest point at

(-1, 2)

and the lowest at

(1, -2)

Answer: The graph is vertical at

(-2, 1)

and

(2, -1)

, where the rate of change of

x

with respect to

y

0

. It's horizontal at

(-1, 2)

and

(1, -2)

, where the rate of change of

y

with respect to

x

0

. The short red segments in the graph mark these four spots.

(d) Read the upper function from left to right using the points

(-2, 1)

(-1, 2)

(0, \sqrt{3})

(1, 1)

, and

(2, -1)

. From

x = -2

x = -1

y

rises from

1

2

, so

\frac{\Delta y}{\Delta x} > 0

. From

x = -1

x = 2

y

falls from

2

-1

, so

\frac{\Delta y}{\Delta x} < 0

Answer: On the upper function,

x

and

y

increase together for

-2 < x < -1

. For

-1 < x < 2

y

decreases as

x

increases. The switch happens at the highest point,

(-1, 2)

, where the graph is horizontal.

Common mistakes

Dropping the $\pm$ . Solving $y^2 = 25 - x^2$ gives two functions. Writing only $y = \sqrt{25 - x^2}$ graphs half the curve.

Calling the whole curve a function. The graph of the equation usually isn't a function of $x$ . The pieces you get by solving are.

Mixing up horizontal and vertical. A rate of change of $y$ with respect to $x$ of $0$ means horizontal. A rate of change of $x$ with respect to $y$ of $0$ means vertical.

Using points that are far apart. The sign of $\frac{\Delta y}{\Delta x}$ tells you how $x$ and $y$ vary together only for points that are close together on the same piece of the graph.

Missing solutions. For a chosen $x$ , an implicit equation can have zero, one, or several $y$ -values. Find all of them.

Back Next

4.5 - Implicitly Defined Functions

Overview

Explicit vs. implicit

Graphing by finding solutions

Solving for one variable

How xxx and yyy change together

Where the graph is horizontal or vertical

Example 1: The circle

Example 2: A tilted curve

Common mistakes

How $x$ and $y$ change together